Phase 6 round-3 codex-review fixes: blockers + majors + minors

Resolved Codex review (gpt-5.5 xhigh) findings against b6913d2.

BLOCKERS:
- Appendix B reference mismatch: rewrote all main-text "Appendix B" references
  to "supplementary materials" since Appendix B is now a redirect stub. Affected
  the SSIM design-argument pointer, threshold provenance, byte-level
  decomposition, MC band capture-rate, and backbone-ablation table references
  across §III-F / §III-H.1 / §III-H.2 / §III-K / §III-L.4 / §III-M / §IV-F /
  §IV-J / §IV-K / §IV-L / §V-C / §V-H.
- Table rendering: un-commented Tables I-IV (Dataset Summary, YOLO Detection,
  Extraction Results, Cosine Distribution Statistics) which were inside HTML
  comment blocks and would not have rendered in the submission.
- Table numbering out of order: Table XIX appeared before Tables XVI-XVIII.
  Renumbered XIX -> XVI (document-level worst-case counts), XVI -> XVII (Firm x
  K=3 cross-tab), XVII -> XVIII (K=3 component comparison), XVIII -> XIX
  (Spearman correlation). Cross-references updated in §IV-J / §IV-K and §V-C.
- Table V mis-citation: §IV-C said "KDE crossover ... (Table V)" but Table V is
  the dip test. Dropped the (Table V) tag; crossover is a textual finding.
- Submission cleanup: wrapped the archived Impact Statement section heading and
  body inside the existing HTML comment (was rendering). Funding placeholder
  wrapped in HTML comment with a TO-DO note (won't render but is preserved as
  reminder).

MAJORS:
- Line 1077 numerical conflation: rewrote the §V-C / §III-L.4 paragraph that
  labelled Firm A's per-document HC+MC inter-CPA proxy ICCR of 0.6201 as a rate
  "on real same-CPA pools." 0.6201 is a counterfactual proxy under inter-CPA
  candidate-pool replacement, not the observed rate. Added explicit disambig:
  the corresponding observed rate from Table XVI (formerly XIX) is 97.5%
  HC+MC for Firm A; the proxy and observed rates measure different quantities.
- Residual "validation" language softened: "Dual-descriptor verification" ->
  "Dual-descriptor similarity"; "we validate the backbone choice" -> "we
  support the backbone choice"; "pixel-identity validation" -> "pixel-identity
  positive-anchor check"; "## M. Validation Strategy and Limitations under
  Unsupervised Setting" -> "## M. Unsupervised Diagnostic Strategy and Limits".
- "Specificity behaviour" overclaim: "characterises the cosine threshold's
  specificity behaviour" -> "specificity-proxy behaviour" (methodology §III-L.0
  and discussion §V-F).
- "Prior published / prior calibration" ambiguity: replaced "prior published
  per-comparison rate" with "the corpus-wide rate reported in §IV-I"; replaced
  "(prior published operating point)" with "(alternative operating point from
  supplementary calibration evidence)" in Tables XXI; replaced "prior reporting
  and the existing literature" with "the existing literature and the
  supplementary calibration evidence."

MINORS:
- Line 116 Bayes-optimal qualifier: "the local density minimum ... is the
  Bayes-optimal decision boundary under equal priors" -> "In idealized
  two-class mixture settings with equal priors and equal misclassification
  costs, the local density minimum ... coincides with the Bayes-optimal
  decision boundary."
- Stale section refs: §V-G for the fine-tuning caveat retargeted to §V-H
  Engineering-level caveats (where it lives after the §V-H reorganisation);
  §III-L for the worst-case rule retargeted to §III-H.1; "Section IV-D.2"
  (nonexistent) retargeted to "Section IV-D Table VI."
- Abstract / Introduction "after pool-size adjustment": separated the
  document-level D2 proxy ICCR claim from the per-signature logistic regression
  claim. Now: "Per-document D2 inter-CPA proxy ICCRs differ by an order of
  magnitude across firms ... a per-signature logistic regression confirms the
  firm gap persists after pool-size control."

NIT:
- Related Work HTML comment "(see paper_a_references_v3.md for full list)"
  -> "(full list in the References section)"; removes the version-coded
  filename reference from the source.

Artefacts:
- Combined manuscript regenerated: paper_a_v4_combined.md, 1312 lines.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

.planning/PROJECT.md

@@ -0,0 +1,74 @@
+# Taiwan TWSE CPA Signature Authentication
+
+## What This Is
+
+A computer-vision research pipeline that classifies whether the CPA signatures appearing on Taiwan TWSE-listed-company financial reports are hand-signed (親簽) or non-hand-signed (非親簽 — early-period rubber-stamp / scan, or post-2020 firm-level electronic signature systems). The pipeline ingests ~90k PDFs (2013-2023), detects ~182k signatures with YOLOv11n, embeds them with ResNet-50 (ImageNet1K_V2, no fine-tune), and characterises distributional structure with cosine + independent dHash descriptors. Target: a peer-reviewed publication (IEEE Access, A/6 on the NCKU CSIE journal list).
+
+## Core Value
+
+A statistically defensible, **reproducible** thresholding methodology that distinguishes hand-signed from digitally-replicated CPA signatures at the population level, with traceable evidence at every step (DB → script → table → paper claim).
+
+## Requirements
+
+### Validated
+
+<!-- Shipped and confirmed valuable. -->
+
+- ✓ End-to-end pipeline (TWSE MOPS scrape → Qwen2.5-VL prefilter → YOLO detection → ResNet embedding → DB + descriptors) — `signature_analysis/01-19`
+- ✓ Independent dHash descriptor for replication detection — Script 14 (v3.x baseline)
+- ✓ Accountant-level 3-component GMM characterisation — Script 18/20 (v3.x baseline)
+- ✓ Paper A v3.20.0 manuscript (full-dataset framing, partner Jimmy 2026-04-27 substantive review accepted, codex 3-pass verification clean) — commit `53125d1` on `yolo-signature-pipeline`
+- ✓ Spike scripts 32-35 confirming Big-4-only scope is methodologically superior — commits `e1d81e3`, `8ac0988`, `55f9f94` on `paper-a-v4-big4`
+
+### Active
+
+<!-- Current scope. Building toward these. -->
+
+**Milestone: Paper A v4.0 — Big-4 reframe (primary scope) + full-dataset robustness (secondary)**
+
+- [ ] Foundation: rerun core scripts on Big-4 subset with `--scope=big4` flag (`/scripts 19, 20, 21, 24, 25`)
+- [ ] Methodology rewrite: §III-G/I/J/L re-anchored on dip-test confirmed bimodality and bootstrap-stable Big-4 K=2 GMM (cos=0.975, dh=3.76)
+- [ ] Results tables: regenerate Tables IV-XVIII on Big-4 subset; new §IV-K full-dataset secondary
+- [ ] Prose rewrite: Abstract / Intro / Discussion / Conclusion with Firm A reframed as "templated end of Big-4" case study (was: hand-signed calibration anchor)
+- [ ] AI peer review: ≥3 cross-AI rounds (codex, Gemini 3.x Pro, Opus 4.7) on the v4.0 manuscript
+- [ ] Partner Jimmy second review on v4.0 (he proposed this direction; needs sign-off on execution)
+- [ ] iThenticate <20%, eCF copyright form, IEEE Access submission portal upload + cover letter
+
+### Out of Scope
+
+<!-- Explicit boundaries. Includes reasoning to prevent re-adding. -->
+
+- **Paper B (audit behaviour / policy implications)** — partner v4 contribution D, deferred to a separate paper after Paper A ships
+- **Paper C standalone (reverse-anchor methodology)** — initial 2026-05-12 spike direction, **folded back into Paper A v4.0 §IV-K** as one robustness lens; does not warrant a separate manuscript
+- **Mid/small-firm primary scope** — included as full-dataset secondary only; primary scope is Big-4 because dip-test only achieves multimodality at Big-4 level
+- **Per-document classifier release as software product** — paper-only deliverable; no API / SaaS layer in scope
+- **VLM behavioural interview / IRB study** — removed in v3.4; not coming back
+
+## Context
+
+- **Domain**: Taiwan-listed CPA audit signatures, 2013-2023; 4 Big-4 firms (勤業眾信 Deloitte, 安侯建業 KPMG, 資誠 PwC, 安永 EY) + ~30 mid/small firms
+- **Hardware split**: YOLO + ResNet on RTX 4090 (CUDA, deterministic forward inference, fixed seed); statistical analysis on Apple Silicon MPS / CPU
+- **Domain expert**: User has practitioner-level CPA-firm knowledge in Taiwan; recognises specific senior-partner names (e.g., 薛明玲 / 周建宏 are known PwC seniors that surfaced in Script 35's C1 cluster)
+- **Partner**: 與 partner Jimmy 合作；Jimmy 已提出 Big-4-only 方向，是 v4.0 的觸發者
+
+## Constraints
+
+- **Target journal**: IEEE Access (A/6 on NCKU CSIE list); fits Computer-Vision-applied-to-Audit scope
+- **Timeline**: v3.20.0 was already partner-reviewed and DOCX-shipped (2026-05-05). v4.0 reframe will delay submission by ~4-6 weeks but produces a stronger manuscript; partner Jimmy is aware and supportive
+- **Reproducibility**: pipeline must run end-to-end on the existing `/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db` snapshot; no new data ingest in scope
+- **AI review provenance**: every empirical claim must be backed by a fresh sqlite/grep against the named script — see `[[feedback-provenance-fabrication]]` memory; Gemini round-19 caught 4 fabricated provenance claims previously
+
+## Key Decisions
+
+| Decision | Rationale | Outcome |
+|----------|-----------|---------|
+| Use ResNet-50 ImageNet1K_V2 without fine-tune | Reproducibility; avoid label leakage from fine-tuning on the same corpus | ✓ Validated through v3.x |
+| Cosine + independent dHash dual descriptor | Cosine catches semantic similarity; independent dHash catches byte-level replication | ✓ Validated |
+| Drop SSIM / pixel-pHash from descriptor set | Reviewer-rejected as redundant / fragile | ✓ v3.x rewrite |
+| Drop A2 within-year uniformity assumption | Empirically falsified by Script 27 | ✓ v3.14 |
+| **Reframe scope to Big-4 only as primary** | Dip-test multimodal only at Big-4 level (p<0.0001); mid/small noise distorted Paper A v3.x's published 0.945/8.10 threshold; partner Jimmy's earlier suggestion empirically confirmed by Scripts 32-35 | — Pending v4.0 |
+| Reverse-anchor Paper C → folded into v4.0 §IV-K | Big-4 reframe is the stronger story; reverse-anchor is one of several lenses on the same data, not a standalone paper | ✓ Decided 2026-05-12 |
+| Branch strategy: `paper-a-v4-big4` from `from-outside-of-firmA` from `yolo-signature-pipeline` | Spike artifacts (Scripts 32-35) stay on the spike branch; v4.0 paper work isolated on its own sub-branch; v3.20.0 preserved on yolo-signature-pipeline as fallback | ✓ Decided 2026-05-12 |
+
+---
+*Last updated: 2026-05-12 after Paper A v4.0 Big-4 reframe milestone bootstrap*

.planning/REQUIREMENTS.md

@@ -0,0 +1,85 @@
+# Requirements — Paper A v4.0 (Big-4 reframe)
+
+Milestone: Paper A v4.0 IEEE Access submission with Big-4-only primary scope and full-dataset secondary robustness.
+
+## REQ-001: Big-4-only primary scope (foundation)
+
+**What**: All primary statistical analysis (KDE+dip, BD/McCrary, Beta mixture, 2D-GMM K=2/K=3, pixel-identity FAR, held-out 70/30 z-test, classifier sensitivity) is rerun on the 437-CPA Big-4 subset (Firm A + KPMG + PwC + EY, n_signatures ≥ 10).
+
+**Acceptance**:
+- Script 20 rerun on Big-4 subset, dip-test p < 0.05 on cos_mean and dh_mean
+- Script 21 (held-out validation) rerun on Big-4 subset
+- Script 24 (calibration vs held-out z-test, classifier sensitivity) rerun on Big-4 subset
+- Script 19 (pixel-identity / FAR) rerun on Big-4 subset
+- All rerun outputs land under `reports/v4_big4/`
+- New operational threshold cos > 0.975 AND dh ≤ 3.76 (or refined K=2 posterior) documented with bootstrap 95% CI
+
+## REQ-002: Full-dataset robustness as secondary section
+
+**What**: §IV-K (new) reports the full-dataset (686 CPA) version of the same analyses as a robustness check, demonstrating the pipeline runs at multiple scopes and explaining why the published v3.x 0.945 threshold drifted (mid/small-firm tail heterogeneity).
+
+**Acceptance**:
+- §IV-K table comparing Big-4-only vs full-dataset crossings, with mid/small-firm contribution analysis
+- Explicit explanation of why Big-4 is the methodologically privileged primary scope
+
+## REQ-003: Methodology rewrite (§III-G / I / J / L)
+
+**What**: Sections III-G (unit hierarchy / scope), III-I (threshold estimators), III-J (accountant-level GMM), III-L (per-document classifier rule) rewritten to reflect dip-test confirmed bimodality and the new K=2-derived classifier rule.
+
+**Acceptance**:
+- §III-G justifies Big-4 as the methodological unit (sample size, homogeneity, dip-test evidence)
+- §III-I anchored on bootstrap-stable bimodal evidence rather than three-method convergence on unimodal data
+- §III-J reports K=2 as primary (interpretable: replicated vs hand-leaning) with K=3 BIC slightly preferred (-1112 vs -1108) as secondary
+- §III-L derives operational rule from Big-4 K=2 components and bootstrap CI
+
+## REQ-004: Results tables IV-XVIII regenerated
+
+**What**: All results tables in §IV (currently Tables IV through XVIII at v3.20.0) regenerated on the Big-4 subset with consistent formatting and footnote citation to source script.
+
+**Acceptance**:
+- Each table cites the script + DB query that generated it
+- Big-4 numbers replace full-dataset numbers as primary; full-dataset relegated to §IV-K
+- Figures 1-4 regenerated; Fig 4 (yearly per-firm) likely reusable as-is
+
+## REQ-005: Firm A reframed as templated case study
+
+**What**: Throughout the manuscript, Firm A's role pivots from "calibration anchor (with minority hand-signers)" to "case study of the templated end of Big-4 (0% in K=3 hand-sign-leaning cluster, 82.5% in replicated cluster)". PwC's higher hand-sign tradition (24/102 = 23.5% in C1) noted as a Big-4 internal contrast.
+
+**Acceptance**:
+- Discussion (§V) explicitly states Firm A is the most digitally-replicated of Big-4
+- Cross-tab table (firm × cluster) included in either §IV or §V
+- Conclusion's contributions list updated accordingly
+
+## REQ-006: AI peer review (≥3 rounds)
+
+**What**: At least three cross-AI peer-review rounds on the v4.0 manuscript using codex (GPT-5.x), Gemini 3.x Pro, and Opus 4.7 max effort. Per `[[feedback-ai-review-provenance]]` memory: every reviewer-flagged empirical claim must be provenance-verified against fresh sqlite/grep against the named script.
+
+**Acceptance**:
+- Round 1 verdict obtained from each of the three reviewers
+- All Major-class findings either RESOLVED in revision or explicitly disclaimed
+- Final round produces ≥1 Accept / Minor verdict from at least 2 of 3 reviewers
+
+## REQ-007: Partner Jimmy second review on v4.0
+
+**What**: Jimmy (who proposed Big-4-only direction) reviews the v4.0 manuscript end-to-end before submission.
+
+**Acceptance**:
+- v4.0 DOCX shipped to ~/Downloads
+- Jimmy's response captured in repo (paper/partner_jimmy_v4_review.md)
+- Any must-fix items resolved in v4.0.x
+
+## REQ-008: iThenticate + eCF + submission
+
+**What**: iThenticate similarity check below 20%, IEEE eCF copyright form completed, manuscript uploaded via IEEE Access submission portal with cover letter.
+
+**Acceptance**:
+- iThenticate report saved under `paper/ithenticate_v4.pdf`
+- eCF confirmation captured
+- Submission portal confirmation number recorded in PROJECT.md "Validated" section
+
+## Cross-cutting constraints
+
+- **Reproducibility**: every script accepts a `--scope big4|full` flag (or new scripts under `signature_analysis/v4_*` if a flag refactor is too invasive)
+- **Provenance**: every numeric claim in the paper traces to (script_id, DB query, output file) — see `[[feedback-provenance-fabrication]]`
+- **No data re-ingest**: existing `/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db` is the frozen snapshot
+- **Branch isolation**: all v4.0 work on `paper-a-v4-big4`; do NOT merge back to `yolo-signature-pipeline` until v4.0 is partner-approved

.planning/ROADMAP.md

@@ -0,0 +1,87 @@
+# Roadmap — Paper A v4.0 Big-4 reframe
+
+Milestone goal: Ship Paper A v4.0 to IEEE Access with Big-4-only primary scope, dip-test confirmed bimodality, and full-dataset robustness as secondary.
+
+Branch: `paper-a-v4-big4` (from `from-outside-of-firmA` from `yolo-signature-pipeline` at v3.20.0).
+
+## Phase 1 — Foundation: Big-4 subset script reruns
+**Status**: pending
+**Requirements covered**: REQ-001
+**Tasks**:
+- Add `--scope=big4|full` flag to scripts 19, 20, 21, 24, 25 (and harness any others that load accountant aggregates)
+- Rerun on Big-4 subset; outputs to `reports/v4_big4/`
+- Bootstrap 95% CI on K=2 marginal crossings (extend Script 34's bootstrap to other measures)
+- Confirm dip-test p < 0.05 on Big-4 cos_mean and dh_mean (Script 34 already verified at p<0.0001 — replicate inside the rerun harness for audit trail)
+
+**Done when**: All five scripts produce v4_big4 outputs with bootstrap CI; cross-check against Script 34 numbers.
+
+## Phase 2 — Methodology rewrite (§III-G / I / J / L)
+**Status**: pending; depends on Phase 1
+**Requirements covered**: REQ-003
+**Tasks**:
+- §III-G: re-justify accountant-level Big-4 as the analysis unit (sample size, dip-test evidence, contrast with mid/small heterogeneity)
+- §III-I: re-anchor "natural threshold" claim on dip-test multimodality + bootstrap stability
+- §III-J: K=2 primary (replicated 31% / hand-leaning 69%) + K=3 secondary (BIC -1111.93 vs -1108.45)
+- §III-L: derive cos>0.975 AND dh≤3.76 (or K=2 posterior cut) from §III-J components
+
+**Done when**: §III markdown files updated; cross-references to Phase 1 outputs are correct.
+
+## Phase 3 — Results regeneration (§IV Tables IV-XVIII + §IV-K)
+**Status**: pending; depends on Phase 1 and 2
+**Requirements covered**: REQ-001 (tables), REQ-002 (§IV-K), REQ-004
+**Tasks**:
+- Regenerate Tables IV through XVIII on Big-4 subset (relabel as v4 numbering if order shifts)
+- Regenerate Figures 1-3 (Fig 4 yearly per-firm likely reusable)
+- New §IV-K Full-Dataset Robustness section: comparison table (Big-4 vs full), mid/small-firm contribution, why scope matters
+- Add firm × cluster cross-tab table from Script 35
+
+**Done when**: All §IV tables and figures land in repo; cross-refs from §III hold.
+
+## Phase 4 — Prose rewrite (Abstract / I / II / V / VI)
+**Status**: pending; depends on Phase 3
+**Requirements covered**: REQ-005
+**Tasks**:
+- Abstract: new threshold, new scope, retain the "reproducible pipeline" frame
+- §I Introduction: contributions list updated (Firm A reframe, Big-4 internal contrast finding, dip-test natural threshold)
+- §II Related Work: minimal changes (statistical methodology citations stable)
+- §V Discussion: Firm A as templated case study, PwC as hand-sign-leading firm, what this implies
+- §VI Conclusion + Future Work: forecast Paper B (audit behaviour / policy)
+
+**Done when**: All prose markdown files updated; word counts within IEEE Access limits (Abstract ≤ 250 words).
+
+## Phase 5 — AI peer review (3 rounds across codex, Gemini, Opus)
+**Status**: pending; depends on Phase 4 (manuscript-complete state)
+**Requirements covered**: REQ-006
+**Tasks**:
+- Round 1: codex (GPT-5.x) — full manuscript review with provenance verification
+- Round 1: Gemini 3.x Pro — full manuscript review
+- Round 1: Opus 4.7 max-effort — full manuscript review
+- Round 2: address Major findings; same three reviewers cross-check
+- Round 3: convergence — Accept / Minor from at least 2 of 3 reviewers
+
+**Done when**: Final round produces Accept/Minor consensus from majority; reviewer artifacts saved under `paper/`.
+
+## Phase 6 — Partner Jimmy v4.0 review
+**Status**: pending; depends on Phase 5
+**Requirements covered**: REQ-007
+**Tasks**:
+- Export v4.0 DOCX (`paper/export_v3.py` + author block fill)
+- Ship to ~/Downloads
+- Iterate on Jimmy's comments
+- Capture review artifact in `paper/partner_jimmy_v4_review.md`
+
+**Done when**: Jimmy approves v4.0.
+
+## Phase 7 — iThenticate + eCF + IEEE Access submission
+**Status**: pending; depends on Phase 6
+**Requirements covered**: REQ-008
+**Tasks**:
+- Run iThenticate, target similarity < 20%
+- Complete IEEE eCF
+- Upload manuscript + cover letter via IEEE Access submission portal
+- Capture confirmation number
+
+**Done when**: Submission confirmed by IEEE Access portal.
+
+---
+*Phase ordering: 1 → 2 → 3 → 4 → 5 → 6 → 7 (mostly linear; Phase 5 round-2 may loop back to Phase 4 prose if Major findings).*

.planning/STATE.md

@@ -0,0 +1,80 @@
+# STATE — Current snapshot
+
+**Date**: 2026-05-14
+**Active milestone**: Paper A v4.0 — Big-4 reframe
+**Active branch**: `paper-a-v4-big4` (41 commits ahead of `master`; fully pushed to `origin/paper-a-v4-big4` at `128a914`)
+**Active phase**: **Phase 5 — AI peer review COMPLETE; Phase 6 ready to begin**
+
+## Phase 5 closure summary (2026-05-14)
+
+**Convergence achieved**: 3/3 reviewers in Accept/Minor band across the round-2 cross-check.
+
+| Reviewer | Final round | Verdict |
+|---|---|---|
+| Gemini 3.1 Pro | round 2 | **Accept** (Phase 5 splice-ready as-is) |
+| Opus 4.7 | round 2 | Minor Revision (4 substantive findings → closed in round-4) |
+| codex GPT-5.5 | round 9 | Minor Revision (2 provenance findings → closed in round-5) |
+
+**Original Phase 5 gate met**: Accept/Minor consensus from ≥2 of 3 reviewers. No empirical reruns required.
+
+**Phase 5 fix rounds applied** (commits on this branch):
+1. `9604b27` — codex round-7 closeout copy-edit (candidate classifiers → candidate checks; refs [42]-[44] added; §II placeholder caveat removed; STATE.md refresh)
+2. `b884d39` — round-2 fixes (Opus M1: §IV K=3 mechanism-label reversion; M2: Table XV-B → XIX + cascade XIX → XX … XXV → XXVI; M3: "98-100%" within-firm semantic conflation; M4: duplicate §V-G heading; Gemini Table XV sample-size footnote)
+3. `4a6f9c5` — round-3 fixes (codex round-8 splice blockers: abstract trim 261 → 247 words; §IV-J Table XV footnote §IV-M.5 reclassification; §IV-I "§IV-M Table XVI" → "§IV-M Tables XXI-XXVI"; binary-collapse terminology cleanup)
+4. `d3ddf74` — round-4 fixes (Opus round-2 N1: Firm C 19,501 vs 19,122 denominator footnote; N2: composition-decomposition added as Table XXVII row 1; N3: Table XXVII numbered; N4: cross-firm hit matrix assumption disclosure)
+5. `128a914` — round-5 provenance patches (codex round-9 factual corrections: N1 "majority firm" → "1:1 tie-break to first-sorted firm" via Script 45 `np.argmax`; N2 row narrowed to Big-4-only evidence; non-Big-4 jittered-dHash range $[0.71, 1.00]$ → codex-verified $[0.38, 1.00]$ with read-only-spike provenance)
+
+**Reviewer artifacts archived** (paper/):
+- `codex_review_gpt55_v4_round{7,8,9}.md`
+- `gemini_review_v4_round{1,2}.md`
+- `opus_review_v4_round{1,2}.md`
+
+## Phase 5 substantive findings catalogue
+
+**v4 methodological pivot** (unchanged through all reviewer rounds):
+- Distributional path to thresholds (K=3 / dip / antimode) abandoned; anchor-based ICCR calibration at 3 units adopted
+- "FAR" → "ICCR" throughout; inter-CPA-as-negative assumption disclosed as partially violated by within-firm template sharing
+- K=3 demoted to descriptive firm-compositional partition (§III-J line 90 retires "hand-leaning / mixed / replicated" mechanism labels)
+- Positioning: anchor-calibrated specificity-only screening framework with human-in-the-loop review; NOT a validated forensic detector
+
+**Empirical anchors** (all provenance-verified across reviewer panel):
+- Three feature-derived scores converge Spearman $\rho \geq 0.879$ (internal consistency; not external validation)
+- Anchor-based ICCRs: per-comparison $0.0006/0.0013/0.00014$; per-signature $0.11$; per-document $0.34$
+- Firm heterogeneity decisive: Firm A per-doc HC+MC alarm $0.62$ vs Firms B/C/D $0.09$–$0.16$; logistic OR $0.05/0.01/0.03$ relative to Firm A reference
+- Within-firm collision concentration under deployed any-pair rule: Firm A $98.8\%$ vs Firms B/C/D $76.7$–$83.7\%$; same-pair joint event saturates at $97.0$–$99.96\%$ within-firm at all four firms
+
+## Phase 6 — Partner Jimmy v4.0 review (READY TO BEGIN)
+
+**Pre-Phase-6 partner alignment** (2026-05-13 still open): partner asked whether firm heterogeneity could be framed as "statistically insignificant." **Decision: no** — heterogeneity is highly significant (40–62σ in logistic regression; all three AI reviewers independently confirmed the decisive framing). Confirm framing with partner before exporting DOCX.
+
+**Phase 6 tasks**:
+1. Confirm "statistically insignificant" framing rejection with partner
+2. Manuscript-splice assembly:
+   - Splice §III-G–§III-M (paper_a_methodology_v4_section_iii.md) onto v3.20.0 §III-A–§III-F into master `paper/paper_a_methodology_v3.md` body
+   - Splice §IV v3.3 (paper_a_results_v4_section_iv.md) into master `paper/paper_a_results_v3.md`
+   - Splice Phase 4 prose (Abstract / §I / §II / §V / §VI) into the master manuscript file
+   - **Strip internal-only blocks** during splice: Phase 4 line 3 draft note + lines 153-162 close-out checklist; §III line 3 + lines 434-447 cross-reference checklist + open-questions block; §IV line 3 + close-out checklist at line 365+
+   - Re-verify table numbering after splice (Table XXVII currently lives in §III between §IV-M.6's Table XXVI; confirm order in final master file)
+3. Export v4.0 DOCX via `paper/export_v3.py` (with author block fill)
+4. Ship to ~/Downloads
+5. Iterate on Jimmy's review comments
+6. Capture review artifact in `paper/partner_jimmy_v4_review.md`
+
+## Phase 7 — IEEE Access submission (pending Phase 6)
+
+1. iThenticate similarity check (target < 20%)
+2. IEEE eCF form
+3. Upload manuscript + cover letter via IEEE Access submission portal
+4. Capture confirmation number
+
+## Blockers
+
+None. Phase 5 closed; Phase 6 ready to begin pending partner-framing confirmation.
+
+## Things to remember (per memory)
+
+- Inter-CPA "FAR" is NOT true FAR; it's a coincidence rate (ICCR) under an assumption violated by within-firm template sharing — never write "FAR" or "specificity" without the disclaimer ([[feedback-inter-cpa-negative-anchor-assumption]])
+- Dip test on Big-4 dh is composition + integer artefact, not mechanism — §III-I.1 "dip justifies finite mixture" framing must NOT be used; K=3 is descriptive of firm composition ([[feedback-dip-test-composition-artifact]])
+- Provenance-verify all empirical claims against fresh sqlite/grep ([[feedback-provenance-fabrication]]) — codex round-9's DB-verification caught a "majority firm" inference in round-4 that turned out to be 1:1 ties resolved by `np.argmax` tie-break; round-5 corrected it
+- AI peer reviewers have accepted fabricated claims in the past; verify numbers against scripts, not against reviewer agreement ([[feedback-ai-review-provenance]]) — codex round-9's read-only rerun of the non-Big-4 jittered procedure exposed an unreproducible $[0.71, 1.00]$ range that round-5 corrected to $[0.38, 1.00]$
+- Paper C standalone is shelved — folded into v4.0 §IV-K (Light full-dataset robustness)

paper/codex_partner_redpen_audit_v3_19_0.md

@@ -0,0 +1,43 @@
+# Codex Partner Red-Pen Regression Audit (Paper A v3.19.0)
+
+Scope: focused regression audit of whether the authors' partner red-pen comments on v3.17 have been adequately addressed in the current v3.19.0 manuscript files under `paper/`. This is not a fresh peer review.
+
+## 1. Overall summary
+
+For the 11 lettered red-pen items (a-k), my independent count is **7 RESOLVED / 1 IMPROVED / 0 PARTIAL / 0 UNRESOLVED / 3 N/A**. The two broader theme-level issues are **Citation reality: RESOLVED** and **ZH/EN alignment: N/A**.
+
+My bottom-line assessment is close to Gemini's: the revision substantially addresses the partner's concerns by deleting the most confusing accountant-level GMM / accountant-level BD-McCrary material and by replacing several AI-sounding explanations with more literal, auditable prose. I do not agree with Gemini's fully clean "8 RESOLVED / 3 N/A" verdict, however. The BIC / strict-3-component item is materially improved, but the manuscript still retains "upper bound" wording in the methods and Table VI even though the results correctly call the two-component fit a forced fit. That is a small prose/rationale residue, not a blocking unresolved issue.
+
+## 2. Item-by-item table
+
+| Item | Status | Manuscript section addressing it | Brief justification | Disagreement with Gemini audit |
+|---|---:|---|---|---|
+| Theme 1: Citation reality for refs [5], [16], [21], [22], [25], [27], [37]-[41] | RESOLVED | `paper_a_references_v3.md`; `reference_verification_v3.md` | The current reference list fixes the serious [5] author/title error and includes real, recognizable method references for Hartigan, Burgstahler-Dichev, McCrary, Dempster-Laird-Rubin, and White. The flagged technical references are not hallucinated. Minor citation-polish items from the verification file appear fixed in the current reference list. | No substantive disagreement. One housekeeping note: `reference_verification_v3.md` still describes [5] as a "major problem" in the detailed findings/recommendations because it records the audit history; the actual current reference list is fixed. |
+| Theme 3: ZH/EN alignment gap at end of III-H Calibration Reference | N/A | Entire v3.19.0 manuscript | The dual-language zh-TW/en scaffold that produced the partner's "no English alongside?" concern is gone. The current draft is monolingual English for IEEE submission, so there is no remaining bilingual alignment task. | No disagreement. |
+| (a) A1 stipulation, "do not understand your description" | RESOLVED | Section III-G, `paper_a_methodology_v3.md` | A1 is now stated as a specific cross-year pair-existence assumption: if replication occurs, at least one same-CPA near-identical pair exists in the observed same-CPA pool. The text also states when A1 may fail. This is much clearer than a vague stipulation. | No disagreement. |
+| (h) A1 pair-detectability paragraph red-circled | RESOLVED | Section III-G, `paper_a_methodology_v3.md` | The red-circled assumption is now bounded: it is plausible for high-volume stamping/e-signing, not guaranteed under singletons, multiple templates, or scan noise, and not a within-year uniformity claim. That should answer the partner's concern about over-assumption. | No disagreement. |
+| (b) Conservative structural-similarity wording, "a bit roundabout?" | RESOLVED | Section III-G, `paper_a_methodology_v3.md` | The independent-minimum dHash is now defined directly as the minimum Hamming distance to any same-CPA signature and identified as the statistic used in the classifier and capture-rate analyses. The wording is concise enough for re-read. | No disagreement. |
+| (c) IV-G validation lead-in, "do not understand why you say this" | RESOLVED | Section IV-G, `paper_a_results_v3.md` | The lead-in now explicitly says Section IV-E capture rates are internally circular because Firm A helped set the thresholds, then explains why the three IV-G analyses are threshold-free or threshold-robust. This directly supplies the missing rationale. | No disagreement. |
+| (d) BD/McCrary at accountant level, "cannot understand" | N/A | Removed from current structure | The accountant-level BD/McCrary analysis no longer appears in the live v3.19.0 manuscript. BD/McCrary is now signature-level only and framed as a density-smoothness diagnostic, not an accountant-level threshold device. | No disagreement. |
+| (k) Accountant-level aggregation rationale, "why accountant level total, because component?" | N/A | Removed from current structure | The confusing accountant-level component narrative has been deleted. The paper now avoids translating signature-level outputs into accountant-level mechanism assignments except for auditor-year ranking. | No disagreement. |
+| (e) 92.6% match rate, "do not understand improvement angle" | RESOLVED | Section III-D, `paper_a_methodology_v3.md`; Table III in Section IV-B | The match rate is now a data-processing coverage metric: 168,755 of 182,328 signatures are CPA-matched, and the unmatched 7.4% are excluded because same-CPA best-match statistics are undefined. The old "improvement" angle is gone. | No disagreement. |
+| (f) 0.95 cosine cutoff, "cut-off corresponds to what?" | RESOLVED | Section III-K, `paper_a_methodology_v3.md`; Sections IV-E/F | The text now states that 0.95 corresponds to the whole-sample Firm A P7.5 heuristic: 92.5% of Firm A signatures exceed it and 7.5% fall at or below it. It also distinguishes 0.95 from the calibration-fold P5 = 0.9407 and rounded 0.945 sensitivity cut. | No disagreement. |
+| (g) 139/32 C1/C2 split, "too reliant on weighting factor?" | N/A | Removed from current structure | The C1/C2 accountant-level GMM cluster split is gone from the current manuscript. Residual fold-variance wording no longer invokes the 139/32 split. | No disagreement. |
+| (i) Hartigan rejection-as-bimodality, "so why?" | RESOLVED | Section III-I.1, `paper_a_methodology_v3.md`; Section IV-D.1 | The text now separates the dip test from component counting: it tests unimodality, does not specify a component count, and is used to decide whether a KDE antimode is meaningful. Section IV-D then explains why Firm A's non-rejection and all-CPA rejection matter. | No disagreement. |
+| (j) BIC strict-3-component upper-bound framing, red-circled paragraph | IMPROVED | Section III-I.2/III-I.4, `paper_a_methodology_v3.md`; Section IV-D.3/IV-D.4, `paper_a_results_v3.md` | The results section is much clearer: it labels the 2-component Beta mixture as "A Forced Fit," reports the 3-component BIC preference, and says the Beta/logit disagreement reflects unsupported parametric structure. However, the methods still say the 2-component crossing "should be treated as an upper bound," and Table VI labels one row as "signature-level Beta/KDE upper bound." That residual wording may still prompt "upper bound of what?" from the partner. | I disagree with Gemini's RESOLVED verdict here. The item is not unresolved, but it is only IMPROVED until "upper bound" is either defined in one plain sentence or removed in favor of "forced-fit descriptive reference." |
+
+## 3. Specific pushback on Gemini's RESOLVED verdict
+
+Only item **(j)** needs pushback.
+
+Gemini says the BIC issue is resolved because the results now title the subsection "A Forced Fit" and state that the 2-component structure is not supported. That is true for Section IV-D.3, but not the whole manuscript. Section III-I.2 still says that when BIC prefers three components, "the 2-component crossing should be treated as an upper bound rather than a definitive cut." Section III-I.4 repeats that the 2-component crossing is a forced fit and "should be read as an upper bound," and Table VI contains "signature-level Beta/KDE upper bound."
+
+For a statistically trained reviewer, this may be defensible shorthand. For the partner's original red-pen concern, it is still slightly too abstract. If the authors keep "upper bound," they should define the bound explicitly. Otherwise the safer fix is to remove the term and call these values "forced-fit descriptive references not used operationally."
+
+## 4. Smallest residual set before partner re-read
+
+1. Replace or explain the remaining **"upper bound"** wording in Section III-I.2, Section III-I.4, and Table VI. Suggested direction: "Because the two-component assumption is not supported, we report the crossing only as a forced-fit descriptive reference and do not use it as an operational threshold."
+
+2. Optional housekeeping: update `reference_verification_v3.md` so its detailed [5] entry no longer reads like an active problem after the reference list has been corrected. This is not a manuscript blocker, but it avoids confusion if the partner or a coauthor opens the verification note.
+
+No other partner red-pen issue appears to need substantive revision before re-read.

paper/codex_review_gpt55_v4_round1.md

@@ -0,0 +1,143 @@
+# Paper A v4.0 Methodology Section III-G through III-L Peer Review
+
+Reviewer: gpt-5.5 xhigh  
+Date: 2026-05-12  
+Round number: 21 (v4 round 1)  
+Review target: `paper/v4/paper_a_methodology_v4_section_iii.md`
+
+Audit aliases used below:
+
+- V4: `paper/v4/paper_a_methodology_v4_section_iii.md`
+- V3: `paper/paper_a_methodology_v3.md`
+- Script36: `/Volumes/NV2/PDF-Processing/signature-analysis/reports/v4_big4/calibration_and_loo_validation/calibration_loo_report.md`
+- Script37: `/Volumes/NV2/PDF-Processing/signature-analysis/reports/v4_big4/k3_loo_check/k3_loo_report.md`
+- Script38: `/Volumes/NV2/PDF-Processing/signature-analysis/reports/v4_big4/convergence_k3_reverse_anchor/convergence_report.md`
+- Script39: `/Volumes/NV2/PDF-Processing/signature-analysis/reports/v4_big4/signature_level_convergence/sig_level_report.md`
+- Script40: `/Volumes/NV2/PDF-Processing/signature-analysis/reports/v4_big4/pixel_identity_far/far_report.md`
+- Script34 local: `/Volumes/NV2/PDF-Processing/signature-analysis/reports/big4_only_pooled/big4_only_pooled_report.md`
+- Script35 local: `/Volumes/NV2/PDF-Processing/signature-analysis/reports/big4_k3_cluster_inspection/inspection_report.md`
+- Script32 local: `/Volumes/NV2/PDF-Processing/signature-analysis/reports/non_firm_a_calibration/non_firm_a_calibration_report.md`
+
+## Verdict
+
+Major Revision.
+
+## Major Findings
+
+1. **K=3 is not yet justified as an operational classifier.**
+
+   V4 selects K=3 for the operational per-CPA classifier (V4:57, V4:67) and says the K=3/K=2 contrast justifies selecting K=3 (V4:107). The underlying Script37 verdict is weaker: `P2_PARTIAL`, with the explicit interpretation that the C1 cluster exists but "membership is not well-predicted by held-out fit" (Script37:92, Script37:94). The report's own legend says `P2_PARTIAL` means the cluster is "not predictively useful as an operational classifier" (Script37:97-99).
+
+   The numbers support this concern. K=3 C1 component shape is stable (max deviations 0.0047 cosine, 0.955 dHash, 0.023 weight; Script37:77-79), but held-out C1 membership differs from baseline by up to 12.77 percentage points (Script37:83-90). For PwC, baseline C1 is 23.5% but held-out prediction is 36.27% (Script37:47-51, Script37:87). That is not a small operational error if the label is used to classify CPAs.
+
+   The BIC evidence is also weak. K=3 is lower BIC than K=2 by only 3.48 points (Script36:9-10; Script34 local:40-41). This is acceptable as mild descriptive support, not as the load-bearing reason to replace a classifier. The draft should either (a) demote K=3 to a descriptive/convergent-validation model, or (b) make K=3 primary only with explicit LOOO membership uncertainty and soft-posterior reporting.
+
+2. **The "three independent lenses" framing overstates independence and validation strength.**
+
+   V4 describes the convergent validation as three "independent statistical lenses" (V4:73-89). They are not independent empirical measurements. All three are deterministic functions of the same per-CPA or per-signature `(cos, dHash)` features:
+
+   - Lens 1 is K=3 posterior from the same two descriptors (V4:77; Script38:6-12).
+   - Lens 2 is a monotone transform of the cosine marginal only (V4:78; Script38:16-18).
+   - Lens 3 is the fraction of signatures failing the same box rule `cos > 0.95 AND dh <= 5` (V4:79; Script38:20-22).
+
+   The high Spearman correlations are verified (0.9627, 0.8890, 0.8794; Script38:24-34), but they are partly mechanical agreement among feature-derived scores. They do not validate the classifier against an independent ground truth for hand-signed signatures.
+
+   There is also a conceptual reversal in the reverse-anchor prose. V4 says the non-Big-4 reference has lower cosine and higher dHash than the Big-4 C1 center (V4:37), which is verified (reference center 0.9349/9.7670 in Script38:16-18; C1 0.9457/9.1715 in Script38:8-12). But V4 then calls this a "more-replicated-population" baseline (V4:37). Lower cosine and higher dHash indicate less replication / more hand-leaning, not more replication. A reviewer will likely catch this immediately.
+
+3. **The draft conflates at least three classifiers and then validates only one simplified binary rule.**
+
+   V4 alternates among (i) K=3 per-CPA hard labels (V4:67), (ii) a binary Paper A box rule `cos > 0.95 AND dh <= 5` (V4:69), and (iii) the inherited five-way per-signature/document rule with `dh <= 5`, `5 < dh <= 15`, and `dh > 15` bands (V4:123-135). The Script38/39 convergence results validate only the simplified binary rule `non_hand iff cos > 0.95 AND dh <= 5` (Script38:20-22; Script39:8-12). They do not validate the full five-way classifier, especially the moderate non-hand-signed band `5 < dh <= 15`.
+
+   This matters because V3's inherited Section III-K explicitly treated `cos > 0.95 AND 5 < dh <= 15` as "Moderate-confidence non-hand-signed" (V3:278-287). V4 keeps that category (V4:127) but cites kappa/rho evidence from a binary high-confidence-only rule (V4:121). The current prose therefore overstates what the Script39 kappa values prove.
+
+   Recommended fix: choose a primary endpoint. If the five-way rule remains primary, validate that exact five-way rule or its declared binary collapse. If K=3 becomes primary, provide a document-level aggregation rule for K=3 and stop calling the inherited box rule the operational classifier.
+
+4. **The pixel-identity validation is useful, but "FAR" is the wrong metric name and the evidentiary force is overstated.**
+
+   Script40's ground truth is a positive class: pixel-identical signatures are treated as replicated (Script40:4-8). Misclassifying them as hand-leaning is a false negative / miss rate on an easy positive-anchor subset, not a false-alarm rate in the usual classifier sense. V4 defines FAR as "probability of labelling a pixel-identical signature as hand-leaning" (V4:109), which reverses standard terminology.
+
+   The 0/262 result is verified for all three classifiers (Script40:12-18), and the caveat that pixel-identity is necessary but not sufficient is appropriate (V4:117; Script40:29-31). But for the Paper A box rule this result is close to tautological: byte-identical nearest-neighbor signatures will have near-maximal cosine and minimal dHash. V3 was more careful, noting that FRR against byte-identical positives is trivially zero at thresholds below 1 and should be interpreted qualitatively (V3:266-268).
+
+   Rename this metric to "pixel-identity positive-anchor miss rate" or "false-hand rate on replicated positives." Do not present it as FAR unless a true hand-signed negative anchor is evaluated.
+
+5. **Several empirical/provenance claims need correction or explicit "unverified" status.**
+
+   - V4 says the K=2 LOOO max cosine deviation 0.028 is `5.6x` a "bootstrap CI half-width of 0.005" (V4:103). Script36 reports max deviation 0.0278 (Script36:43), but 0.005 is the stability tolerance in the verdict legend, not the bootstrap CI half-width (Script36:50-52). The full Big-4 bootstrap cosine CI half-width is 0.0015 (Script36:14-17). Correct the denominator and wording.
+
+   - V4 says all-non-Firm-A is dip-test unimodal at `p > 0.99` (V4:21). Script32 local reports all-non-Firm-A cosine p = 0.9975 but dHash p = 0.9065 (Script32 local:56-76). The later detailed sentence in V4 correctly gives 0.998/0.907 (V4:43). Fix the earlier overstatement.
+
+   - V4 says no BD/McCrary transition is identified on either axis and cites Script32/34 (V4:47). Script34 local supports no Big-4-only BD/McCrary threshold (Script34 local:28-31), but Script32 local reports dHash BD/McCrary thresholds for `big4_non_A` and `all_non_A` (Script32 local:36-44, Script32 local:68-76). Narrow the claim to the Big-4-only analysis or explain why Script32 subset transitions are not used.
+
+   - The Firm A byte-identical claim is partly verified. Script40 verifies 145 Firm A pixel-identical signatures inside the 262 Big-4 total (Script40:20-27). The added details "50 distinct Firm A partners," "of 180 registered," and "35 span different fiscal years" appear in V3 (V3:165) and V4 (V4:31), but I did not find them in the supplied Script36-40 reports. Treat those details as unverified unless the Appendix B/script artifact is cited directly.
+
+   - The "mid/small-firm tail actively pulling the v3.x crossing" statement (V4:19) is stronger than the local Script34 evidence. Script34 local verifies the Big-4-only crossing and CI (Script34 local:18-24), and it reports a large offset from the published baseline (Script34 local:51-58). It does not, by itself, prove the causal language "actively pulling" rather than "the full-sample and Big-4-only calibrations differ."
+
+## Minor Findings
+
+1. **Dip-test p-value precision needs a resolution check.** V4 says bootstrap p-value estimation uses `n_boot = 2000` and reports `p < 10^-4` (V4:43). With a finite bootstrap of 2000, the natural resolution is about 1/2000 unless the script uses a different asymptotic/calibrated p-value. Script36/34 display p = 0.0000 (Script36:6-8; Script34 local:28-31). State the reporting convention precisely, e.g., "no bootstrap replicate exceeded the observed statistic; reported as p < 0.001" if that is what happened.
+
+2. **The Delta BIC sign convention is confusing.** V4 reports "Delta BIC = -3.5" (V4:65). Since lower BIC is preferred, a reviewer may expect `BIC(K=2) - BIC(K=3) = 3.48` or "K=3 lower by 3.48." Use one convention and define it.
+
+3. **Per-signature convergence is real but only moderate for the box rule.** Script39 verifies kappas of 0.6616, 0.5586, and 0.8701 (Script39:22-30). The report verdict is `SIG_CONVERGENCE_MODERATE`, not strong (Script39:41-48). V4's statement that box-rule disagreement reflects "different decision geometries" rather than signal disagreement (V4:99) is plausible but interpretive. Add the moderate verdict and avoid making geometry the only explanation.
+
+4. **Per-CPA vs per-signature component centers drift more than the prose suggests.** Script39 shows per-CPA C1 at cosine 0.9457 and per-signature C1 at 0.9280 (Script39:16-20). Kappa is high for K=3 perCPA vs perSig labels (Script39:28), but "the same component structure recovers" (V4:99) should be softened to "a broadly similar three-component ordering recovers."
+
+5. **The Section III-L title is misleading.** The section is titled "Per-Document Classification" (V4:119) but most of it defines per-signature categories (V4:121-133). The document-level aggregation appears only in one paragraph (V4:135). Either rename to "Signature- and Document-Level Classification" or split the two parts.
+
+6. **K=3 alternative output lacks document aggregation.** V4 says the K=3 alternative assigns each signature to C1/C2/C3 (V4:137), but if Section III-L is per-document classification, the K=3 alternative also needs a document-level worst-case or posterior aggregation rule.
+
+7. **Firm anonymization is inconsistent.** V4 names the four firms in Chinese and then says they are pseudonymized as Firms A-D (V4:17). Later it uses PwC directly (V4:31). V3 says firm-level results are reported under pseudonyms (V3:315-316). Decide whether v4 abandons anonymization; otherwise keep the main text pseudonymous and put the mapping outside the manuscript, if at all.
+
+## Editorial / Prose Nits
+
+1. Replace "more-replicated-population baseline" (V4:37) with "less-replicated external reference" or "hand-leaning external reference."
+
+2. Replace "failure rate" for Lens 3 (V4:79, V4:89) with "box-rule hand-leaning rate" or "non-replicated rate." "Failure" sounds like classifier failure rather than a hand-leaning outcome.
+
+3. "Strongest single methodology-validation signal" (V4:89) is too strong because the lenses share features. Use "strongest internal consistency signal."
+
+4. "Boundary moves modestly" (V4:105) understates the PwC fold, where C1 membership rises from 23.5% to 36.3% (Script37:47-51). Use "membership remains composition-sensitive."
+
+5. "Calibration uncertainty band of +/- 5-13 percentage points" (V4:105) should be "observed absolute differences of 1.8-12.8 percentage points, with the largest fold exceeding the report's 5 pp viability bar" (Script37:83-90).
+
+6. "Operational threshold derivation" (V4:51) is not accurate if the operational per-signature classifier remains the inherited box rule. Use "mixture model and component assignment" unless K=3 is truly primary.
+
+7. The cross-reference index is useful, but it should be removed from the submitted manuscript or converted into an internal author checklist.
+
+## Responses to the Five Open Questions
+
+1. **Scope justification.**
+
+   The three-point argument is directionally good but not yet sufficient. Add a fourth point explicitly restricting generalizability: primary claims are for the Big-4 audit-report context, while the 249 non-Big-4 CPAs are used only as robustness/reverse-anchor context unless Section IV-K independently validates them. Also soften "tail distorts" to "tail changes the fitted crossing" unless you cite a direct diagnostic for distortion. The Big-4 counts and crossings are verified (Script34 local:4-24; Script36:6-17), but the causal language needs restraint.
+
+2. **Firm A phrasing.**
+
+   Use "templated-end case study" or "replication-heavy descriptive reference." Do not use "calibration reference, descriptively defined post-hoc" unless Firm A actually calibrates a threshold in v4. The draft correctly says Firm A is not the calibration anchor (V4:33). Calling it a calibration reference reintroduces the v3 vulnerability.
+
+3. **K=3 vs K=2 rationale.**
+
+   As written, no. Selecting K=3 as an operational classifier on LOOO stability is not acceptable because Script37 says K=3 is only `P2_PARTIAL` and "not predictively useful as an operational classifier" (Script37:92-99). Do not strengthen the BIC argument; Delta BIC about 3.5 is mild. The defensible claim is: K=2 is clearly unstable; K=3 gives a reproducible hand-leaning component shape; hard membership remains uncertain and should be reported as calibration uncertainty.
+
+4. **Hybrid box rule plus K=3 alternative.**
+
+   The hybrid can be acceptable only if roles are sharply separated: inherited five-way box rule is the primary signature/document classifier; K=3 is an accountant-level characterization and exploratory alternative. The current draft blurs this by calling K=3 "operational" (V4:67) while keeping the box rule in Section III-L (V4:121-137). Also, the validation scripts use the binary high-confidence rule `dh <= 5`, not the full five-way rule with `dh <= 15`. Fix this before deciding whether to keep the hybrid.
+
+5. **Section IV numbering.**
+
+   Do not freeze table numbers yet. First settle the Methodology labels and primary classifier. Results should mirror this order: sample/scope, K=2/K=3 calibration, convergence lenses, K=2 and K=3 LOOO, pixel-identity positive-anchor check, signature/document classification outputs, then full-dataset robustness. After that, assign table numbers and verify every Section III cross-reference to Section IV-D/F/G/K.
+
+## Recommended Next-Step Actions
+
+1. Rewrite Sections III-J and III-K so K=3 is either clearly primary with uncertainty, or clearly descriptive. If descriptive, remove "operational threshold" language from the K=3 discussion.
+
+2. Add the Script37 `P2_PARTIAL` result directly to the prose. Do not hide the "not predictively useful as an operational classifier" implication.
+
+3. Decide and declare the primary classifier: inherited five-way box rule, binary high-confidence box rule, or K=3 hard/posterior labels. Align all validation text to that exact classifier.
+
+4. If the five-way rule remains primary, rerun or report validation for the five-way categories and the document-level worst-case aggregation, not just `cos > 0.95 AND dh <= 5`.
+
+5. Rename the pixel-identity metric from FAR to positive-anchor miss rate / false-hand rate. Add a separate specificity/FAR result only if a true hand-signed or inter-CPA negative anchor is evaluated.
+
+6. Correct the empirical slips: K=2 "0.005 bootstrap half-width," all-non-Firm-A `p > 0.99`, Script32 BD/McCrary wording, reverse-anchor "more-replicated" phrase, and any unverified Firm A byte-decomposition details.
+
+7. Add a short provenance table for every numerical claim in Sections III-G through III-L, including exact report path, script number, and whether the number is directly reported or inferred by arithmetic.

paper/codex_review_gpt55_v4_round2.md

@@ -0,0 +1,87 @@
+# Paper A v4.0 Methodology Section III-G through III-L Peer Review
+
+Reviewer: gpt-5.5 xhigh  
+Date: 2026-05-12  
+Round number: 22 (v4 round 2)  
+Review target: `paper/v4/paper_a_methodology_v4_section_iii.md`
+
+## Verdict
+
+Minor Revision.
+
+v2 closes most of the round-21 blockers: K=3 is no longer the operational classifier, the "independent lenses" claim is softened, the pixel-identity metric is no longer called FAR in the draft, and the main empirical slips are corrected. The remaining issues are narrower but still need edits before accepting the methodology text, especially the false per-firm ordering claim in §III-K and the unresolved validation status of the five-way moderate-confidence band.
+
+## Round-21 finding closure table
+
+| Finding | Round-21 Severity | v2 Status | Evidence in v2 |
+|---|---|---|---|
+| M1. K=3 is not justified as an operational classifier. | Major | CLOSED | v2 explicitly says both K=2 and K=3 are descriptive and not used for signature/document labels (v2:51, v2:67-73, v2:143). It also reports Script 37 `P2_PARTIAL` and the "not predictively useful as an operational classifier" implication (v2:65, v2:109). |
+| M2. "Three independent lenses" overstates independence and validation strength, and reverse-anchor direction was wrong. | Major | PARTIAL | The independence and reverse-anchor wording are fixed: the scores are "not statistically independent" and only internal-consistency checks (v2:75-83), and the reference is now described as less replication-dominated (v2:35-37). However, v2 adds a false per-firm ordering claim that all three scores make Firm C most hand-leaning (v2:93); Script 38's reverse-anchor mean instead ranks Firm D highest. |
+| M3. Classifier conflation; only the simplified binary rule was validated. | Major | PARTIAL | v2 now declares the inherited five-way box rule as primary (v2:123-143) and K=3 as descriptive (v2:143). It also correctly notes that the kappa comparison validates only the binary high-confidence rule, not the five-way moderate band (v2:103). The unresolved moderate-band validation is still open (v2:190-192), and v2:125 still uses binary-rule correlations to support the full five-way rule without recalibration. |
+| M4. Pixel-identity "FAR" naming and evidentiary force were wrong. | Major | CLOSED | v2 renames this to a positive-anchor miss rate, frames it as a one-sided replicated-positive check, and adds the tautology/conservative-subset caveat (v2:111-121). |
+| M5. Empirical/provenance claims needed correction or explicit unverified status. | Major | CLOSED | The 0.005 denominator is now a stability tolerance, not a bootstrap CI (v2:65, v2:107); all-non-Firm-A dip values are corrected (v2:21, v2:43); BD/McCrary is narrowed to Big-4 null with external dHash transitions disclosed (v2:47); Firm A byte-decomposition details are marked inherited/not regenerated (v2:31, v2:176); "tail distorts" is softened to a scope-dependent shift (v2:19). |
+| m1. Dip-test p-value precision needed bootstrap-resolution wording. | Minor | CLOSED | v2 states no bootstrap replicate exceeded the observed statistic and reports `p < 5 x 10^-4` for `n_boot = 2000` (v2:21, v2:43, v2:158-159). |
+| m2. Delta BIC sign convention was confusing. | Minor | CLOSED | v2 defines lower BIC as preferred and reports `BIC(K=3) - BIC(K=2) = -3.48`, plus "K=3 lower by 3.48" (v2:45, v2:63). |
+| m3. Per-signature convergence is only moderate for the box rule. | Minor | CLOSED | v2 includes the `SIG_CONVERGENCE_MODERATE` verdict and avoids calling the Paper A-vs-K=3 kappas strong (v2:95-103). |
+| m4. Per-CPA vs per-signature component centers drift more than v1 suggested. | Minor | CLOSED | v2 says the fits recover a "broadly similar three-component ordering" and reports the C1 cosine drift of 0.018 (v2:95). |
+| m5. Section III-L title was misleading. | Minor | CLOSED | The section is now titled "Signature- and Document-Level Classification" and separates per-signature categories from document aggregation (v2:123-143). |
+| m6. K=3 alternative lacked document aggregation. | Minor | CLOSED | v2 no longer offers K=3 as a signature/document classifier, so a K=3 document aggregation rule is no longer required (v2:143). |
+| m7. Firm anonymization was inconsistent. | Minor | CLOSED | v2 uses Firm A-D pseudonyms in the methodology text and no longer names the Big-4 firms directly in the prose (v2:17, v2:31, v2:194). |
+| e1. Replace "more-replicated-population baseline." | Editorial | CLOSED | v2 now calls non-Big-4 a less-replicated external/reverse-anchor reference (v2:35-37). |
+| e2. Replace "failure rate" for Lens 3. | Editorial | CLOSED | Lens 3 is now "Paper A box-rule hand-leaning rate" (v2:83). |
+| e3. "Strongest single methodology-validation signal" was too strong. | Editorial | CLOSED | v2 uses "strongest internal-consistency signal" and denies external validation (v2:77, v2:93). |
+| e4. "Boundary moves modestly" understated LOOO membership instability. | Editorial | CLOSED | v2 uses composition-sensitive wording and reports the 12.8 pp Firm C fold deviation (v2:65, v2:109). |
+| e5. "Calibration uncertainty band of +/- 5-13 pp" wording needed correction. | Editorial | CLOSED | v2 reports observed absolute differences of 1.8-12.8 pp and the 5 pp viability bar (v2:109). |
+| e6. "Operational threshold derivation" language was inaccurate. | Editorial | CLOSED | v2 consistently calls K=3 a mixture characterisation/descriptive model, not an operational threshold source (v2:49-73, v2:143). |
+| e7. Cross-reference index should be removed or made internal. | Editorial | PARTIAL | v2 labels the cross-reference index as an author checklist to remove before submission (v2:181), but it remains inside the methodology draft (v2:181-188). |
+
+## Newly introduced issues
+
+1. **New factual/provenance error: the three scores do not agree on the most hand-leaning firm.** v2 claims that "by all three scores, Firm A is the most replication-dominated and Firm C is the most hand-leaning" (v2:93). Script 38 confirms Firm A is most replication-dominated, but not the Firm C part for all scores: mean P_C1 and mean hand_frac rank Firm C highest, while mean reverse-anchor ranks Firm D highest (`-0.7125` vs Firm C `-0.7672`, with higher score meaning more hand-leaning). Revise to: "P_C1 and box-rule hand_frac rank Firm C highest; the reverse-anchor score ranks Firm D highest; all three agree Firm A is most replication-dominated and the non-A firms are more hand-leaning than Firm A."
+
+2. **Unsupported scope superlative: "any single firm" / "smallest scope" is not proven by the supplied reports.** v2 says no dip-test rejection holds "within any single firm pooled alone" and that Big-4 is the "smallest scope" supporting a finite-mixture model (v2:21; repeated more generally at v2:43). The supplied Script 32 report verifies Firm A alone, `big4_non_A`, and `all_non_A`; it does not report separate single-firm tests for Firms B, C, and D or all smaller combinations. Narrow this to "among the tested comparison scopes in Script 32" or add the missing single-firm tests.
+
+3. **K=3 hard labels are incorrectly described as used in the Spearman correlations.** v2:143 says the "K=3 hard label" is used for the internal-consistency Spearman correlations. Script 38's Spearman table uses the K=3 posterior score `P_C1`, not hard labels. Change v2:143 to "K=3 posterior score is used for the Spearman correlations; hard labels are used for the cluster cross-tabulation."
+
+4. **Provenance table over-cites Script 38 for the Big-4 signature count.** v2:17 and v2:152 attribute the 150,442 signature count partly/directly to Script 38. In the supplied markdown report, Script 39 directly reports the 150,442 signature-level cloud; Script 38's visible report does not directly state that count. Keep Script 39 as the direct source unless the JSON artifact is also cited.
+
+5. **"Max fold-to-fold deviation" wording is imprecise.** v2 reports a K=2 "max fold-to-fold deviation" of 0.028 (v2:65, v2:107). Script 36's 0.0278 is the max absolute deviation across folds as reported in the stability summary, not the pairwise fold range; the fold cut range is about 0.0376 (0.9756 - 0.9380). Use the report's exact wording or explicitly define the statistic.
+
+## Provenance re-verification
+
+| v2 numerical claim | v2 lines | Spike-report check | Status |
+|---|---:|---|---|
+| Big-4 has 437 CPAs split 171 / 112 / 102 / 52. | v2:17, v2:151 | Script 36 reports 437 CPAs; Script 34 reports the four firm counts. | CONFIRMED |
+| Big-4 signature-level cloud has 150,442 signatures. | v2:17, v2:95, v2:152 | Script 39 reports fitting on 150,442 signature-level points. | CONFIRMED, but source should be Script 39 rather than Script 38 in the provenance table. |
+| Big-4 K=2 crossings are cos 0.9755 and dHash 3.7549, with CIs [0.9742, 0.9772] and [3.4762, 3.9689]. | v2:45, v2:53, v2:154-156 | Script 36 and Script 34 report these point estimates and bootstrap CIs. | CONFIRMED |
+| K=3 components are C1 0.9457/9.1715/0.143, C2 0.9558/6.6603/0.536, C3 0.9826/2.4137/0.321. | v2:55-63, v2:163 | Scripts 35, 37, and 38 report the same centers and weights. | CONFIRMED |
+| K=3 LOOO membership deviations are 1.8-12.8 pp, with `P2_PARTIAL`. | v2:65, v2:109, v2:168 | Script 37 reports diffs 1.76, 4.68, 5.81, 12.77 pp and verdict `P2_PARTIAL`. | CONFIRMED |
+| Spearman correlations are 0.963, 0.889, and 0.879. | v2:85-91, v2:169 | Script 38 reports 0.9627, 0.8890, and 0.8794. | CONFIRMED |
+| All three scores rank Firm C as most hand-leaning. | v2:93 | Script 38 per-firm summary ranks Firm C highest on mean P_C1 and mean hand_frac, but Firm D highest on mean reverse-anchor. | FLAGGED |
+| Per-signature kappas are 0.662, 0.559, and 0.870; verdict moderate. | v2:95-103, v2:170 | Script 39 reports 0.6616, 0.5586, 0.8701 and `SIG_CONVERGENCE_MODERATE`. | CONFIRMED |
+| Pixel-identical subset is n=262 split 145 / 8 / 107 / 2, with 0% miss rate and Wilson upper 1.45%. | v2:111-119, v2:172-173 | Script 40 reports total 262, the per-firm split, and 262/262 correct for all three candidate classifiers with Wilson [0.00%, 1.45%]. | CONFIRMED |
+| Non-Firm-A dip values are 0.998/0.906 for `big4_non_A` and 0.998/0.907 for `all_non_A`. | v2:21, v2:43, v2:161-162 | Script 32 reports 0.9985/0.9055 and 0.9975/0.9065, matching v2 rounded values. | CONFIRMED |
+
+## Outstanding open questions
+
+1. **Five-way moderate-confidence validation still needs a decision.** v2 is honest that the v4 kappa evidence covers only the high-confidence binary rule (v2:103, v2:190-192). If the five-way classifier remains primary, the cleanest next step is a Big-4-specific capture/FAR/cross-tab analysis for the moderate band and the document-level worst-case aggregation. If not rerun, the manuscript should explicitly state that the moderate band remains inherited from v3.x and is not newly validated by Scripts 38-40.
+
+2. **Firm anonymisation policy still needs confirmation for §IV-V.** v2 itself is pseudonymous, but the open question at v2:194 remains real: once §IV-V discuss within-Big-4 contrasts, the manuscript should consistently use Firm A-D and keep any real-name mapping out of the paper body.
+
+3. **Section IV numbering can remain deferred.** v2:196 is procedural and does not block §III acceptance; resolve after the methodology claims and result-table sequence are frozen.
+
+## Recommended next-step actions
+
+1. Correct v2:93's per-firm ordering claim against Script 38.
+
+2. Decide whether to add a Big-4-specific validation for the five-way moderate band and document-level aggregation. If not, narrow v2:125 so binary-rule correlations do not appear to validate the full five-way classifier.
+
+3. Narrow the dip-test scope language at v2:21 and v2:43, or add missing individual-firm dip tests for Firms B-D.
+
+4. Fix v2:143 so Spearman correlations are tied to K=3 posterior scores, not K=3 hard labels.
+
+5. Correct the provenance table entry for the 150,442 signature count to cite Script 39 as the direct markdown-report source.
+
+6. Replace "max fold-to-fold deviation" with the exact Script 36 statistic or report the actual pairwise fold range.
+
+7. Remove the author checklist and open-question block from the manuscript version after these decisions are resolved.

paper/codex_review_gpt55_v4_round3.md

@@ -0,0 +1,143 @@
+# Paper A Round 23 Review - v4 round 3
+
+Reviewer: gpt-5.5 xhigh  
+Date: 2026-05-12  
+Target: `paper/v4/paper_a_results_v4_section_iv.md` (§IV v2)  
+Cross-checked against: `paper/v4/paper_a_methodology_v4_section_iii.md` (§III v3), round-21/22 reviews, `paper/paper_a_results_v3.md`, and the supplied spike reports.
+
+## Verdict
+
+Major Revision.
+
+The empirical core of §IV v2 is much stronger than the earlier methodology drafts: most new Big-4 numerical tables match the spike reports. The blockers are presentation and provenance risks that reviewers will catch quickly: table numbering is not coherent, several §III cross-references now point to the wrong §IV material, the inherited detection count is misstated, and the draft says firm anonymisation is maintained while repeatedly printing real firm names.
+
+## Major findings
+
+1. **Table numbering is not coherent enough for partner review.**
+
+   §IV v2 says provisional numbering covers Tables IV-XVIII (line 3), and line 13 says v3.20.0 Table IV is "retained as Table IV here." But the file does not actually include a Table IV block; the first displayed v4 table is Table V at line 23. Line 17 also cites the inherited all-pairs analysis as "v3.20.0 §IV-C, Table V," while line 23 reuses Table V for the new Big-4 dip test. That is acceptable only if the inherited table is explicitly not a v4 table; otherwise Table V is duplicated.
+
+   The same issue recurs at the end: line 240 assigns current Table XVIII to the full-dataset Spearman robustness table, while line 254 says the inherited backbone ablation is "Table XVIII in v3.20.0." If the ablation is retained in the v4 manuscript, it cannot also be current Table XVIII. Fix by deciding which inherited v3 tables are reprinted/renumbered versus cited only as v3.x provenance.
+
+2. **§III v3 contains stale cross-references that §IV v2 does not support as written.**
+
+   §III line 13 says signature-level capture-rate analyses are in §IV-D, §IV-F, and §IV-G. In §IV v2, those are accountant-level distributional characterisation, internal-consistency checks, and LOOO reproducibility. This is a direct cross-reference failure.
+
+   §III line 23 says "all §IV results except §IV-K" are Big-4 restricted. §IV v2 itself is narrower and more accurate at line 9: §IV-D through §IV-J are Big-4 primary, while §IV-K is full-dataset robustness. But §IV-A-C are inherited full-corpus setup/detection/all-pairs material, §IV-I is inherited full-corpus inter-CPA FAR, and §IV-L is an inherited corpus-wide ablation. §III must be changed to match the actual results section.
+
+   §III line 109 says the moderate-confidence band retains v3.x capture-rate evaluation in "§IV-F"; in current §IV, §IV-F is not the inherited v3 capture-rate section. It should cite v3.x Tables IX/XI/XII/XII-B or current §IV-J's inheritance note, not current §IV-F.
+
+3. **The inherited detection-count sentence is numerically wrong / ambiguous.**
+
+   §IV line 13 says "182,328 detected signatures across 86,072 prefiltered audit-report PDFs." The v3 baseline distinguishes these counts: VLM screening identified 86,072 documents with signature pages, 12 corrupted PDFs were excluded, and batch YOLO inference ran on 86,071 documents; v3 Table III then reports 85,042 documents with detections and 182,328 extracted signatures. Current line 13 collapses these stages and assigns the 182,328 signatures to the wrong denominator.
+
+   Suggested rewrite: "VLM screening identified 86,072 signature-page documents; after 12 corrupted PDFs were excluded, YOLO batch inference processed 86,071 documents, with 85,042 yielding detections and 182,328 extracted signatures."
+
+4. **The draft claims firm anonymisation is maintained, but the §IV tables reveal real firm names.**
+
+   §III line 23 says the Big-4 firms are pseudonymously labelled Firm A-D. §IV line 265 says firm anonymisation is "maintained throughout §IV (Firm A-D used consistently)." That is false: real names appear in the displayed result tables at lines 93-96, 120-123, 132-135, 179-182, 204-207, and 217-220.
+
+   Either remove the parenthetical real names everywhere in §IV or explicitly abandon the pseudonym policy in §III and the close-out checklist. Given prior review history, this should be fixed before partner review.
+
+5. **Some interpretive claims overstate what the spike results prove.**
+
+   The main false one is line 211: it says the non-Firm-A moderate-confidence proportions are "consistent with the §III-K Spearman convergence on the per-CPA hand-leaning ranking." The MC ordering is C (41.44%), B (35.88%), D (29.33%), while Table X's hand-leaning scores rank D above B on all three score summaries and rank D above C on the reverse-anchor score. MC-band occupancy is not a monotone proxy for the per-CPA hand-leaning ranking; D's mass moves heavily into Uncertain instead.
+
+   Line 184 also compares Firm A's signature-level HC rate (81.70%) to its accountant-level C3 rate (82.46%). The numbers are close and the qualitative reading is reasonable, but they are different units. State this as qualitative alignment, not as a like-for-like consistency check.
+
+   Line 43 calls off-Big-4 dHash transitions "consistent with histogram-resolution artefacts." Script 32 verifies varying dHash transitions; it does not by itself prove a bin-width artefact analysis for those accountant-level subsets. "Scope-dependent and not used operationally" is safer.
+
+6. **The moderate-confidence band is honestly disclosed as inherited, but the support language still needs narrowing.**
+
+   §IV line 211 correctly states that Scripts 38-40 do not separately validate the MC band. That is good. But §III line 131 still says the binary-rule internal-consistency checks support continued use of the inherited five-way rule "without recalibration." That is stronger than the evidence: the v4 kappa/Spearman checks cover the binary high-confidence box rule, not the MC band or document-level worst-case aggregation. The defensible wording is: v4 reports Big-4 outputs for the inherited five-way rule; the MC band remains v3-calibrated and not newly validated in Scripts 38-42.
+
+## Minor findings
+
+1. **K=3 LOOO C1 weight drift is rounded away from the report.** §IV line 137 reports max C1 weight deviation as 0.025. Script 37's report says 0.023, and the JSON gives 0.023489. Use 0.023 or 0.0235.
+
+2. **Seed coverage statement stops at Script 41.** §IV line 7 says seeds are fixed across Scripts 32-41, but v2 depends on Script 42 for Tables XV and XV-B. Either include Script 42 if true or say "stochastic v4 spike scripts" rather than implying a complete script range.
+
+3. **Inclusivity of the low-cosine cutoff should match Script 42.** §IV line 17 says cosine `< 0.837` implies Likely-hand-signed; Script 42 defines LH as `cos <= 0.837`. Align §III-L and §IV-C/J exactly.
+
+4. **The "round-22 open question 1, Light scope" process note is not traceable to the round-22 review file.** §IV line 228 may reflect an author decision outside the supplied review, but it should be removed from manuscript prose or backed by an internal note.
+
+5. **The ablation section pointer is wrong.** §IV line 252 says the inherited feature-backbone ablation is from v3.x §IV-H.3, but in `paper/paper_a_results_v3.md` it is §IV-I, beginning at line 461.
+
+6. **Line 73's "component recovery ... across Scripts 35, 37, and 38" can be misread.** Script 37's full-baseline block replicates Script 35, but the LOOO fold components vary by design. Say "the full-fit baseline is reproduced in Scripts 35, 37, and 38" if that is the intended claim.
+
+## Editorial nits
+
+1. Remove the draft note and Phase 3 close-out checklist before submission, or move them to an internal author note.
+
+2. Line 110: "This convergent-checks evidence" should be "These convergence checks" or "This convergence evidence."
+
+3. Line 3: "is finalised" should be "will be finalised" while numbering remains provisional.
+
+4. Standardise "dHash" versus "dh" in tables and prose; the spike reports use `dh`, but the paper body mostly uses dHash.
+
+5. Avoid mixing "replicated," "templated," and "non-hand-signed" as if they are exact synonyms. The paper's caveats rely on preserving those distinctions.
+
+## Provenance verification table
+
+| §IV v2 claim | §IV lines | Source checked | Status |
+|---|---:|---|---|
+| Big-4 primary scope: 437 CPAs and 150,442 signatures with both descriptors. | 9 | Script 36 report lines 6, 32-37; Script 39 report line 12. | Confirmed. |
+| Detection inheritance: 182,328 signatures across 86,072 PDFs. | 13 | v3 results lines 14, 17-25; v3 methodology search hits distinguish 86,072 VLM-positive, 86,071 processed, 85,042 with detections. | Needs correction; denominator conflated. |
+| All-pairs KDE crossover at 0.837. | 17 | v3 results lines 49 and 118; Script 42 rule lines 6-10 uses 0.837. | Confirmed; fix `<` vs `<=` wording. |
+| Big-4 dip-test p-values reported as `< 5 x 10^-4`. | 27, 32 | Script 36 report lines 6-8; Script 34 report lines 28-31; bootstrap resolution stated in §IV line 32. | Confirmed with reporting convention. |
+| Firm A / Big4-non-A / all-non-A dip p-values: 0.992/0.924, 0.998/0.906, 0.998/0.907. | 28-30 | Script 32 report lines 30, 40, 62, 72, 94, 104. | Confirmed after rounding. |
+| BD/McCrary Big-4 null and non-A dHash transitions at 10.8 and 6.6. | 38-41 | Script 34 report lines 28-31; Script 32 report lines 40-41 and 72-73. | Confirmed; artefact interpretation not directly proven. |
+| K=2 components, crossings, bootstrap CIs, and BIC. | 53-63 | Script 34 report lines 23-41; Script 36 report lines 12-28. | Confirmed. |
+| K=3 component centers/weights and BIC lower by 3.48. | 69-73 | Script 35 report lines 6-10; Script 34 report lines 40-49; Script 36 report lines 9-10. | Confirmed. |
+| Spearman correlations 0.9627, 0.8890, 0.8794 and non-Big-4 reference center 0.935/9.77. | 83-87 | Script 38 report lines 16-18 and 24-30. | Confirmed. |
+| Per-firm score summaries in Table X. | 93-98 | Script 38 report lines 43-48. | Confirmed; anonymisation violation. |
+| Cohen kappas 0.662, 0.559, 0.870 and per-signature K=3 centers. | 106-110 | Script 39 report lines 16-28. | Confirmed after rounding. |
+| K=2 LOOO fold rules and all-or-none held-out classifications. | 120-125 | Script 36 report lines 32-44 and JSON stability summary. | Confirmed. |
+| K=3 LOOO C1 fold rates and `P2_PARTIAL`. | 131-137 | Script 37 report lines 16-19, 25-90, 92-99; JSON exact drift values. | Confirmed, except weight drift should be 0.023/0.0235 not 0.025. |
+| Pixel-identity subset n=262, split 145/8/107/2, 0/262 miss rate, Wilson upper 1.45%. | 147-153 | Script 40 report lines 8, 12-18, 22-27. | Confirmed. |
+| Inter-CPA FAR 0.0005 with Wilson [0.0003, 0.0007] inherited from v3. | 157 | v3 results lines 182-190 and 263-275. | Confirmed as inherited, not v4-regenerated. |
+| Five-way per-signature counts and 11 excluded signatures. | 167-173 | Script 42 report lines 14-26. | Confirmed. |
+| Per-firm five-way percentages. | 179-184 | Script 42 report lines 30-44. | Confirmed; line 211 interpretation is not supported. |
+| Document-level overall counts, n=75,233, mixed-firm PDFs n=379. | 188-198 | Script 42 report lines 46-57; JSON `document_level`. | Confirmed. |
+| Single-firm per-document rows. | 204-209 | Script 42 report lines 59-66. | Confirmed. |
+| Full-dataset robustness components, BIC, Spearman rho. | 234-248 | Script 41 report lines 8-31. | Confirmed. |
+| Feature-backbone ablation inherited from v3.x Table XVIII. | 252-254 | v3 results lines 461-475. | Inherited content confirmed, but v3 section pointer and current v4 table numbering collide. |
+
+## Cross-reference checks (§III -> §IV)
+
+| §III v3 claim | §III lines | §IV v2 support | Status |
+|---|---:|---|---|
+| Signature-level capture-rate analyses are in §IV-D/F/G. | 13 | Current §IV-D/F/G are accountant-level dip/mixture, internal consistency, and LOOO. | Fails; stale v3 cross-reference. |
+| All §IV results except §IV-K are Big-4 restricted. | 23 | §IV-A-C, §IV-I, and §IV-L are inherited full-corpus/corpus-wide material. | Fails; narrow to "primary v4 analyses §IV-D-J except inherited §IV-I." |
+| Big-4 scope is 437 CPAs / 150,442 signatures. | 23 | §IV lines 9, 163 and Script 39. | Supported. |
+| Dip-test and BD/McCrary distributional characterisation. | 47-53 | §IV Tables V-VI, lines 23-43. | Supported. |
+| K=2 and K=3 mixture components and mild BIC preference. | 51, 59-73 | §IV Tables VII-VIII, lines 49-73. | Supported. |
+| K=2 unstable and K=3 descriptive only under LOOO. | 71-79, 111-115 | §IV Tables XII-XIII, lines 116-137. | Supported. |
+| Three-score internal consistency and per-firm ranking nuance. | 83-100 | §IV Tables IX-X, lines 79-100. | Supported. |
+| Per-signature K=3 convergence kappas. | 101-109 | §IV Table XI, lines 102-110. | Supported. |
+| Pixel-identity positive-anchor miss rate. | 117-127 | §IV Table XIV, lines 141-153. | Supported. |
+| Five-way signature/document classifier retained as primary; K=3 not used for operational labels. | 131-149 | §IV-J, lines 159-224. | Mostly supported; the MC band remains inherited and current wording should not imply v4 validation. |
+| Moderate-confidence band retains v3.x capture-rate evaluation. | 109, 145, 198 | §IV line 211 cites v3 Tables IX/XI/XII but not XII-B; §III line 109's "§IV-F" is now wrong. | Needs citation cleanup. |
+| Firm anonymisation maintained. | 23 and open question 200 | §IV repeatedly includes real firm names in parentheses. | Fails unless policy changes. |
+
+## Recommended next-step actions
+
+1. Freeze the v4 table scheme before any prose edits: decide whether inherited v3 tables are reprinted as current v4 tables, cited only as v3 tables, or moved to appendix/supplement. Then renumber Tables IV-XVIII and remove Table XV-B if the journal style cannot handle letter suffixes.
+
+2. Fix §III cross-references after the table scheme is frozen, especially §III line 13, §III line 23, and §III lines 109/119/145.
+
+3. Correct §IV line 13's detection denominator and restate the VLM-positive / corrupted-excluded / YOLO-processed / with-detections sequence.
+
+4. Remove all real firm names from §IV or explicitly change the anonymisation policy. Do not leave line 265 claiming anonymisation while tables reveal names.
+
+5. Delete or rewrite line 211's MC-ordering claim. If the MC band remains inherited, present the per-firm MC proportions descriptively only.
+
+6. Narrow the support claim for the five-way rule: Scripts 38-40 validate only the binary high-confidence rule, while Script 42 reports five-way output counts. Either add a Big-4-specific MC/document validation or state plainly that MC/document validation is inherited from v3.x.
+
+7. Fix small numeric/provenance issues: K=3 weight drift 0.023/0.0235, Script 42 seed wording, cutoff inclusivity, v3 ablation section pointer, and the unsupported "round-22 Light scope" process note.
+
+## Phase 4 readiness assessment
+
+Not ready for partner review without Phase 4 revisions.
+
+The spike-script provenance for the new Big-4 result tables is mostly sound, so I do not see a need to rerun the main v4 empirical scripts solely to fix §IV. But the current section would invite reviewer attacks on table identity, stale cross-references, anonymisation, and overinterpretation of the inherited MC band. After those are corrected, §IV should be close to partner-review ready; the only substantive open decision is whether to add a new Big-4-specific validation for the moderate-confidence/document-level rule or keep it explicitly inherited from v3.x.

paper/codex_review_gpt55_v4_round4.md

@@ -0,0 +1,108 @@
+# Paper A Round 24 Review - v4 round 4
+
+Reviewer: gpt-5.5 xhigh  
+Date: 2026-05-12  
+Target: `paper/v4/paper_a_results_v4_section_iv.md` (§IV v3)  
+Paired methodology: `paper/v4/paper_a_methodology_v4_section_iii.md` (§III v4)  
+Rubric: `paper/codex_review_gpt55_v4_round3.md` (6 Major, 6 Minor, 5 Editorial)
+
+## Verdict
+
+Minor Revision.
+
+The round-23 blockers are substantially reduced. The §IV v3 result tables are now mostly provenance-faithful, the inherited-v3 table identity problem is largely resolved, detection counts are corrected, §IV firm rows are pseudonymised, and the moderate-confidence band is now described honestly as inherited rather than newly validated.
+
+I do not recommend Accept yet because several cleanup issues remain visible in the paired §III/§IV package: §III v4 still leaks real firm names despite the pseudonym policy, §III still carries the stale K=3 LOOO weight-drift value of 0.025 where the report and §IV v3 use 0.023, and the internal draft notes/checklists still contain stale round/version/table-numbering language.
+
+## Round-23 Finding Closure Table
+
+| Round-23 finding | Status | v3/v4 evidence |
+|---|---|---|
+| Major 1. Table numbering was incoherent and inherited v3 tables collided with current v4 tables. | PARTIAL | Core collision is fixed: §IV v3 says inherited v3.x tables are cited only as `v3.20.0 Table N` and not renumbered (§IV:3), and detection/all-pairs/ablation are cited as v3.20.0 Tables III/IV/V/XVIII (§IV:13, 19, 256). Residual: the same draft note still says "Tables IV-XVIII" even though the new v4 sequence starts at Table V (§IV:3), and the close-out checklist repeats "Tables IV-XVIII" plus `Table XV-B` (§IV:265). |
+| Major 2. §III v3 contained stale cross-references not supported by §IV v2. | PARTIAL | Main cross-refs are repaired: §III now points signature-level classification to §IV-J and inherited inter-CPA FAR to §IV-I (§III:13), and accurately scopes §IV-D through §IV-J as v4-new Big-4 analyses while excluding §IV-A-C/I/L and full-dataset §IV-K (§III:23). Residual stale/internal references remain: §III says the corresponding FAR evidence comes from "§III-J inherited; Table X" (§III:119), and the open question still proposes adding a moderate-band analysis in current §IV-F even though §IV-F is convergence checks (§III:198; §IV:77-112). |
+| Major 3. Inherited detection-count sentence was numerically wrong / ambiguous. | CLOSED | §IV v3 now distinguishes VLM-positive documents, corrupted exclusions, YOLO-processed documents, detected-document count, and extracted signatures (§IV:13), matching the v3 baseline's Table III sequence (v3:14, 20-22). |
+| Major 4. Draft claimed anonymisation while §IV tables revealed real firm names. | PARTIAL | §IV v3 uses Firm A-D in tables and prose (§IV:91-100, 120-125, 131-137, 179-184, 204-209, 217-222), so the §IV-specific failure is closed. But the paired §III v4 still leaks real names/aliases: "held-out-EY" (§III:71) and "Firms B (KPMG) and D (EY)" (§III:99), contradicting the pseudonym policy in §III:23 and §IV:3. |
+| Major 5. Interpretive claims overstated what the spike results prove. | CLOSED | The off-Big-4 dHash transition language is now scope-dependent rather than an artefact claim (§IV:45). The Firm A HC vs C3 comparison is explicitly qualitative and cross-unit (§IV:186). MC-band ordering is now explicitly descriptive and not treated as Spearman validation (§IV:213). |
+| Major 6. Moderate-confidence band support language needed narrowing. | CLOSED | §III v4 now states that Scripts 38-42 do not separately validate the MC/style/document components and that v4 only supports the binary high-confidence sub-rule (§III:131). §IV v3 repeats this limitation and cites v3.20.0 Tables IX/XI/XII/XII-B as inherited support (§IV:213). |
+| Minor 1. K=3 LOOO C1 weight drift should be 0.023/0.0235, not 0.025. | PARTIAL | §IV v3 is corrected to 0.023 (§IV:139), matching Script 37. §III v4 still says 0.025 in prose and provenance (§III:71, 115, 173). |
+| Minor 2. Seed coverage statement stopped at Script 41 although §IV used Script 42. | CLOSED | §IV v3 now says seeds are fixed across Scripts 32-42 (§IV:7). |
+| Minor 3. Low-cosine cutoff inclusivity should match Script 42 (`cos <= 0.837`). | PARTIAL | §IV v3 is explicit: cosine `<= 0.837` maps to Likely-hand-signed (§IV:19), matching Script 42. §III-L still says "Cosine below" the crossover (§III:143), which is less precise than the inherited rule; make it "at or below 0.837." |
+| Minor 4. "Round-22 open question 1, Light scope" process note was not traceable. | CLOSED | The §IV-K body now describes the full-dataset robustness scope directly, without the round-22 process-note wording (§IV:230). The remaining stale process text is confined to the internal checklist (§IV:260-267). |
+| Minor 5. Ablation section pointer was wrong. | CLOSED | §IV v3 correctly identifies the inherited feature-backbone ablation as v3.20.0 §IV-I and distinguishes v3 Table XVIII from current v4 Table XVIII (§IV:254-256). |
+| Minor 6. "Component recovery across Scripts 35, 37, and 38" could be misread. | CLOSED | §IV v3 now says the full-fit K=3 baseline is reproduced in Scripts 35, 37, and 38, while Script 37 fold components differ by design and are separately reported (§IV:75). |
+| Editorial 1. Remove draft note and Phase 3 close-out checklist before submission. | OPEN | Both files still include internal draft notes and author checklists/open questions (§III:3-9, 187-202; §IV:3, 260-267). §IV's checklist also says the section is being prepared for "codex round 23" even though this is round 24 (§IV:262). |
+| Editorial 2. "This convergent-checks evidence" grammar. | CLOSED | §IV v3 uses "These convergence checks" (§IV:112). |
+| Editorial 3. "is finalised" should be "will be finalised." | CLOSED | §IV v3 uses future/provisional wording (§IV:3, 265). |
+| Editorial 4. Standardise `dHash` versus `dh`. | CLOSED | Manuscript prose/tables consistently use `dHash`; raw spike-script `dh` appears only inside source descriptions or quoted rule names (§III:13, 133-145; §IV:36, 53-63, 167-184). |
+| Editorial 5. Avoid mixing "replicated," "templated," and "non-hand-signed" as exact synonyms. | CLOSED | Current usage mostly preserves distinctions: replicated is used for positive-anchor / C3 contexts (§IV:143-155), non-hand-signed for the operational five-way categories (§IV:167-173), and templated mainly for K=2 fold-rule wording (§IV:120-127). No remaining overclaim depends on treating them as exact synonyms. |
+
+## Newly Introduced Or Remaining Issues
+
+1. **§III v4 still violates the anonymisation policy.** §III says firms are pseudonymously labelled Firm A-D throughout the manuscript (§III:23), but line 71 says "held-out-EY" and line 99 names KPMG and EY. §IV v3 fixed this; §III now needs the same scrub.
+
+2. **§III v4 has a stale K=3 LOOO weight-drift number.** Script 37 reports max C1 weight deviation 0.023, and §IV v3 uses 0.023 (§IV:139). §III still reports 0.025 in two prose locations and the provenance table (§III:71, 115, 173).
+
+3. **Two §III internal references are stale.** The positive-anchor paragraph cites "§III-J inherited; Table X" for inter-CPA FAR (§III:119), but the paired result location is §IV-I and the inherited source is v3.20.0 §IV-F.1/Table X (§IV:157-159). The open question asks whether to add a moderate-band analysis in §IV-F (§III:198), but current §IV-F is the convergence section.
+
+4. **Internal notes are stale enough to confuse a handoff.** §III's draft note says "(2026-05-12, v3)" although the file title is v4 (§III:1, 3). §IV's close-out checklist says "before §IV is sent for codex round 23" even though round 23 has already happened (§IV:262), and item 4 says issues are addressed in "this v2" inside a v3 file (§IV:267).
+
+5. **§III mentions the full-dataset `n = 686` but does not list it in the §III provenance table.** §III:23 states that §IV-K reports a full-dataset cross-check at 686 CPAs; Script 41 directly reports full dataset `N CPAs = 686`. Add that row if the number remains in §III.
+
+6. **The table-numbering note still has a small self-contradiction.** §IV:3 says the new v4 sequence is Table V through Table XVIII, then says "Tables IV-XVIII" remain provisional. Either add a current Table IV, or make all provisional references "Tables V-XVIII" and decide whether `Table XV-B` is acceptable for the target style.
+
+## Cross-Reference Checks (§III v4 <-> §IV v3)
+
+| Claim / linkage | §III v4 line evidence | §IV v3 line evidence | Status |
+|---|---:|---:|---|
+| Big-4 scope and inherited/non-Big-4 exceptions. | §III:23 | §IV:9, 13, 19, 157-159, 230, 254-256 | Supported. |
+| Big-4 sample size: 437 CPAs and 150,442 classified signatures. | §III:23, 157-158 | §IV:9, 15, 165, 175 | Supported. |
+| Dip-test and BD/McCrary accountant-level characterisation. | §III:49-53 | §IV:25-45 | Supported. |
+| K=2/K=3 mixture components and mild BIC preference. | §III:59-69 | §IV:51-75 | Supported. |
+| K=2 unstable; K=3 descriptive, not operational, under LOOO. | §III:71-79, 111-115 | §IV:116-139 | Mostly supported; align §III's 0.025 weight drift to §IV's/report's 0.023. |
+| Three-score internal-consistency correlations and per-firm ranking nuance. | §III:83-99 | §IV:79-102 | Supported, except §III anonymisation leak in line 99. |
+| Per-signature K=3 convergence and binary kappa values. | §III:101-109 | §IV:104-112 | Supported. |
+| Pixel-identity positive-anchor miss rate. | §III:117-127 | §IV:141-155 | Supported, but §III:119 should cite §IV-I/v3 §IV-F.1 for inter-CPA FAR, not "§III-J inherited." |
+| Five-way classifier retained as primary and MC band inherited. | §III:131-149 | §IV:161-213 | Supported; make §III:143 inclusive for `cos <= 0.837`. |
+| K=3 hard label vs K=3 posterior roles. | §III:149 | §IV:215-224 and 81-89 | Supported: hard labels for cluster cross-tab, posterior P(C1) for Spearman. |
+| Full-dataset robustness is light scope only. | §III:23, 31 | §IV:228-252 | Supported, but add provenance for `n = 686` to §III table or remove the number from §III. |
+| Internal author/open-question checklist. | §III:187-202 | §IV:260-267 | Not manuscript-ready; stale references remain. |
+
+## Provenance Re-Verification Of Changed Numerics
+
+| Changed numerical claim | Manuscript line(s) | Source checked | Status |
+|---|---:|---|---|
+| Detection sequence: 86,072 VLM-positive; 12 corrupted; 86,071 YOLO-processed; 85,042 with detections; 182,328 signatures. | §IV:13 | v3 baseline reports 86,071 processed, 85,042 with detections, and 182,328 signatures (v3:14, 20-22). The 86,072/12 sequence is inherited from the v3 narrative already cited in round 23. | Confirmed; round-23 denominator conflation is fixed. |
+| Big-4 signature sample: 150,453 loaded, 150,442 classified, 11 missing descriptors. | §IV:175 | Script 42 reports loaded 150,453, classified 150,442, unclassified 11 (five_way_report:14-16). | Confirmed. |
+| K=2 marginal crossings and bootstrap CIs: cos 0.9755, dHash 3.755, CIs [0.9742, 0.9772] and [3.476, 3.969]. | §IV:62-65; §III:51, 59-60 | Script 36 reports cos point 0.9755 and dHash point 3.7549 with those CIs (calibration_loo_report:14-17). | Confirmed. |
+| K=3 components: C1 0.9457/9.17/0.143; C2 0.9558/6.66/0.536; C3 0.9826/2.41/0.321. | §IV:67-75; §III:61-69 | Scripts 35/37/38 report the same baseline (inspection_report:6-10; k3_loo_report:6-10; convergence_report:8-12). | Confirmed. |
+| K=3 lower than K=2 by 3.48 BIC points. | §IV:75; §III:69 | Script 36 reports K=2 BIC -1108.45 and K=3 BIC -1111.93 (calibration_loo_report:9-10). | Confirmed by arithmetic. |
+| Spearman correlations: 0.9627, 0.8890, 0.8794, with p-values bounded in manuscript. | §IV:81-89; §III:91-99 | Script 38 reports 0.9627 / 3.92e-249, 0.8890 / 1.09e-149, 0.8794 / 2.73e-142 (convergence_report:26-30). | Confirmed. |
+| Per-firm score nuance: Firm C highest on P(C1)=0.3110 and hand_frac=0.7896; Firm D higher on reverse-anchor score -0.7125 vs Firm C -0.7672. | §IV:95-102; §III:99 | Script 38 per-firm summary reports those values (convergence_report:43-48). | Confirmed; §III should anonymise KPMG/EY parentheticals. |
+| K=3 LOOO C1 weight drift is 0.023, not 0.025. | §IV:139; §III:71, 115, 173 | Script 37 reports max C1 weight deviation 0.023 (k3_loo_report:77-79). | §IV confirmed; §III mismatch remains. |
+| Pixel-identical Big-4 subset n=262, split 145/8/107/2, all classifiers 0% miss with Wilson upper 1.45%. | §IV:145-153; §III:117-127 | Script 40 reports total 262, 262/262 correct for all three classifiers, and per-firm split 145/8/107/2 (far_report:8, 12-18, 22-27). | Confirmed. |
+| Five-way per-signature counts: HC 74,593; MC 39,817; HSC 314; UN 35,480; LH 238. | §IV:165-175 | Script 42 reports the same counts and percentages (five_way_report:20-26). | Confirmed. |
+| Per-firm five-way percentages: Firm A 81.70/10.76/0.05/7.42/0.07; Firm B 34.56/35.88/0.29/29.09/0.18; Firm C 23.75/41.44/0.38/34.21/0.22; Firm D 24.51/29.33/0.22/45.65/0.29. | §IV:181-186, 213 | Script 42 reports the same percentages (five_way_report:39-44). | Confirmed; interpretation is now appropriately descriptive. |
+| Document-level counts: n=75,233 PDFs; HC 46,857; MC 19,667; HSC 167; UN 8,524; LH 18; mixed-firm PDFs n=379. | §IV:190-200 | Script 42 reports n=75,233, mixed-firm n=379, and those category counts (five_way_report:46-57). | Confirmed. |
+| Full-dataset robustness: full n=686; component rows; full rho 0.9558; drift 0.0069. | §IV:232-250; §III:23 | Script 41 reports Big-4 n=437, full n=686, component drifts, BICs, rho 0.9558, and drift 0.0069 (fulldataset_report:8-31). | Confirmed; add §III provenance row for n=686. |
+
+## Phase 4 Readiness
+
+Partial.
+
+The empirical tables are close to partner-review ready and I do not see a need to rerun the main v4 scripts for §IV. The remaining issues are mostly manuscript hygiene, pseudonym consistency, and cross-reference/provenance alignment. They are small edits, but they are visible enough that I would not send the paired §III/§IV package to partner review until they are fixed.
+
+## Recommended Next-Step Actions
+
+1. Scrub §III v4 for real firm names/aliases. Replace "held-out-EY" and "Firms B (KPMG) and D (EY)" with Firm A-D language, or explicitly abandon the pseudonym policy everywhere.
+
+2. Align K=3 LOOO weight drift to Script 37 throughout §III: use 0.023 (or 0.0235 if exact precision is preferred), matching §IV:139.
+
+3. Fix the remaining stale cross-references: §III:119 should point to current §IV-I / inherited v3.20.0 §IV-F.1 Table X; §III:198 should not refer to current §IV-F for a possible moderate-band analysis.
+
+4. Make the §III-L low-cosine rule inclusive: Likely hand-signed is `cos <= 0.837`, matching Script 42 and §IV:19.
+
+5. Remove or move internal draft notes, cross-reference indices, open questions, and the §IV Phase 3 close-out checklist before partner review. At minimum, fix stale "v2/v3/round 23" text.
+
+6. Finalise table numbering after deciding whether `Table XV-B` is acceptable. If the current v4 sequence starts at Table V, remove residual "Tables IV-XVIII" wording.
+
+7. Add §III provenance for the full-dataset `n = 686` claim if it remains in §III-G; cite Script 41 / `fulldataset_report.md`.

paper/codex_review_gpt55_v4_round5.md

@@ -0,0 +1,79 @@
+# Paper A Round 25 Review - v4 round 5
+
+Reviewer: gpt-5.5 xhigh  
+Date: 2026-05-12  
+Target: `paper/v4/paper_a_results_v4_section_iv.md` (§IV v3.1 target; file header still says Draft v3)  
+Paired methodology: `paper/v4/paper_a_methodology_v4_section_iii.md` (§III v5)  
+Rubric: `paper/codex_review_gpt55_v4_round4.md` (3 Major-PARTIAL, 2 Minor-PARTIAL, 1 Editorial-OPEN, plus 7 next-step actions)
+
+## Verdict
+
+Minor Revision.
+
+The round-24 empirical and cross-reference residuals have mostly converged. §III v5 now aligns the K=3 LOOO weight drift to 0.023, fixes the §IV-I / v3.20.0 Table X FAR pointer, makes the low-cosine rule inclusive at `cos <= 0.837`, and adds the full-dataset `n = 686` provenance row. §IV v3.1 remains numerically/provenance-faithful.
+
+I do not recommend Accept yet because the partner-facing package still contains internal draft notes/checklists and unresolved table-numbering/version residues. There is also a small anonymisation regression in §III's v5 changelog: the body now uses Firm A-D, but the internal note itself reprints two real firm names (§III:11).
+
+## Round-24 Finding Closure Table
+
+| Round-24 item | v5/v3.1 status | v5/v3.1 line evidence |
+|---|---|---|
+| Major 1. Table numbering was incoherent and inherited v3 tables collided with current v4 tables. | PARTIAL | Core collision remains fixed: §IV says fresh v4 tables are V-XVIII and inherited v3 tables keep `v3.20.0 Table N` (§IV:3); inherited detection/all-pairs/ablation are cited as v3.20.0 Tables III/IV/V/XVIII (§IV:13, 19, 256). Residual remains: the same note still says "Tables IV-XVIII" despite the v4 sequence starting at Table V (§IV:3), and the close-out checklist repeats "Tables IV-XVIII" with `Table XV-B` (§IV:265). |
+| Major 2. §III stale cross-references not supported by §IV. | CLOSED | §III now points signature-level classification to §IV-J and inherited inter-CPA FAR to §IV-I (§III:18), scopes v4-new vs inherited §IV sections accurately (§III:28), cites the FAR evidence as §IV-I / v3.20.0 §IV-F.1 Table X (§III:124), and no longer sends the moderate-band open question to current §IV-F (§III:204). |
+| Major 4. Anonymisation leak in paired §III/§IV package. | PARTIAL | The manuscript body is repaired: §III uses Firm A-D in the score discussion (§III:104), and §IV tables/prose use Firm A-D (§IV:95-98, 181-184, 217-222). However §III's internal v5 changelog reprints real names while saying they were removed (§III:11). This is not a body-table leak, but it keeps the file-level anonymisation cleanup incomplete until draft notes are stripped. |
+| Minor 1. K=3 LOOO C1 weight drift should be 0.023/0.0235, not 0.025. | CLOSED | §III now reports 0.023 in the K=3 LOOO discussion (§III:76, 120) and provenance table (§III:178); §IV reports 0.023 (§IV:139). This matches Script 37 (`k3_loo_report.md`:79). |
+| Minor 3. Low-cosine cutoff inclusivity should match Script 42 (`cos <= 0.837`). | CLOSED | §III-L now defines Likely hand-signed as "Cosine at or below" the crossover with `cos <= 0.837` (§III:148); §IV repeats `cosine <= 0.837 => Likely-hand-signed` and explicitly ties it to Script 42 (§IV:19). |
+| Editorial 1. Remove draft notes and Phase 3 close-out checklist before submission. | OPEN | Internal notes remain in both files: §III has a draft note, cross-reference index, and open questions (§III:3, 193-208); §IV has a draft note and Phase 3 checklist (§IV:3, 260-269). §IV also still identifies itself as Draft v3 / post rounds 21-23 (§IV:1, 3) despite this round targeting v3.1. |
+| Action 1. Scrub §III real firm names/aliases. | PARTIAL | The old body leaks are gone, but §III:11 now quotes two real firm names in the v5 changelog. Replace with "real firm names/aliases" or remove the changelog before partner review. |
+| Action 2. Align K=3 LOOO weight drift to Script 37 throughout §III. | CLOSED | §III:76, §III:120, and §III:178 all use 0.023; §IV:139 matches. |
+| Action 3. Fix stale §III refs: FAR pointer and moderate-band open question. | CLOSED | FAR pointer now cites §IV-I / v3.20.0 §IV-F.1 Table X (§III:124); the moderate-band open question now points to v3.20.0 Tables IX/XI/XII/XII-B and §IV-J, not current §IV-F (§III:204). |
+| Action 4. Make §III-L low-cosine rule inclusive. | CLOSED | §III:148 says `cos <= 0.837`; §IV:19 and Script 42 agree. |
+| Action 5. Remove/move internal notes and fix stale v2/v3/round-23 text. | OPEN | Notes remain (§III:3, 193-208; §IV:3, 260-269). Some stale text is still visible: §IV title and draft note say Draft v3 / post rounds 21-23 (§IV:1, 3), and the checklist says "this v3 of §IV" (§IV:267). |
+| Action 6. Finalise table numbering and remove residual "Tables IV-XVIII" if sequence starts at Table V. | PARTIAL | The current body table sequence is internally usable (V-XVIII with XV-B), but the finalisation note still says Tables IV-XVIII (§IV:3, 265), and §III leaves table numbering open (§III:208). |
+| Action 7. Add §III provenance for full-dataset `n = 686`. | CLOSED | §III now states §IV-K uses `n = 686` (§III:28) and adds a provenance row citing Script 41 / `fulldataset_report.md` (§III:184). §IV reports the same full-dataset count (§IV:230, 247). |
+
+## Newly Introduced Issues
+
+1. **§III v5 changelog reintroduces real firm names.** The body anonymisation fix succeeded, but §III:11 quotes two real names in the internal changelog. If the note is stripped before partner review, this disappears; if the file is circulated as-is, anonymisation is still not clean.
+
+2. **§III empirical-anchor range is stale after the Script 41/42 additions.** §III:14 says empirical anchors reference Scripts 32-40, but the same file now cites Script 41 for full-dataset `n = 686` (§III:184) and references Scripts 38-42 in the classifier-validation caveat (§III:136). §IV's anchor statement already uses Scripts 32-42 (§IV:3). Align §III:14 to Scripts 32-42.
+
+3. **§IV v3.1 is not labelled as v3.1 in the file.** The requested target is §IV v3.1, but the file title and draft note still say v3 / post rounds 21-23 (§IV:1, 3). This is editorial, but it will confuse the Phase 4 handoff.
+
+## Cross-Reference Checks (§III v5 <-> §IV v3.1)
+
+| Linkage | §III v5 evidence | §IV v3.1 evidence | Status |
+|---|---:|---:|---|
+| Big-4 scope and inherited/full-dataset exceptions. | §III:28, 36 | §IV:9, 15, 230, 254-256 | Tight. |
+| K=2/K=3 mixtures are descriptive, not operational. | §III:62, 76-84, 154 | §IV:75, 139, 224 | Tight. |
+| Three-score internal-consistency and per-firm ranking nuance. | §III:88-104 | §IV:79-102 | Tight in body; anonymisation note issue remains outside body (§III:11). |
+| Positive-anchor miss rate and inherited inter-CPA FAR. | §III:122-132, 186 | §IV:143-159 | Tight; the old bad "§III-J inherited; Table X" pointer is gone. |
+| Five-way classifier retained; MC band inherited only. | §III:136-150, 204 | §IV:163, 213 | Tight. |
+| Inclusive LH cutoff at `cos <= 0.837`. | §III:148 | §IV:19 | Tight and matches Script 42. |
+| Full-dataset robustness is light scope only. | §III:28, 184, 204 | §IV:230-252 | Tight. |
+| Internal notes / table-numbering handoff. | §III:193-208 | §IV:260-269 | Not partner-ready; remaining editorial open items are all here. |
+
+## Provenance Spot-Checks Of v5 Changes
+
+| v5 change checked | Manuscript evidence | Spike-report evidence | Status |
+|---|---:|---:|---|
+| K=3 LOOO C1 weight drift is 0.023, not 0.025. | §III:76, 120, 178; §IV:139 | `k3_loo_report.md`:76 lists fold C1 weights; `k3_loo_report.md`:79 reports max C1 weight deviation 0.023. | Confirmed. |
+| Full-dataset `n = 686` provenance row added. | §III:28, 184; §IV:230, 247 | `fulldataset_report.md`:10-13 reports Big-4 437 and full dataset 686; lines 29-31 report full rho 0.9558 and drift 0.0069, matching §IV:246-248. | Confirmed. |
+| Low-cosine Likely-hand-signed rule is inclusive at `cos <= 0.837`. | §III:148; §IV:19 | `five_way_report.md`:6-10 defines HC/MC/HSC/UN/LH and gives `LH : cos <= 0.837`. | Confirmed. |
+| Full-dataset component rows in §IV-K. | §IV:236-240 | `fulldataset_report.md`:19-23 reports the same full component centers, drifts, and BIC values after rounding. | Confirmed. |
+
+## Phase 4 Readiness
+
+Partial.
+
+The empirical content and §III-§IV technical cross-references are ready for Phase 4 technical review. The package is not yet clean enough for partner-facing circulation because the internal notes/checklists remain, §IV still carries v3/round-23 labels, table numbering is still provisional, and §III:11 reprints real firm names inside the changelog.
+
+## Recommended Next-Step Actions
+
+1. Strip or move all internal draft notes, cross-reference indices, open questions, and the §IV Phase 3 checklist before partner review. This also removes the §III:11 anonymisation regression if the changelog is deleted.
+
+2. If any changelog remains, replace the real names in §III:11 with "real firm names/aliases" and update §III:14 from Scripts 32-40 to Scripts 32-42.
+
+3. Finalise §IV table numbering: either make the current v4 sequence explicitly Tables V-XVIII with XV-B accepted, or renumber to remove XV-B; in either case remove residual "Tables IV-XVIII" wording (§IV:3, 265).
+
+4. Update the §IV header/draft note to the actual target version and round status, or remove the draft note entirely (§IV:1, 3, 267).

paper/codex_review_gpt55_v4_round6.md

@@ -0,0 +1,157 @@
+# Paper A Round 26 Review - v4 round 6
+
+Reviewer: gpt-5.5 xhigh  
+Date: 2026-05-12  
+Target: `paper/v4/paper_a_prose_v4_phase4.md` (Phase 4 prose draft v1)  
+Foundation checked: `paper/v4/paper_a_methodology_v4_section_iii.md` (§III v6) and `paper/v4/paper_a_results_v4_section_iv.md` (§IV v3.2)  
+Trajectory checked: rounds 21-25 plus v3.20.0 Abstract / §I / §II / §V / §VI baselines
+
+## Verdict
+
+Major Revision.
+
+The technical core in §III v6 and §IV v3.2 is stable, but the new Phase 4 prose introduces several reviewer-visible regressions. The most important are: (i) the Abstract and Introduction revive the "independent scores" overclaim even though §III/§IV repeatedly say the three scores are not statistically independent; (ii) §I and §V overstate the Big-4 scope evidence by claiming unsupported single-firm and full-dataset dip-test non-rejections; (iii) §II is still a placeholder with `[add citation]`, not a submission-ready related-work section; and (iv) §V-G drops several inherited limitations from v3.20.0.
+
+## Section-By-Section Findings
+
+### Abstract
+
+1. **Major - line 11: "Three independent feature-derived scores" contradicts the converged methodology.** §III-K states that the three scores are "not statistically independent measurements" because all are deterministic functions of the same descriptor means (§III:90), and §IV-F repeats the caveat (§IV:79). The Abstract should say "three feature-derived scores" or "three non-identical feature-derived summaries" and, if space allows, add the shared-feature caveat.
+
+2. **Minor - line 11: "candidate classifiers" can be read as operational-classifier language.** One of the three "candidate classifiers" is the K=3 per-CPA hard label, which §III-J/§III-L explicitly demotes to descriptive characterisation, not operational signature/document classification (§III:64, §III:156). Use "candidate rules/scores" or explicitly reserve "operational classifier" for the inherited five-way box rule.
+
+3. **Minor - line 11: the Abstract passes IEEE Access form but has no margin.** It is one paragraph and `wc -w` counts 247 words, so it satisfies the <=250-word target. Any added caveat will require trimming elsewhere.
+
+4. **Minor - line 11: the Abstract does not name the primary operational output.** The abstract describes the pipeline and the K=3 / convergence / anchor checks, but it does not state that the primary operational output remains the inherited five-way per-signature classifier with worst-case document aggregation (§III-L; §IV-J). This omission makes the K=3 and reverse-anchor checks look more central operationally than §III/§IV allow.
+
+### §I Introduction
+
+1. **Major - line 31: the Big-4 scope claim is overbroad and partly unsupported.** The sentence says "neither any single firm pooled alone nor the broader full-dataset variant rejects unimodality." §III and §IV only report comparison dip tests for Firm A alone, Firms B+C+D pooled, and all non-Firm-A pooled (§III:34, §III:56; §IV:27-34). They explicitly state that single-firm dip tests for Firms B, C, and D were not separately computed (§III:34, §III:56; §IV:34). §IV-K is a light full-dataset K=3 + Spearman robustness check and does not report a full-dataset dip test (§IV:230-252). Rewrite this as "no narrower comparison scope tested in Script 32..." and remove the full-dataset dip-test claim unless a spike report is added.
+
+2. **Major - line 29: the section cross-reference for accountant-level distributional characterisation is wrong.** The prose points to "§III-D" for the Big-4 accountant-level distributional characterisation. In the converged methodology, this material is §III-G through §III-J, especially §III-I and §III-J (§III:18-86). §IV-D/§IV-E are correct.
+
+3. **Major - line 35: the Introduction repeats the "independent feature-derived scores" error.** The next sentence correctly says the scores are not statistically independent, but the opening clause still hands reviewers an avoidable contradiction. This was a central round-21/22 issue and should not reappear in the front matter.
+
+4. **Minor - line 47: contribution 4 again overstates "not at narrower scopes."** The defensible phrase is "not in the narrower comparison scopes tested" because B/C/D single-firm dip tests were not computed.
+
+5. **Minor - line 55: contribution 8 overclaims the full-dataset check.** §IV-K deliberately re-runs only K=3 + Paper A box-rule Spearman convergence at full `n = 686`; it does not re-run LOOO, five-way moderate-band validation, or operational threshold calibration (§IV:230). "Pipeline reproducibility at multiple scopes" should be narrowed to "the K=3 + box-rule rank-convergence check reproduces at the full-CPA scope."
+
+6. **Minor - line 25: the methodological safeguards paragraph uses "external validation" too broadly.** The pixel-identity anchor is a conservative positive-subset check, the inter-CPA FAR is inherited corpus-wide, and LOOO is descriptive composition-sensitivity evidence. The paragraph should avoid implying full external validation of the operational classifier.
+
+### §II Related Work
+
+1. **Major - lines 63-65: §II is not submission-ready prose if inserted as written.** The section says v3.20.0 §II is retained "without substantive change," but the target Phase 4 file is supposed to replace the §II block. As written, it is a meta-summary rather than an actual Related Work section. Either the master manuscript must keep the full v3.20.0 §II text and splice in the LOOO paragraph, or this file must contain the full revised §II.
+
+2. **Major - line 67: unresolved citation placeholder.** "`[add citation]`" is still present. This must be replaced before Phase 5; otherwise a reviewer can attack the only new Related Work content as uncited.
+
+3. **Minor - line 67: "calibration uncertainty band on the operational rule" conflicts with the converged classifier framing.** §III-J says neither K=2 nor K=3 is used as an operational classifier (§III:64), and §III-L reserves operational classification for the inherited five-way box rule (§III:138-156). If the LOOO paragraph is about K=2/K=3 mixture fits, call it a composition-sensitivity or calibration-uncertainty check on the candidate mixture boundary/characterisation, not on "the operational rule."
+
+### §V Discussion
+
+1. **Major - line 81: the prose reifies mechanism labels at the CPA level.** "Some CPAs are templated, some are hand-leaning, some are mixed" is stronger than §III allows. §III-G says a per-CPA mean is a summary statistic, not a claim that all signatures for that CPA share a mechanism (§III:22). Use component-membership wording: "some CPAs' observed signatures place their per-CPA means in the templated, mixed, or hand-leaning regions."
+
+2. **Major - line 81: the within-CPA unimodality explanation is speculative.** The claim that occasional template reuse "produces a unimodal per-signature distribution within the CPA but a multimodal per-CPA distribution across CPAs" is not directly tested in §III/§IV. v3.x tested Firm A and all-CPA signature-level distributions, and v4.0 adds per-signature K=3 consistency (§IV-F), but there is no per-CPA distributional test for individual CPAs.
+
+3. **Major - lines 103-119: limitations are incomplete relative to v3.20.0 and the inherited pipeline.** The v4 limitations keep the Big-4 scope, missing hand-signed ground truth, pixel-identity subset, inherited-rule, A1, K=3 composition, and no-intent caveats. They drop v3 limitations that still apply: ImageNet-pretrained ResNet-50 without signature-domain fine-tuning (v3 §V:90-92), HSV red-stamp removal artifacts (v3 §V:93-95), longitudinal scanning/PDF/compression confounds (v3 §V:97-99), source-exemplar misattribution in max/min pair logic (v3 §V:100-102), and legal/regulatory interpretation limits (v3 §V:108-109). If these are intentionally retired, the draft needs a reason; otherwise they should be restored.
+
+4. **Major - line 107: the scope limitation repeats the unsupported full-dataset dip-test implication.** The sentence says dip-test multimodality is "not available at narrower or broader scopes." §III/§IV do not report full-dataset dip-test results; §IV-K is explicitly a light Spearman robustness check (§IV:230-252). Keep the LOOO broader-scope caveat, but do not claim full-dataset dip-test non-availability without evidence.
+
+5. **Minor - line 79: "v4.0 inherits and confirms" is too strong for the per-signature continuous-spectrum reading.** The exact v3 per-signature diagnostic package is inherited; v4.0's new per-signature evidence is mostly the K=3 consistency check (§IV-F) and five-way output (§IV-J). Safer: "v4.0 inherits this signature-level reading and remains consistent with it."
+
+6. **Minor - line 85: inherited Firm A byte-level details need provenance language.** The 145 Firm A pixel-identical signatures are verified in Script 40, but the "50 distinct partners" and "35 cross-year" details are explicitly inherited from v3 / Script 28 and not regenerated in v4.0 (§III:44, §III:190). The discussion should mark that provenance, especially because the spike reports provided for v4 only verify the 145 count.
+
+7. **Minor - line 87: Firm A does not alone anchor §IV-H.** §IV-H's positive-anchor subset is all Big-4 byte-identical signatures, `n = 262`, split 145 / 8 / 107 / 2 across Firms A-D (§IV:145-153). Firm A is the largest subset and the case-study evidence, but not the whole anchor.
+
+8. **Minor - line 97: "published box rule" is not traceable.** §III/§IV call this the inherited Paper A / v3.x box rule, not a published external rule (§III:96, §III:138; §IV:85-87). Use "inherited box rule" unless there is a publication citation.
+
+9. **Minor - line 97: "produce the same per-CPA ranking" is stronger than the evidence.** The scores are highly correlated, but §III/§IV note a residual non-Firm-A disagreement: reverse-anchor ranks Firm D fractionally above Firm C while P(C1) and box-rule hand-leaning rate rank Firm C highest (§III:106; §IV:102). Say "broadly concordant ranking."
+
+10. **Minor - line 101: "candidate classifiers" again blurs operational status.** K=3 hard labels remain descriptive. This can be fixed together with the Abstract wording.
+
+### §VI Conclusion And Future Work
+
+1. **Major - line 127: "cross-scope pipeline reproducibility" overstates §IV-K.** The full-dataset result verifies only that K=3 P(C1) and Paper A hand-leaning-rate Spearman convergence remains high at `n = 686` with drift `0.0069` (§IV:242-250; full-dataset report:25-31). It does not reproduce the pipeline, the five-way classifier, the moderate-confidence band, LOOO, or operational thresholds at full scope.
+
+2. **Minor - line 129: the future-work audit-quality contrast must stay explicitly descriptive.** "Firm A's 82% templated concentration vs Firm C's 23.5% hand-leaning concentration" comes from K=3 hard-posterior accountant-level assignment (§IV:215-224), whose membership is composition-sensitive (§IV:129-139). The future-work sentence is acceptable if it says these are descriptive component concentrations and that current Paper A provides no audit-quality correlation evidence.
+
+3. **Minor - lines 125-127: the conclusion underplays the actual operational output.** It names the pipeline and methodological checks, but it does not mention the inherited five-way per-signature/document-level classifier that §III-L and §IV-J define as the operational output. This is not a numerical error, but it leaves the operational-vs-descriptive distinction less clear at closure.
+
+## Reviewer-Attack Vulnerabilities Specific To The Prose
+
+1. A reviewer can quote line 11 or line 35 ("independent feature-derived scores") against §III-K/§IV-F's non-independence caveat and argue that the paper exaggerates validation strength.
+
+2. A reviewer can attack the Big-4 scope claim because the prose says "any single firm" and "full-dataset variant" even though B/C/D single-firm dip tests and full-dataset dip tests are not reported.
+
+3. The current §II can be rejected as incomplete because it is a placeholder, not a related-work section, and includes `[add citation]`.
+
+4. "Published box rule" invites a citation challenge. The body only supports "inherited Paper A / v3.x box rule."
+
+5. The discussion sometimes turns descriptive component labels into apparent mechanism claims about CPAs. This conflicts with the §III-G rule that per-CPA means are summaries, not partner-level mechanism assignments.
+
+6. The phrase "candidate classifiers" for K=3 and reverse-anchor checks can be read as walking back the round-21 convergence that K=3 is descriptive and the five-way box rule is operational.
+
+7. The limitations section is vulnerable because it drops inherited limitations that still apply to the pipeline: feature backbone transfer, red-stamp preprocessing, longitudinal document-generation shifts, source-exemplar misattribution, and legal interpretation limits.
+
+8. The full-dataset robustness claim is easy to overread. §IV-K is intentionally "light scope"; calling it pipeline reproducibility or cross-scope operational reproducibility exceeds the evidence.
+
+## Provenance Verification Table
+
+| # | Phase 4 numerical claim | Phase line(s) | Provenance checked | Status |
+|---:|---|---:|---|---|
+| 1 | Abstract is <=250 words | 11 | `sed -n '11p' ... \| wc -w` returned 247 | Confirmed, but close to limit |
+| 2 | 90,282 reports, 182,328 signatures, 758 CPAs | 11, 37, 125 | §IV:7 gives 90,282 PDFs; §IV:13 gives 182,328 extracted signatures; v3 §I:62 gives 758 CPAs | Confirmed with inherited full-corpus CPA source |
+| 3 | Big-4 sub-corpus: 437 CPAs, 150,442 signatures | 11, 37, 125 | §III:30; §IV:9, §IV:15; five-way report:14-15 | Confirmed |
+| 4 | Big-4 dip-test multimodality, `p < 5 x 10^-4` on both axes | 11, 31, 81, 127 | §III:34, §III:56, §III:171-172; §IV:27-34 | Confirmed for Big-4 |
+| 5 | "Neither any single firm pooled alone nor broader full-dataset variant rejects" | 31 | §III:34/56 and §IV:34 say only Firm A alone was tested among single firms; §IV-K has no full-dataset dip test | Not verified / overclaimed |
+| 6 | K=2 crossings `cos*=0.9755`, `dHash*=3.755`, cosine CI half-width 0.0015 | 31 | calibration report:16-17; §III:58, §III:166-170; §IV:60-63 | Confirmed |
+| 7 | K=2 LOOO max cosine-crossing deviation `0.028`, `5.6x` tolerance, Firm A held-out 100% vs non-A 0% | 31, 91 | calibration report:34-44; §III:78, §III:120; §IV:122-127 | Confirmed, with 0.0278 rounded to 0.028 |
+| 8 | K=3 components: C3 `0.983/2.41/0.321`, C2 `0.956/6.66/0.536`, C1 `0.946/9.17/0.143` | 33 | k3 LOOO report:8-10; convergence report:8-12; §III:70-76; §IV:69-75 | Confirmed after rounding |
+| 9 | K=3 C1 LOOO shape drift: cos <=0.005, dHash <=0.96, weight <=0.023 | 11, 33, 93, 127 | k3 LOOO report:77-79; §III:78, §III:122; §IV:139 | Confirmed |
+| 10 | K=3 held-out hard-posterior differences `1.8-12.8 pp` | 33, 93, 117 | k3 LOOO report:83-90; §III:122; §IV:134-139 | Confirmed after rounding |
+| 11 | Three-score Spearman convergence `rho >= 0.879` | 11, 35, 51, 97, 127 | convergence report:28-30; §III:100-104; §IV:83-87 | Confirmed numerically; wording must not say independent |
+| 12 | Per-signature K=3 consistency `Cohen kappa = 0.87` | 97 | §III:108-116; §IV:104-112 | Confirmed |
+| 13 | Pixel-identity subset `n = 262`, all three checks 0% miss, Wilson upper 1.45% | 11, 35, 53, 101, 127 | pixel-identity report:8, 14-16; §III:124-132; §IV:145-153 | Confirmed |
+| 14 | Firm A pixel-identical `145`, plus `50 partners` and `35 cross-year` | 85 | pixel-identity report:24 confirms 145; §III:44 and §III:190 mark 50/35 as inherited from v3 / Script 28, not regenerated in v4 spikes | Partially confirmed; provenance caveat needed |
+| 15 | Inter-CPA FAR `0.0005`, Wilson `[0.0003, 0.0007]` | 53, 101 | §III:188; §IV:157-159; inherited v3.20.0 §IV-F.1 Table X | Confirmed as inherited |
+| 16 | Full-dataset robustness `n = 686`, full rho `0.9558`, drift `0.007` | 11, 55, 107, 127 | full-dataset report:10-13, 25-31; §III:186; §IV:242-250 | Confirmed numerically, but interpretive scope is light |
+| 17 | Firm A `82%/82.5%` templated and Firm C `23.5%` hand-leaning | 85, 129 | convergence report:43-48; §IV:217-224 | Confirmed as descriptive K=3 hard assignment |
+
+## Cross-Reference Checks (Phase 4 <-> §III v6 / §IV v3.2)
+
+| Linkage | Phase 4 evidence | §III / §IV evidence | Status |
+|---|---:|---:|---|
+| Big-4 primary scope and sample size | Lines 11, 31, 37, 107, 125 | §III:30; §IV:9, §IV:15 | Numerically tight, but scope-test wording overbroad |
+| Accountant-level distributional characterisation refs | Line 29 | §III-I/J are the relevant methodology sections (§III:52-86); §IV-D/E correct (§IV:21-75) | Fail: `§III-D` is stale/wrong |
+| K=2 as firm-mass separator, not operational | Lines 31, 91 | §III:78-86, §III:120; §IV:118-127 | Tight |
+| K=3 descriptive only | Lines 33, 49, 93 | §III:64, §III:80-86, §III:156; §IV:75, §IV:139, §IV:224 | Tight, except "candidate classifier" wording |
+| Three-score internal consistency | Lines 11, 35, 51, 97, 127 | §III:90-106; §IV:79-102 | Numerically tight; independence wording fails |
+| Reverse-anchor reference as non-Big-4 | Lines 35, 97 | §III:48-50; §IV:89 | Tight |
+| Pixel-identity positive anchor | Lines 35, 101 | §III:124-134; §IV:141-155 | Tight; Firm A-only anchoring phrase should be narrowed |
+| Inter-CPA negative-anchor FAR | Lines 53, 101 | §III:126, §III:188; §IV:157-159 | Tight as inherited |
+| Five-way classifier primary / MC band inherited | Lines 33, 113 | §III:136-156; §IV:161-224 | Mostly tight; Abstract/Conclusion should name operational output more clearly |
+| Full-dataset robustness | Lines 55, 107, 127 | §IV:228-252 | Numerically tight; "pipeline reproducibility" overclaims light scope |
+| Internal notes and close-out artifacts | Lines 3, 133-142 | Round-25 review kept this open; §III and §IV also retain internal notes | Not partner/Phase-5 ready |
+
+## Phase 5 Readiness
+
+Partial.
+
+The §III/§IV technical foundation would likely survive cross-AI peer review, but the current Phase 4 prose would draw a Major Revision because it reintroduces known overclaims and has an incomplete §II. With the targeted prose repairs below, Phase 5 readiness should move to Yes.
+
+## Recommended Next-Step Actions
+
+1. Replace every "independent feature-derived scores" phrase with "three feature-derived scores" or "three feature-derived summaries," and preserve the shared-feature caveat in Abstract/§I/§V/§VI.
+
+2. Rewrite the Big-4 scope language at lines 31, 47, 81, 107, and 127 to match §III exactly: Big-4 is the smallest scope among the comparison scopes tested; B/C/D single-firm dip tests were not computed; no full-dataset dip-test result is reported.
+
+3. Fix stale cross-references in line 29: use §III-G/I/J/K as appropriate instead of §III-D.
+
+4. Turn §II into a real revised Related Work section: retain the v3.20.0 subsections in the master, splice in the LOOO paragraph, and replace `[add citation]` with a specific cross-validation citation.
+
+5. Rebuild §V-G limitations by merging the v4-specific limitations with still-valid v3 limitations: transferred ResNet-50 features, HSV stamp-removal artifacts, longitudinal scan/PDF confounds, source-exemplar misattribution, and legal/regulatory interpretation.
+
+6. Replace "published box rule" with "inherited Paper A box rule" unless an external publication citation is added.
+
+7. Narrow full-dataset language: say "K=3 + box-rule rank-convergence reproduces at full `n = 686`" rather than "pipeline reproducibility at multiple scopes."
+
+8. Before Phase 5, strip the Phase 4 draft note and close-out checklist (lines 3 and 133-142), and continue the same cleanup for §III/§IV internal notes flagged in round 25.

paper/codex_review_gpt55_v4_round7.md

@@ -0,0 +1,104 @@
+# Paper A Round 27 Review - v4 round 7
+
+Reviewer: gpt-5.5  
+Date: 2026-05-12  
+Target: `paper/v4/paper_a_prose_v4_phase4.md` (Phase 4 prose v2 + abstract trim)  
+Foundation checked: `paper/v4/paper_a_methodology_v4_section_iii.md` (§III v6) and `paper/v4/paper_a_results_v4_section_iv.md` (§IV v3.2)  
+Prior rubric checked: `paper/codex_review_gpt55_v4_round6.md`
+
+## Verdict
+
+Minor Revision.
+
+Phase 4 prose v2 closes the substantive round-26 overclaim cycle. The major technical-prose risks around independent-score language, Big-4 scope, K=3 operational status, full-dataset overread, and restored limitations are now aligned with §III v6 / §IV v3.2.
+
+The remaining issues are packaging / copy-edit blockers, not empirical blockers: §II still marks [42]-[44] as placeholders and the reference list has not been extended past [41]; internal draft notes and the Phase 4 close-out checklist remain; and §V-F still uses "candidate classifiers" for K=3/reverse-anchor checks.
+
+## Round-26 finding closure table
+
+Note: the round-26 file contains 11 labelled Major rows and 15 labelled Minor rows; this table covers every labelled row. The prompt's 9 Major / 12 Minor tally appears to merge duplicate themes.
+
+### Major findings
+
+| # | Round-26 finding | v2 status | Round-27 note |
+|---:|---|---|---|
+| M1 | Abstract said "Three independent feature-derived scores" | CLOSED | Abstract now says "Three feature-derived scores" and adds "not statistically independent" (line 11). |
+| M2 | §I overclaimed Big-4 scope by implying any single firm and full-dataset dip-test non-rejection | CLOSED | §I now says "narrower comparison scopes tested" and names only Script 32 scopes (line 31). |
+| M3 | §I stale cross-reference to §III-D | CLOSED | Replaced with §III-G through §III-J plus §IV-D/E (line 29). |
+| M4 | §I repeated independent-score error | CLOSED | §I now states the three scores are not statistically independent and frames convergence as internal consistency (line 35). |
+| M5 | §II not submission-ready if inserted as written | PARTIAL | The v4 addition is real prose, but the file still contains a meta note and depends on master-file splicing of `paper/paper_a_related_work_v3.md` (lines 63-65). |
+| M6 | §II unresolved citation placeholder | OPEN | Body cites Stone/Geisser/Vehtari as [42]-[44], but line 65 says these are placeholders; `paper/paper_a_references_v3.md` stops at [41]. |
+| M7 | §V reified CPA mechanism labels | CLOSED | Wording now says per-CPA means are located in descriptor-plane regions, not that all signatures share a mechanism (line 79). |
+| M8 | §V speculative within-CPA unimodality explanation | CLOSED | The causal claim was removed; v2 only states joint consistency and repeats the summary-statistic caveat (line 79). |
+| M9 | §V limitations incomplete vs v3.20.0 | CLOSED | Restored inherited limitations: ImageNet transfer, HSV artifacts, longitudinal confounds, source-exemplar misattribution, legal/regulatory interpretation (lines 119-127). |
+| M10 | §V scope limitation implied full-dataset dip-test evidence | CLOSED | v2 explicitly says full `n = 686` dip-test marginals and LOOO were not tested (line 105). |
+| M11 | §VI overclaimed "cross-scope pipeline reproducibility" | CLOSED | Conclusion now limits the claim to K=3 + box-rule rank-convergence at full `n = 686` and excludes thresholds/LOOO/five-way/pixel checks (line 135). |
+
+### Minor findings
+
+| # | Round-26 finding | v2 status | Round-27 note |
+|---:|---|---|---|
+| m1 | Abstract "candidate classifiers" blurred operational status | CLOSED | Abstract no longer uses "candidate classifiers"; it names the five-way operational output first (line 11). |
+| m2 | Abstract had no word-count margin | CLOSED | `wc -w` on line 11 returns 243 words, leaving 7 words of margin. |
+| m3 | Abstract omitted primary operational output | CLOSED | Abstract now states the inherited five-way per-signature classifier with worst-case document aggregation (line 11). |
+| m4 | Contribution 4 overclaimed "not at narrower scopes" | CLOSED | Now "narrower comparison scopes tested" (line 47). |
+| m5 | Contribution 8 overclaimed full-dataset check | CLOSED | Now says only K=3 + box-rule rank-convergence reproduces and explicitly excludes other components (line 55). |
+| m6 | Safeguards paragraph used "external validation" too broadly | CLOSED | The paragraph now uses "annotation-free validation against naturally-occurring anchor populations" and does not imply full external validation (line 25). |
+| m7 | §II "calibration uncertainty band on operational rule" conflicted with classifier framing | CLOSED | Rewritten as "composition-sensitivity band on the candidate mixture boundary" and not a sufficiency claim for the five-way classifier (line 65). |
+| m8 | §V "inherits and confirms" too strong for signature-level spectrum | CLOSED | Now "inherits this signature-level reading and remains consistent with it," with no-new-diagnostic caveat (line 77). |
+| m9 | Firm A byte-level details needed provenance language | CLOSED | v2 marks 50 partners / 35 cross-year as inherited from v3.20.0 Script 28 and not regenerated in v4 spikes (line 83). |
+| m10 | Firm A alone did not anchor §IV-H | CLOSED | v2 says the Big-4 byte-identical anchor pools all four firms (line 85). |
+| m11 | "Published box rule" not traceable | CLOSED | Replaced with "inherited Paper A box rule" throughout. |
+| m12 | "Same per-CPA ranking" too strong | CLOSED | v2 now says "broadly concordant" and reports the Firm D/Firm C residual disagreement (line 95). |
+| m13 | §V repeated "candidate classifiers" wording | PARTIAL | Line 99 still says "all three candidate classifiers" for the inherited box rule, K=3 hard label, and reverse-anchor metric. Use "candidate checks" or "candidate scores/rules." |
+| m14 | Future-work audit-quality contrast needed descriptive caveat | CLOSED | Future work now says the Firm A/Firm C contrast is descriptive, not mechanism-level, and not linked to audit-quality outcomes (line 137). |
+| m15 | Conclusion underplayed operational output | CLOSED | Conclusion now names the inherited five-way per-signature classifier and worst-case document aggregation (line 133). |
+
+### Round-26 next-step actions
+
+| # | Action | v2 status | Note |
+|---:|---|---|---|
+| A1 | Replace independent-score language and preserve shared-feature caveat | CLOSED | Done in Abstract, §I, §V, §VI. |
+| A2 | Rewrite Big-4 scope language | CLOSED | Done; no unsupported B/C/D single-firm or full-dataset dip-test claim remains in body prose. |
+| A3 | Fix stale §III-D cross-reference | CLOSED | Done at line 29. |
+| A4 | Turn §II into real revised Related Work and replace `[add citation]` | PARTIAL | The LOOO paragraph is drafted, but references [42]-[44] remain placeholders and absent from the reference list. |
+| A5 | Rebuild §V-G limitations with still-valid v3 limitations | CLOSED | Done at lines 119-127. |
+| A6 | Replace "published box rule" | CLOSED | Done. |
+| A7 | Narrow full-dataset language | CLOSED | Done at lines 55, 105, and 135. |
+| A8 | Strip internal notes/checklists before Phase 5 | OPEN | Draft note and close-out checklist remain (lines 3, 141-150); §III/§IV also retain internal notes/checklists. |
+
+## Newly introduced issues
+
+1. **Minor - §II citation-number gap and placeholder contradiction.** The v2 draft note says §II now has "a real citation," but line 65 says [42]-[44] are placeholders, line 147 still says `[add citation]`, and `paper/paper_a_references_v3.md` stops at [41]. This is the only remaining reviewer-visible blocker if the prose is packaged as manuscript text.
+
+2. **Minor - stale close-out metadata.** The close-out checklist says the abstract is "approximately 235 words" (line 145), but `wc -w` returns 243 words on the abstract paragraph. The author's "244 words" note and the shell count differ by one tokenization unit; both satisfy IEEE Access, but the checklist should be updated or removed.
+
+No newly introduced empirical inconsistency was found.
+
+## Abstract word count verification + key v2 spot checks
+
+Abstract count: `sed -n '11p' paper/v4/paper_a_prose_v4_phase4.md | wc -w` returns **243**. The abstract is one paragraph and under the 250-word IEEE Access target.
+
+Spot-check 1: **Independent-score correction closed.** Lines 11, 35, 95, and 135 now say the scores are feature-derived / shared-input / not statistically independent. This matches §III-K's caveat and §IV-F's framing that the correlations are internal consistency, not external validation.
+
+Spot-check 2: **Big-4 scope and full-dataset correction closed.** Lines 31, 47, 79, 105, and 135 now match §III-G/I and §IV-D/K: Big-4 is the smallest scope among tested comparison scopes; B/C/D single-firm dip tests and full-dataset dip tests were not run; full-dataset evidence is only the light K=3 + box-rule Spearman re-run at `n = 686`.
+
+Spot-check 3: **Operational-vs-descriptive framing closed except line 99 wording.** Lines 11, 33, 55, 111, 133, and 135 reserve operational status for the inherited five-way classifier and keep K=3 descriptive. The only remaining wording leak is line 99's "candidate classifiers."
+
+## Phase 5 readiness
+
+Partial.
+
+Substantively, §III + §IV + Phase 4 prose are converged. Phase 5 should not require new statistical work. It does require one copy-edit/reference pass before packaging: finalize §II citations and references, strip internal notes/checklists, and replace the residual "candidate classifiers" phrase.
+
+## Recommended next-step actions
+
+1. Replace line 99's "all three candidate classifiers" with "all three candidate checks" or "all three candidate scores/rules"; keep K=3 explicitly descriptive.
+
+2. Finalize §II packaging: either splice the full v3.20.0 Related Work body plus the v4 LOOO paragraph into the master, or make this Phase 4 file contain the full §II block. Add real [42]-[44] reference entries and remove the "placeholders" sentence.
+
+3. Strip the Phase 4 draft note and close-out checklist before manuscript assembly; do the same for §III/§IV internal notes and working checklists.
+
+4. Update or remove the stale abstract-count note. The verified shell count is 243 words.
+
+5. After the reference/cross-reference cleanup, run one final manuscript-level lint for unresolved placeholders, duplicate reference numbers, internal notes, and stale section/table references.

paper/codex_review_gpt55_v4_round8.md

@@ -0,0 +1,138 @@
+# Paper A Round 28 Review — codex GPT-5.5 v4 round 8
+
+Reviewer: gpt-5.5
+Date: 2026-05-14
+Target: paper/v4/paper_a_prose_v4_phase4.md + paper/v4/paper_a_methodology_v4_section_iii.md + paper/v4/paper_a_results_v4_section_iv.md (post round-2 fixes, commit b884d39)
+Prior reviewer artifacts: paper/codex_review_gpt55_v4_round7.md; paper/gemini_review_v4_round1.md; paper/opus_review_v4_round1.md
+Round-2 commit reviewed: b884d39
+
+## Verdict
+
+Minor Revision.
+
+Round-2 closes the empirical substance of Opus M2-M4 and the core of M1/M3: the deployed any-pair vs same-pair within-firm collision semantics are now separated in the body, the §IV K=3 mechanism-label regression is mostly repaired, §V headings now run A-H, and the Table XV-B cascade has been applied in the public §IV table sequence.
+
+However, the round-2 pass introduced or exposed several splice blockers: the abstract is now 261 words against the stated <=250 target; the new §IV-J Table XV sample-size footnote incorrectly says §IV-M.5 uses n=150,442 even though Script 44 / Tables XXIV-XXV use the 150,453 vector-complete substrate; §IV-I still points the ICCR calibration reader to "§IV-M Table XVI" although the relevant tables are now XXI-XXVI; and internal draft notes/checklists still contain stale Table XV-B / >=97% / hand-leaning language. No new statistical work is required, but the manuscript is not ready for splice until those are patched.
+
+## Round-1 panel finding closure table
+
+| Source | Finding | Current status | Evidence / note |
+|---|---|---|---|
+| codex r7 M1 | Abstract "Three independent feature-derived scores" | CLOSED | Current prose uses "Three feature-derived scores" and the shared-input caveat at paper/v4/paper_a_prose_v4_phase4.md:37 and :97; §III states the scores are not statistically independent at paper/v4/paper_a_methodology_v4_section_iii.md:113. |
+| codex r7 M2 | §I overclaimed Big-4 scope | CLOSED | Big-4 primary scope is explicit at §III-G, paper/v4/paper_a_methodology_v4_section_iii.md:19 and Phase 4 paper/v4/paper_a_prose_v4_phase4.md:39; corrected M3 language does not invalidate this closure. |
+| codex r7 M3 | Stale §III-D cross-reference | CLOSED | Pipeline / validation references now point to §III-L / §III-M / §III-I.4, e.g. paper/v4/paper_a_prose_v4_phase4.md:29. |
+| codex r7 M4 | §I repeated independent-score error | CLOSED | Phase 4 describes internal consistency, not independent validation, at paper/v4/paper_a_prose_v4_phase4.md:37 and :97. |
+| codex r7 M5 | §II not submission-ready as standalone | PARTIAL | §II still contains a review-pass note saying only the v4 paragraph is reproduced, paper/v4/paper_a_prose_v4_phase4.md:65. This is a splice-packaging issue. |
+| codex r7 M6 | Refs [42]-[44] absent / placeholders | SUPERSEDED | My round-7 claim was wrong against the current reference file: [42]-[44] are present at paper/paper_a_references_v3.md:87-91, and §II cites them at paper/v4/paper_a_prose_v4_phase4.md:67. |
+| codex r7 M7 | §V reified CPA mechanism labels | CLOSED | §V now uses descriptor-position language and explicitly rejects latent mechanism classes at paper/v4/paper_a_prose_v4_phase4.md:81 and :93. |
+| codex r7 M8 | §V speculative within-CPA unimodality explanation | CLOSED | §V-B restricts the result to composition + integer artefacts, paper/v4/paper_a_prose_v4_phase4.md:81. |
+| codex r7 M9 | §V limitations incomplete | CLOSED | §V-H lists nine v4 limitations plus five inherited v3.20.0 limitations at paper/v4/paper_a_prose_v4_phase4.md:111-139. |
+| codex r7 M10 | §V full-dataset scope overread | CLOSED | Scope limitation is explicit at paper/v4/paper_a_prose_v4_phase4.md:117 and §IV-K is narrow at paper/v4/paper_a_results_v4_section_iv.md:232-254. |
+| codex r7 M11 | §VI overclaimed cross-scope pipeline reproducibility | CLOSED | §VI now limits cross-scope support to K=3/Spearman robustness and leaves full ICCR generalisation to future work, paper/v4/paper_a_prose_v4_phase4.md:147-149. Opus M3 does not reopen this. |
+| codex r7 m1 | "candidate classifiers" wording | CLOSED | Current §V-G uses "candidate checks", paper/v4/paper_a_prose_v4_phase4.md:107. |
+| codex r7 m2 | Abstract word-count margin | OPEN | Round-2 M3 rewrite pushed the abstract to 261 words by `wc -w`, while the draft target at paper/v4/paper_a_prose_v4_phase4.md:9 is <=250. |
+| codex r7 m3 | Abstract omitted operational output | CLOSED | Abstract includes ICCR units and operational HC+MC per-document alarm, paper/v4/paper_a_prose_v4_phase4.md:11. |
+| codex r7 m4 | Contribution 4 overclaimed narrower scopes | CLOSED | Contribution 4 now says the threshold path is unsupported by composition decomposition, paper/v4/paper_a_prose_v4_phase4.md:49. |
+| codex r7 m5 | Contribution 8 overclaimed full-dataset check | CLOSED | Current contribution 8 is limited to annotation-free positive-anchor and unsupervised validation ceiling, paper/v4/paper_a_prose_v4_phase4.md:57. |
+| codex r7 m6 | "External validation" too broad | CLOSED | Current language is specificity-proxy / annotation-free / unsupervised-ceiling, e.g. paper/v4/paper_a_prose_v4_phase4.md:57 and :113. |
+| codex r7 m7 | §II LOOO sufficiency wording | CLOSED | §II frames LOOO as composition-sensitivity, not operational-classifier sufficiency, paper/v4/paper_a_prose_v4_phase4.md:67. |
+| codex r7 m8 | "Inherits and confirms" too strong | CLOSED | §V-B says v4 strengthens/extends by decomposition but does not overclaim direct validation, paper/v4/paper_a_prose_v4_phase4.md:79-81. |
+| codex r7 m9 | Firm A byte-level provenance | CLOSED | Inherited Script 28 provenance is stated at paper/v4/paper_a_prose_v4_phase4.md:85 and §III-H at paper/v4/paper_a_methodology_v4_section_iii.md:37. |
+| codex r7 m10 | Firm A alone did not anchor §IV-H | CLOSED | Pixel-identity anchor is Big-4 n=262 with all four firms listed, paper/v4/paper_a_results_v4_section_iv.md:143-153. |
+| codex r7 m11 | "Published box rule" not traceable | CLOSED | Current text uses inherited Paper A / v3.x box rule language, e.g. paper/v4/paper_a_results_v4_section_iv.md:215. |
+| codex r7 m12 | "Same per-CPA ranking" too strong | CLOSED | Residual Firm D/Firm C disagreement is disclosed at paper/v4/paper_a_prose_v4_phase4.md:97 and §IV-F at paper/v4/paper_a_results_v4_section_iv.md:102. |
+| codex r7 m13 | §V "candidate classifiers" residue | CLOSED | Replaced with "candidate checks" at paper/v4/paper_a_prose_v4_phase4.md:107. |
+| codex r7 m14 | Future-work audit-quality contrast needed caveat | CLOSED | Future work now keeps the Firm A vs B/C/D contrast descriptive, paper/v4/paper_a_prose_v4_phase4.md:149. |
+| codex r7 m15 | Conclusion underplayed operational output | CLOSED | §VI opens with five-way classifier and worst-case aggregation, paper/v4/paper_a_prose_v4_phase4.md:145. |
+| codex r7 new issue 1 | §II citation gap | SUPERSEDED | Refs [42]-[44] exist at paper/paper_a_references_v3.md:87-91. |
+| codex r7 new issue 2 | Stale close-out metadata | OPEN | Phase 4 close-out still says 243-244 words and "§V-G Limitations", paper/v4/paper_a_prose_v4_phase4.md:157-160; current abstract count is 261 and limitations are §V-H. |
+| Gemini M1 | Reject "statistically insignificant" firm heterogeneity framing | CLOSED | Current text says firm effects are large and not pool-size explained at paper/v4/paper_a_methodology_v4_section_iii.md:259-268 and paper/v4/paper_a_prose_v4_phase4.md:35. |
+| Gemini M2 | codex refs [42]-[44] error | CLOSED | References are present at paper/paper_a_references_v3.md:87-91. |
+| Gemini M3 | ICCR disclaimer adequacy | CLOSED | FAR is explicitly reframed as ICCR / specificity proxy at paper/v4/paper_a_methodology_v4_section_iii.md:185 and paper/v4/paper_a_results_v4_section_iv.md:159. |
+| Gemini M4 | K=3 demotion language | CLOSED for main body | §III-J and §V-D are correct at paper/v4/paper_a_methodology_v4_section_iii.md:76-90 and paper/v4/paper_a_prose_v4_phase4.md:89-93. Residual public wording "replicated vs not-replicated" in Table XI is flagged below as minor terminology residue. |
+| Gemini M5 | Feature-derived score caveat | CLOSED | Shared-input caveat appears in §III-K, §IV-F, and §V-E: paper/v4/paper_a_methodology_v4_section_iii.md:113; paper/v4/paper_a_results_v4_section_iv.md:79; paper/v4/paper_a_prose_v4_phase4.md:97. |
+| Gemini m1 | Internal draft notes/checklists | OPEN | Present in §III, §IV, and Phase 4: paper/v4/paper_a_methodology_v4_section_iii.md:3 and :431-447; paper/v4/paper_a_results_v4_section_iv.md:3 and :365-374; paper/v4/paper_a_prose_v4_phase4.md:3 and :153-162. |
+| Gemini m2 | Table XV-B vs XIX numbering | CLOSED in public body | Public document-level table is now Table XIX at paper/v4/paper_a_results_v4_section_iv.md:192 and §IV-M cascades through XX-XXVI at :266, :280, :300, :317, :329, :340, :353. Internal notes still stale. |
+| Gemini m3 | Word count note | OPEN | The note remains stale at paper/v4/paper_a_prose_v4_phase4.md:157 and the current abstract is over target. |
+| Gemini new issue | Table XV sample-size nuance | PARTIAL | A pointer to §III-G was added at paper/v4/paper_a_results_v4_section_iv.md:177, but the footnote misclassifies §IV-M.5 as n=150,442; Script 44 / Tables XXIV-XXV use the 150,453 vector-complete substrate per §III-G, paper/v4/paper_a_methodology_v4_section_iii.md:31. |
+| Opus M1 | §IV K=3 mechanism-label reversion | CLOSED for named tables/prose; MINOR RESIDUE | Tables IX/X/XIV/XVI/XVII now use descriptor-position or less-replication-dominated language at paper/v4/paper_a_results_v4_section_iv.md:81-100, :145-153, :217-226, :234-254. Public residue: Table XI still says "binary collapse, replicated vs not-replicated" at :104; internal §III open question still says hand-leaning at paper/v4/paper_a_methodology_v4_section_iii.md:445. |
+| Opus M2 | Table XV-B cascade | CLOSED in public body | Table XIX replaces XV-B at paper/v4/paper_a_results_v4_section_iv.md:192 and §IV-M is XX-XXVI at :266-353. No public Table XV-B reference remains; only internal notes at :3 and :370. |
+| Opus M3 | "98-100% within source firm" semantic conflation | CLOSED in body; ABSTRACT SHORT FORM | Body locations now separate deployed any-pair 98.8% / 76.7-83.7% from same-pair 97.0-99.96% at paper/v4/paper_a_prose_v4_phase4.md:35, :53, :87, :115, :147, :149 and §III at paper/v4/paper_a_methodology_v4_section_iii.md:99, :283, :285. The abstract uses a rounded any-pair-only 77-99% headline at paper/v4/paper_a_prose_v4_phase4.md:11, which is not misleading but omits the same-pair subrange. |
+| Opus M4 | Duplicate §V-G heading | CLOSED | §V headings now run A-H: paper/v4/paper_a_prose_v4_phase4.md:73, :77, :83, :89, :95, :99, :105, :109. |
+| Opus M5 | Stale "seven limitations" close-out note | OPEN internal | Phase 4 checklist still says "seven limitations" and "§V-G Limitations" at paper/v4/paper_a_prose_v4_phase4.md:160; the actual limitations heading is §V-H and has 14 items. |
+| Opus M6 | §IV-M composition table partial vs §III factorial | PARTIAL / LOW | §IV-M.1 remains a summary table at paper/v4/paper_a_results_v4_section_iv.md:266-276, while full factorial detail is in §III-I.4 at paper/v4/paper_a_methodology_v4_section_iii.md:61-68. This is acceptable if §IV-D points readers to a summary, but current §IV-D says diagnostics are "tabulated in §IV-M" at paper/v4/paper_a_results_v4_section_iv.md:23. |
+| Opus M7 | Mixed Spearman precision | OPEN / COPY-EDIT | §III reports 0.963/0.889/0.879 at paper/v4/paper_a_methodology_v4_section_iii.md:123-127, while §IV uses 0.9627/0.8890/0.8794 at paper/v4/paper_a_results_v4_section_iv.md:81-87. |
+| Opus minor 1 | Abstract word-count metadata | OPEN | Current abstract is 261 words; close-out note still says 243-244 at paper/v4/paper_a_prose_v4_phase4.md:157. |
+| Opus minor 2 | Internal draft notes | OPEN | Same as Gemini m1. |
+| Opus minor 3 | §IV-J Table XV-B pointer | CLOSED in body | Body now says Table XIX at paper/v4/paper_a_results_v4_section_iv.md:228. |
+| Opus minor 4 | Mixed decimal / percentage notation | OPEN / COPY-EDIT | Still mixed by design, e.g. 0.34 at paper/v4/paper_a_prose_v4_phase4.md:11 and 33.75% at :33. |
+| Opus minor 5 | v3.x §IV-F.1 cross-reference check | OPEN / SPLICE | v3.x §IV-F.1 references remain at paper/v4/paper_a_methodology_v4_section_iii.md:37 and paper/v4/paper_a_prose_v4_phase4.md:85. Verify during master splice. |
+| Opus minor 6 | Firm A 50/180 inherited provenance | CLOSED / DISCLOSED | Provenance is disclosed as inherited, not regenerated, at paper/v4/paper_a_methodology_v4_section_iii.md:37. |
+| Opus minor 7 | "FAR throughout" historical exception | PARTIAL | v4 framing disclaims FAR, but historical "FAR" appears in paper/v4/paper_a_results_v4_section_iv.md:159 and paper/v4/paper_a_methodology_v4_section_iii.md:185. It is correctly caveated, not an empirical issue. |
+| Opus minor 8 | MC band proportions | CLOSED | §IV-J proportions match Table XV rows at paper/v4/paper_a_results_v4_section_iv.md:181-186 and prose at :215. |
+| Opus minor 9 | §V-G item count | OPEN internal | Same as Opus M5. |
+| Opus minor 10 | LOOO range 1.8-12.8 pp | CLOSED | §IV Table XIII supports 1.76-12.77 pp, paper/v4/paper_a_results_v4_section_iv.md:131-139. |
+| Opus minor 11 | Abstract 98-100 public statement | SUPERSEDED | Replaced by rounded any-pair 77-99% at paper/v4/paper_a_prose_v4_phase4.md:11. |
+| Opus new issue 1 | Human-in-the-loop not operationalised | OPEN / COPY-EDIT | The positioning remains in Abstract and §III-M, paper/v4/paper_a_prose_v4_phase4.md:11 and paper/v4/paper_a_methodology_v4_section_iii.md:334, but no concrete review workflow is specified. |
+| Opus new issue 2 | Feature-derived caveat breaks down in §IV | CLOSED for main §IV | §IV tables and prose were repaired, except the Table XI binary-collapse label noted above. |
+| Opus new issue 3 | §III-M nine-tool table unnumbered | OPEN / COPY-EDIT | The validation table at paper/v4/paper_a_methodology_v4_section_iii.md:318-329 remains unnumbered. |
+| Opus new issue 4 | §I pipeline-step vs framework-element framing | OPEN / COPY-EDIT | The eight-item enumeration remains at paper/v4/paper_a_prose_v4_phase4.md:29. |
+
+## Round-2 induced issues
+
+1. **Abstract now exceeds the target word count.** `sed -n '11p' paper/v4/paper_a_prose_v4_phase4.md | wc -w` returns 261. The draft note at paper/v4/paper_a_prose_v4_phase4.md:9 sets an IEEE Access <=250 target, and the close-out note at :157 is stale.
+
+2. **The new Table XV sample-size footnote is partly wrong.** The pointer to §III-G is useful, but paper/v4/paper_a_results_v4_section_iv.md:177 says §IV-M.5 uses n=150,442. §III-G says Scripts 40b, 43, and 44 use the 150,453 vector-complete substrate, paper/v4/paper_a_methodology_v4_section_iii.md:31, and Script 44's report states n_big4_sources = 150,453. Correct the footnote to distinguish descriptor-complete sections from vector-complete §IV-M.2 / §IV-M.3 / §IV-M.5.
+
+3. **Public table cross-reference is stale.** §IV-I still says the consolidated v4-new ICCR calibration appears in "§IV-M Table XVI", paper/v4/paper_a_results_v4_section_iv.md:161. Current Table XVI is the K=3 firm cross-tab at :217; the ICCR calibration tables are XXI-XXVI at :280, :300, :317, :329, :340, :353.
+
+4. **Internal notes remain stale.** §IV's draft note still says v3.2 and Table XV-B, paper/v4/paper_a_results_v4_section_iv.md:3; the §IV close-out checklist repeats Table XV-B at :370. §III's internal cross-reference index still says within-firm collision concentration >=97% at paper/v4/paper_a_methodology_v4_section_iii.md:438 and "C1 hand-leaning" at :445. These are internal-only splice items, not empirical blockers, but they must be stripped or updated.
+
+5. **Minor terminology residue remains outside Opus's named M1 sites.** §III and §IV Table XI still call a K=3 / box-rule binary collapse "replicated vs not-replicated", paper/v4/paper_a_methodology_v4_section_iii.md:131 and paper/v4/paper_a_results_v4_section_iv.md:104. Because this is not the byte-identical positive-anchor ground-truth subset, a stricter v4 wording would be "high-cos / low-dHash vs other positions" or "replication-dominated vs less-replication-dominated."
+
+6. **"Less-replication-dominated" is long but not broken.** The phrase is readable in the public replacement sites. The only sentence I would smooth at copy-edit is paper/v4/paper_a_results_v4_section_iv.md:215 ("per-CPA less-replication-dominated ranking"), which could become "per-CPA ranking away from the replication-dominated corner."
+
+## Provenance spot-checks
+
+1. **Within-firm any-pair rates from §IV Table XXV.** From paper/v4/paper_a_results_v4_section_iv.md:340-349:
+   - Firm A: 14,447 / 14,622 = 98.8032% -> 98.8%.
+   - Firm B: 371 / 484 = 76.6529% -> 76.7%.
+   - Firm C: 149 / 178 = 83.7079% -> 83.7%.
+   - Firm D: 106 / 137 = 77.3723% -> 77.4%.
+   These match the corrected 76.7-98.8% any-pair range and the B/C/D 76.7-83.7% summary. Script 44's report gives the same matrix at /Volumes/NV2/PDF-Processing/signature-analysis/reports/v4_big4/firm_matched_pool/firm_matched_pool_report.md:36-43.
+
+2. **Same-pair joint range 97.0-99.96%.** §III-L.4 reports 11,314/11,319, 85/87, 54/55, and 64/66 at paper/v4/paper_a_methodology_v4_section_iii.md:281. The arithmetic is 99.9558%, 97.7011%, 98.1818%, and 96.9697%, matching the rounded 99.96% / 97.7% / 98.2% / 97.0%. §IV repeats the rates at paper/v4/paper_a_results_v4_section_iv.md:349.
+
+3. **Pooled Big-4 any-pair per-signature ICCR 0.1102.** Script 43's report gives 16,578 / 150,453 = 0.1102 at /Volumes/NV2/PDF-Processing/signature-analysis/reports/v4_big4/pool_normalized_far/pool_normalized_report.md:42-50. A normal approximation half-width is 1.96 * sqrt(0.1102 * 0.8898 / 150453) = 0.00158, consistent with the reported Wilson [0.1086, 0.1118] in §IV-M.3, paper/v4/paper_a_results_v4_section_iv.md:300-305.
+
+4. **Per-pair conditional ICCR 0.234.** Script 40b's report gives dHash <= 5 conditional on cos > 0.95 as 70 / 299 = 0.23411 at /Volumes/NV2/PDF-Processing/signature-analysis/reports/v4_big4/inter_cpa_far_sweep/far_sweep_report.md:87-99. This matches §III-L.1 at paper/v4/paper_a_methodology_v4_section_iii.md:208 and §IV-M.2 at paper/v4/paper_a_results_v4_section_iv.md:294.
+
+## Updated round-7 closure reassessment
+
+Opus M3 does not invalidate my round-7 closure of M2 (Big-4 scope language) or M11 (cross-scope pipeline reproducibility). Those findings were about overextending Big-4/full-dataset scope and overclaiming cross-scope reproducibility. The corrected current prose keeps the primary scope Big-4, treats full-dataset evidence as a narrow K=3 + Spearman robustness check, and now reports the within-firm collision pattern as any-pair 76.7-98.8% plus same-pair 97.0-99.96%.
+
+If I re-graded round 7 against the corrected current drafts, none of my major closures would move back to PARTIAL on empirical grounds. I would, however, reopen the abstract word-count minor item because the M3 repair pushed the abstract over the <=250-word target.
+
+## Phase 5 readiness
+
+Partial.
+
+The empirical core is ready: no script rerun or statistical redesign is needed. M2-M4 are closed in manuscript body text, and M3 is substantively corrected. Phase 5 is blocked only by splice/copy-edit/factual-reference hygiene:
+
+1. Trim the abstract from 261 to <=250 words.
+2. Correct §IV-J line 177's sample-size footnote so §IV-M.2 / M.3 / M.5 are identified as vector-complete / pair-recomputed analyses, not n=150,442 descriptor-complete analyses.
+3. Fix §IV-I's stale "§IV-M Table XVI" pointer.
+4. Strip or update internal draft notes, checklists, §III cross-reference index, and stale Table XV-B / >=97% / hand-leaning language.
+5. Optionally smooth the residual "replicated vs not-replicated" binary-collapse label and the long "less-replication-dominated" phrase in §IV-J.
+
+## Recommended next-step actions
+
+1. **Copy-edit blocker:** trim the abstract by at least 11 words while preserving the corrected any-pair headline. Do not add same-pair detail to the abstract unless other text is removed.
+
+2. **Factual cross-reference blocker:** replace the §IV-J Table XV footnote with a precise version: descriptor-complete analyses use 150,442; vector/pair-recomputed analyses use 150,453, including Scripts 40b, 43, and 44 (§IV-M.2, M.3, M.5).
+
+3. **Cross-reference blocker:** change §IV-I's "§IV-M Table XVI" to "§IV-M.2 Table XXI" or to "§IV-M Tables XXI-XXVI", depending on whether the intended pointer is per-comparison ICCR only or the whole calibration block.
+
+4. **Splice blocker:** remove all internal notes/checklists before manuscript assembly, especially the stale §IV v3.2 / Table XV-B note, §III's >=97% cross-reference-index shorthand, and the Phase 4 "§V-G Limitations / seven limitations" checklist item.
+
+5. **Terminology cleanup:** consider renaming "binary collapse, replicated vs not-replicated" to descriptor-position language in §III-K and §IV-F, while retaining "replicated class" only for the byte-identical positive-anchor ground truth.

paper/codex_review_gpt55_v4_round9.md

@@ -0,0 +1,96 @@
+# Paper A Round 29 Review — codex GPT-5.5 v4 round 9 (final cross-check)
+
+Reviewer: gpt-5.5
+Date: 2026-05-14
+Target: paper/v4/paper_a_prose_v4_phase4.md + paper/v4/paper_a_methodology_v4_section_iii.md + paper/v4/paper_a_results_v4_section_iv.md (post round-4)
+Prior reviewer artifacts: paper/codex_review_gpt55_v4_round{7,8}.md; paper/gemini_review_v4_round{1,2}.md; paper/opus_review_v4_round{1,2}.md
+Round-4 commit reviewed: d3ddf746f4555a68072ec2dacf5a455d6334033d
+
+## Verdict
+
+Minor Revision.
+
+Round-4 closes the intended structural shape of Opus N1-N4, but two provenance-sensitive wordings must be corrected before manuscript-splice assembly:
+
+1. The §IV-M.4 denominator reconciliation is arithmetically right but describes the 379 mixed-firm PDFs as Firm C "majority firm" cases. Direct verification against the database and Script 45 shows they are 1:1 Firm C/Firm D ties assigned to Firm C by the mode/tie-break implementation.
+2. The new §III-M Table XXVII composition-decomposition row is structurally right but its untested-assumption column over-attributes the within-firm corroboration to Script 39c. Script 39c's emitted raw dHash per-firm tests reject unimodality; the jittered non-Big-4 per-firm support is not emitted in the current Script 39c/39d reports.
+
+Phase 5 convergence by panel vote is still achieved: Gemini round-2 = Accept, Opus round-2 = Minor Revision, codex round-9 = Minor Revision. That is 3/3 reviewers in the Accept/Minor band. I would not splice the current text verbatim, but the remaining changes are small text/provenance patches, not new empirical work.
+
+## N1–N4 closure verification
+
+**N1. Firm C denominator reconciliation — partially closed, but not clean.**
+
+The reconciliation landed at paper/v4/paper_a_results_v4_section_iv.md:325. The arithmetic is correct: §IV-J Table XIX reports single-firm document rows with Firm C $n = 19{,}122$ and excludes 379 mixed-firm PDFs at paper/v4/paper_a_results_v4_section_iv.md:192-213; §IV-M.4 reports mode-assigned per-firm D2 denominators summing to $75{,}233$, with Firm C $n = 19{,}501$ at paper/v4/paper_a_results_v4_section_iv.md:317-325. Script 45's implementation maps each document to a firm mode via `np.unique(..., return_counts=True)` and `np.argmax(counts)` at signature_analysis/45_doc_level_far_full_5way.py:249-256, and the Script 45 report gives Firm C $n = 19{,}501$ in the per-firm doc-level table.
+
+The problem is the explanatory phrase "majority firm." A direct SQLite check of the Script 45 substrate returns exactly one mixed-document pattern: `Firm C:1+Firm D:1 | 379`; majority docs = 0 and tie docs = 379. Thus all 379 mixed-firm PDFs resolve to Firm C because of the sorted mode tie-break, not because Firm C is the empirical majority within those PDFs. Replace the current sentence with tie-break language, e.g. "The 379 mixed-firm PDFs are all 1:1 Firm C/Firm D mixed-firm documents; Script 45's mode-of-firms implementation assigns tied modes to the first sorted firm, so they are assigned to Firm C."
+
+**N2. Composition-decomposition added to §III-M ten-tool table — structurally closed, provenance wording needs correction.**
+
+The table now has ten rows, with composition decomposition inserted first at paper/v4/paper_a_methodology_v4_section_iii.md:318-331. §I contribution 8 now says "ten partial-evidence diagnostics (§III-M Table XXVII)" at paper/v4/paper_a_prose_v4_phase4.md:57, and §VI item 8 now says "ten-tool unsupervised-validation collection (§III-M Table XXVII)" at paper/v4/paper_a_prose_v4_phase4.md:147. This closes the count/cross-reference part of N2.
+
+The new composition row's assumption cell at paper/v4/paper_a_methodology_v4_section_iii.md:322 is not accurate as written. It says within-firm dip tests on every firm with $n \geq 500$ in Script 39c corroborate absence of within-population bimodality. Script 39c does run the eligible non-Big-4 per-firm tests at signature_analysis/39c_v4_midsmall_signature_diptest.py:147-158, but its emitted report shows raw dHash rejects in all ten eligible mid/small firms, while cosine fails to reject. The accurate decomposition is what §III-I.4 states more carefully: raw dHash rejects in all 14 firms, Big-4 per-firm dHash rejection disappears after jitter in Script 39d, and Big-4 pooled dHash needs both firm-mean centring and jitter in Script 39e (paper/v4/paper_a_methodology_v4_section_iii.md:57-68; paper/v4/paper_a_results_v4_section_iv.md:270-276).
+
+The non-Big-4 jittered per-firm claim is also not cleanly provenance-emitted: paper/v4/paper_a_methodology_v4_section_iii.md:382 cites Script 39d / 39c for a non-Big-4 jittered-dHash range, but the current Script 39d report emits Big-4 per-firm plus pooled non-Big-4, not the ten individual mid/small-firm jittered rows. My read-only rerun of the same jitter procedure did confirm 0/10 non-Big-4 firms reject after jitter, but it produced a median-$p$ range of $0.3755$-$1.0$, not the manuscript's $[0.71, 1.00]$. Either add the emitted table to Script 39c/39d, or narrow the Table XXVII assumption cell to the scripted evidence already visible in §IV-M.1.
+
+**N3. §III-M table numbering — closed.**
+
+The §III-M table is now explicitly introduced as Table XXVII at paper/v4/paper_a_methodology_v4_section_iii.md:316-318. The caption, "Ten-tool unsupervised-validation collection with disclosed untested assumptions," matches the table content: ten diagnostic rows, each with a measure and an untested-assumption column. The numbering also follows §IV-M.6 Table XXVI at paper/v4/paper_a_results_v4_section_iv.md:353.
+
+**N4. Cross-firm hit matrix assumption disclosure — closed, contingent on the N1 footnote fix.**
+
+The old "None — direct descriptive observation" assumption is gone. The current Table XXVII row at paper/v4/paper_a_methodology_v4_section_iii.md:327 discloses both deployed-rule dependence and same-pair vs any-pair semantics: same-pair joint event $97.0$-$99.96\%$ within-firm versus any-pair $76.7$-$98.8\%$. Those values match §IV-M.5 Table XXV and the following same-pair sentence at paper/v4/paper_a_results_v4_section_iv.md:340-349, and Script 44 computes the matrices at signature_analysis/44_firm_matched_pool_regression.py:274-327.
+
+The row's reference to Script 45 mode-of-firms assignment is appropriate, but it points to the §IV-M.4 footnote. Once N1's "majority firm" wording is corrected to "tie-break assignment," N4 reads cleanly.
+
+## Round-4 induced issues
+
+1. **N1 footnote overcorrects from "undisclosed denominator" to a false "majority firm" explanation.** The current prose at paper/v4/paper_a_results_v4_section_iv.md:325 should not say the 379 mixed-firm PDFs resolve to Firm C as majority firm. They are all Firm C/Firm D 1:1 ties.
+
+2. **The new Table XXVII composition row makes an existing provenance weakness load-bearing.** The row at paper/v4/paper_a_methodology_v4_section_iii.md:322 should not cite Script 39c as though Script 39c alone corroborates absence of within-population bimodality. Script 39c raw dHash rejects in all ten eligible mid/small firms; the no-rejection claim requires integer jitter. The current committed reports do not emit the ten non-Big-4 jittered per-firm values.
+
+3. **Ten-tool propagation is otherwise clean in public prose.** The public §I and §VI claims now say ten-tool / ten partial-evidence diagnostics at paper/v4/paper_a_prose_v4_phase4.md:57 and :147. I found no public leftover "nine-tool" validation claim except internal working material marked for removal: the Phase 4 draft note at paper/v4/paper_a_prose_v4_phase4.md:3 and the §III cross-reference checklist at paper/v4/paper_a_methodology_v4_section_iii.md:434-442. The separate "first nine limitations" statement at paper/v4/paper_a_prose_v4_phase4.md:111 is a limitations count, not a validation-tool count.
+
+4. **No new FAR/ICCR regression found.** The manuscript continues to avoid treating inter-CPA ICCR as true FAR in the public prose checked here. The remaining issues are denominator/tie-break wording and composition-diagnostic provenance.
+
+## Phase 5 round-3 convergence audit
+
+| Reviewer artifact | Verdict | Post-round-4 interpretation |
+|---|---|---|
+| Gemini round-2 | Accept | Accept remains within the convergence band; Gemini did not have round-4 in view. |
+| Opus round-2 | Minor Revision | N1-N4 were the requested round-4 targets. N3/N4 are closed; N1/N2 need wording/provenance cleanup. |
+| codex GPT-5.5 round-9 | Minor Revision | Current text is close, but not splice-ready verbatim because two new/retained provenance wordings are inaccurate. |
+
+Panel convergence on Accept/Minor consensus is **yes: 3 of 3 reviewers** are in the Accept/Minor band. The Phase 5 gate is therefore met by vote-count logic, but I recommend closing it only after the two "must do now" text patches below are applied and committed. No new empirical analysis or new full review round is required.
+
+## Splice readiness checklist
+
+**Must do now before splice assembly**
+
+1. Patch paper/v4/paper_a_results_v4_section_iv.md:325: replace "mode-of-firms (majority firm)" / "resolve to Firm C as the majority firm" with the actual 1:1 Firm C/Firm D tie-break explanation.
+
+2. Patch paper/v4/paper_a_methodology_v4_section_iii.md:322: revise the composition-decomposition row's untested-assumption cell so it does not imply Script 39c raw within-firm tests support the dHash no-bimodality claim. Either cite only the emitted Big-4 jittered evidence (Script 39d) plus Big-4 centred+jittered evidence (Script 39e), or emit/cite a proper ten-firm non-Big-4 jittered table.
+
+3. If retaining the non-Big-4 jittered per-firm claim, reconcile paper/v4/paper_a_methodology_v4_section_iii.md:59 and :382 plus paper/v4/paper_a_prose_v4_phase4.md:31 and :81 with a committed script/report. If not retaining it, narrow those sentences to the evidence already emitted in §IV-M.1.
+
+4. Re-run a targeted grep after patching: `rg -n "majority firm|9 tools|nine-tool|Script 39c|jittered-dHash" paper/v4`.
+
+**Splice-time mechanical strip**
+
+1. Remove the Phase 4 draft note at paper/v4/paper_a_prose_v4_phase4.md:3, which still contains the internal stale "nine-tool" wording.
+
+2. Remove the Phase 4 close-out notes at paper/v4/paper_a_prose_v4_phase4.md:153 onward before moving prose into the master manuscript.
+
+3. Remove the §III author cross-reference checklist at paper/v4/paper_a_methodology_v4_section_iii.md:434-450; it still says "9 tools" at line 442 and is explicitly marked "remove before submission."
+
+4. During master-file assembly, recheck table numbering after the actual splice, because Table XXVII currently lives in §III while Tables XX-XXVI are in §IV-M.
+
+## Recommended next-step actions
+
+1. Apply the N1 tie-break wording patch in §IV-M.4.
+
+2. Apply the N2 Table XXVII composition-row provenance patch; decide whether to emit the missing non-Big-4 jittered per-firm table or narrow the claim.
+
+3. Run the targeted grep in the checklist and commit the patch as the final Phase 5 text cleanup.
+
+4. Proceed to manuscript-master splice with the internal-note/checklist strip. Partner Jimmy review can then treat the manuscript as Phase 5-converged rather than re-litigating the empirical core.

paper/export_v3.py

@@ -5,9 +5,16 @@ from docx import Document
 from docx.shared import Inches, Pt, RGBColor
 from docx.enum.text import WD_ALIGN_PARAGRAPH
 from pathlib import Path
+import hashlib
 import re
 
+import matplotlib
+matplotlib.use("Agg")
+import matplotlib.pyplot as plt
+
 PAPER_DIR = Path("/Volumes/NV2/pdf_recognize/paper")
+EQUATION_CACHE_DIR = PAPER_DIR / "equations"
+EQUATION_CACHE_DIR.mkdir(exist_ok=True)
 FIG_DIR = Path("/Volumes/NV2/PDF-Processing/signature-analysis/paper_figures")
 EXTRA_FIG_DIR = Path("/Volumes/NV2/PDF-Processing/signature-analysis/reports")
 OUTPUT = PAPER_DIR / "Paper_A_IEEE_Access_Draft_v3.docx"
@@ -48,10 +55,10 @@ FIGURES = {
         "Fig. 3. Firm A per-signature cosine and dHash distributions against the overall CPA population.",
         3.5,
     ),
-    "Fig. 4 visualizes the accountant-level clusters": (
+    "Fig. 4 summarises the per-firm yearly per-signature": (
-        EXTRA_FIG_DIR / "accountant_mixture" / "accountant_mixture_2d.png",
+        EXTRA_FIG_DIR / "figures" / "fig_yearly_big4_comparison.png",
-        "Fig. 4. Accountant-level 3-component Gaussian mixture in the (cosine-mean, dHash-mean) plane.",
+        "Fig. 4. Per-firm yearly per-signature best-match cosine, 2013-2023. (a) Mean per-signature best-match cosine by firm bucket and fiscal year (threshold-free). (b) Share of per-signature best-match cosine ≥ 0.95 (operational cut of Section III-K). Five lines: Firm A, B, C, D, Non-Big-4. Firm A is above the other Big-4 firms in every year; Non-Big-4 is below all four Big-4 firms in every year.",
-        4.5,
+        6.5,
     ),
     "conducted an ablation study comparing three": (
         FIG_DIR / "fig4_ablation.png",
@@ -62,7 +69,321 @@ FIGURES = {
 
 
 def strip_comments(text):
-    return re.sub(r"<!--.*?-->", "", text, flags=re.DOTALL)
+    """Remove HTML comments, but UNWRAP comments whose first non-blank line
+    starts with `TABLE ` (or `TABLE\t`).
+
+    The v3 markdown sources wrap every numerical table in an HTML comment of
+    the form
+
+        <!-- TABLE V: Hartigan Dip Test Results
+        | Distribution | N | ... |
+        |--------------|---|-----|
+        | ...          | … | ... |
+        -->
+
+    The caption (`TABLE V: Hartigan Dip Test Results`) is on the same line as
+    the opening `<!--`, the markdown table body is on the lines following,
+    and `-->` closes the block. The previous implementation wholesale-deleted
+    these comments, which silently dropped every table from the rendered
+    DOCX. We now (i) detect comments whose first non-empty line starts with
+    `TABLE `, (ii) emit a synthetic caption marker line `__TABLE_CAPTION__:
+    <caption>` so process_section can render the caption as a centered
+    bold paragraph above the table, and (iii) keep the table body so the
+    existing markdown-table detector picks it up. Non-TABLE comments
+    (figure placeholders, editorial notes) are stripped as before.
+    """
+    def _replace(match):
+        body = match.group(1)
+        # Find first non-blank line.
+        for line in body.splitlines():
+            stripped = line.strip()
+            if stripped:
+                first = stripped
+                break
+        else:
+            return ""
+        if not first.startswith("TABLE ") and not first.startswith("TABLE\t"):
+            return ""
+        # Split caption (first non-blank line) from the rest.
+        lines = body.splitlines()
+        # Find index of the first non-blank line and use everything after.
+        for idx, line in enumerate(lines):
+            if line.strip():
+                caption = line.strip()
+                rest = "\n".join(lines[idx + 1:])
+                break
+        else:
+            return ""
+        # Emit caption marker + body. Surround with blank lines so the
+        # paragraph/table detector treats the marker as its own paragraph.
+        return f"\n\n__TABLE_CAPTION__:{caption}\n{rest}\n"
+    # Non-greedy match across lines.
+    return re.sub(r"<!--(.*?)-->", _replace, text, flags=re.DOTALL)
+
+
+# ---------------------------------------------------------------------------
+# LaTeX → plain text + Unicode conversion
+# ---------------------------------------------------------------------------
+# The v3 markdown sources contain inline LaTeX ($...$) and a small number of
+# display-math blocks ($$...$$). Pandoc would render these natively; the
+# python-docx pipeline used here does not, so without preprocessing every
+# `\leq`, `\text{dHash}_\text{indep}`, `\Delta\text{BIC}`, `60{,}448`, etc.
+# leaks into the DOCX as raw LaTeX. The helpers below convert the common
+# inline cases to Unicode and split subscripts/superscripts into proper Word
+# runs. Display-math (rare; 3 equations in this paper) gets a best-effort
+# linearisation and is acceptable for a partner-handoff DOCX; final IEEE
+# typesetting is handled by the publisher's LaTeX/MathType pipeline.
+
+LATEX_TOKEN_REPLACEMENTS = [
+    # Greek letters (lower)
+    (r"\\alpha(?![A-Za-z])", "α"), (r"\\beta(?![A-Za-z])", "β"), (r"\\gamma(?![A-Za-z])", "γ"),
+    (r"\\delta(?![A-Za-z])", "δ"), (r"\\epsilon(?![A-Za-z])", "ε"), (r"\\zeta(?![A-Za-z])", "ζ"),
+    (r"\\eta(?![A-Za-z])", "η"), (r"\\theta(?![A-Za-z])", "θ"), (r"\\iota(?![A-Za-z])", "ι"),
+    (r"\\kappa(?![A-Za-z])", "κ"), (r"\\lambda(?![A-Za-z])", "λ"), (r"\\mu(?![A-Za-z])", "μ"),
+    (r"\\nu(?![A-Za-z])", "ν"), (r"\\xi(?![A-Za-z])", "ξ"), (r"\\pi(?![A-Za-z])", "π"),
+    (r"\\rho(?![A-Za-z])", "ρ"), (r"\\sigma(?![A-Za-z])", "σ"), (r"\\tau(?![A-Za-z])", "τ"),
+    (r"\\phi(?![A-Za-z])", "φ"), (r"\\chi(?![A-Za-z])", "χ"), (r"\\psi(?![A-Za-z])", "ψ"),
+    (r"\\omega(?![A-Za-z])", "ω"),
+    # Greek letters (upper, only those distinguishable from Latin)
+    (r"\\Gamma(?![A-Za-z])", "Γ"), (r"\\Delta(?![A-Za-z])", "Δ"), (r"\\Theta(?![A-Za-z])", "Θ"),
+    (r"\\Lambda(?![A-Za-z])", "Λ"), (r"\\Xi(?![A-Za-z])", "Ξ"), (r"\\Pi(?![A-Za-z])", "Π"),
+    (r"\\Sigma(?![A-Za-z])", "Σ"), (r"\\Phi(?![A-Za-z])", "Φ"), (r"\\Psi(?![A-Za-z])", "Ψ"),
+    (r"\\Omega(?![A-Za-z])", "Ω"),
+    # Relations / arrows
+    (r"\\leq(?![A-Za-z])", "≤"), (r"\\geq(?![A-Za-z])", "≥"),
+    (r"\\neq(?![A-Za-z])", "≠"), (r"\\approx(?![A-Za-z])", "≈"),
+    (r"\\equiv(?![A-Za-z])", "≡"), (r"\\sim(?![A-Za-z])", "~"),
+    (r"\\to(?![A-Za-z])", "→"), (r"\\rightarrow(?![A-Za-z])", "→"),
+    (r"\\leftarrow(?![A-Za-z])", "←"), (r"\\Rightarrow(?![A-Za-z])", "⇒"),
+    (r"\\Leftarrow(?![A-Za-z])", "⇐"),
+    # Binary operators
+    (r"\\times(?![A-Za-z])", "×"), (r"\\cdot(?![A-Za-z])", "·"),
+    (r"\\pm(?![A-Za-z])", "±"), (r"\\mp(?![A-Za-z])", "∓"),
+    (r"\\div(?![A-Za-z])", "÷"),
+    # Misc
+    (r"\\infty(?![A-Za-z])", "∞"), (r"\\partial(?![A-Za-z])", "∂"),
+    (r"\\sum(?![A-Za-z])", "∑"), (r"\\prod(?![A-Za-z])", "∏"),
+    (r"\\int(?![A-Za-z])", "∫"),
+    (r"\\ldots(?![A-Za-z])", "…"), (r"\\dots(?![A-Za-z])", "…"),
+    # Spacing commands (drop or replace with single space)
+    (r"\\,", " "), (r"\\;", " "), (r"\\:", " "),
+    (r"\\!", ""), (r"\\ ", " "),
+    (r"\\quad(?![A-Za-z])", "  "), (r"\\qquad(?![A-Za-z])", "    "),
+    # Escaped punctuation
+    (r"\\%", "%"), (r"\\#", "#"), (r"\\&", "&"),
+    (r"\\\$", "$"), (r"\\_", "_"),
+]
+
+
+def _unwrap_command(text, cmd):
+    """Repeatedly replace `\\cmd{X}` → `X` until stable."""
+    pat = re.compile(r"\\" + cmd + r"\{([^{}]*)\}")
+    prev = None
+    while prev != text:
+        prev = text
+        text = pat.sub(r"\1", text)
+    return text
+
+
+MATH_START = ""  # Private Use Area: XML-safe
+MATH_END = ""
+
+
+def latex_to_unicode(text):
+    """Convert a LaTeX-laced markdown paragraph into plain text.
+
+    Math context is preserved with private-use sentinel characters
+    (MATH_START / MATH_END) so the downstream run-splitter only treats
+    `_X` / `^X` as subscript / superscript inside math regions; in body
+    text underscores in identifiers like `signature_analysis` survive.
+    """
+    if "$" not in text and "\\" not in text:
+        return text
+
+    # 1. Strip display-math delimiters first (keep the inner content for
+    #    best-effort linearisation), wrapping math regions with sentinels.
+    #    Then strip inline math delimiters with the same sentinel wrapping.
+    text = re.sub(r"\$\$([\s\S]+?)\$\$",
+                  lambda m: MATH_START + m.group(1) + MATH_END, text)
+    text = re.sub(r"\$([^$]+?)\$",
+                  lambda m: MATH_START + m.group(1) + MATH_END, text)
+
+    # 2. Replace token-level commands with Unicode glyphs *before* unwrapping
+    #    `\text{...}` and friends, so that `\Delta\text{BIC}` becomes
+    #    `Δ\text{BIC}` (then `ΔBIC`) rather than `\DeltaBIC` which would be
+    #    stripped wholesale by the cleanup pass.
+    for pat, repl in LATEX_TOKEN_REPLACEMENTS:
+        text = re.sub(pat, repl, text)
+
+    # 3. Unwrap formatting / text commands (innermost first via _unwrap loop).
+    for cmd in ("text", "mathbf", "mathit", "mathrm", "mathsf", "mathtt",
+                "operatorname", "emph", "textbf", "textit"):
+        text = _unwrap_command(text, cmd)
+
+    # 4. \frac{a}{b} → (a)/(b); \sqrt{x} → √(x). Apply repeatedly to handle
+    #    one level of nesting; deeper nesting is rare in this paper.
+    for _ in range(3):
+        text = re.sub(
+            r"\\t?frac\{([^{}]+)\}\{([^{}]+)\}",
+            r"(\1)/(\2)",
+            text,
+        )
+    text = re.sub(r"\\sqrt\{([^{}]+)\}", r"√(\1)", text)
+
+    # 5. TeX braces used purely for spacing/grouping: K{=}3 → K=3,
+    #    60{,}448 → 60,448, 10{,}175 → 10,175.
+    text = re.sub(r"\{([=<>+\-,])\}", r"\1", text)
+
+    # 6. Strip any remaining `\cmd{...}` (best effort) and `\cmd ` tokens.
+    text = re.sub(r"\\[a-zA-Z]+\{([^{}]*)\}", r"\1", text)
+    text = re.sub(r"\\[a-zA-Z]+(?![A-Za-z])", "", text)
+
+    # 7. Collapse runs of whitespace introduced by command stripping.
+    text = re.sub(r"[ \t]{2,}", " ", text)
+    return text
+
+
+_SUBSUP_PATTERN = re.compile(
+    r"_\{([^{}]*)\}"     # _{...}
+    r"|\^\{([^{}]*)\}"   # ^{...}
+    r"|_([A-Za-z0-9+\-])"  # _X (single token)
+    r"|\^([A-Za-z0-9+\-])"  # ^X (single token)
+)
+
+
+def _emit_plain(paragraph, text, font_name, font_size, bold, italic):
+    if not text:
+        return
+    run = paragraph.add_run(text)
+    run.font.name = font_name
+    run.font.size = font_size
+    run.bold = bold
+    run.italic = italic
+
+
+def _emit_math(paragraph, text, font_name, font_size, bold, italic):
+    """Emit `text` from a math region: split on `_X` / `_{X}` / `^X` / `^{X}`
+    and render those as Word subscripts / superscripts."""
+    if "_" not in text and "^" not in text:
+        _emit_plain(paragraph, text, font_name, font_size, bold, italic)
+        return
+    pos = 0
+    for m in _SUBSUP_PATTERN.finditer(text):
+        if m.start() > pos:
+            _emit_plain(paragraph, text[pos:m.start()],
+                        font_name, font_size, bold, italic)
+        sub_text = m.group(1) or m.group(3)
+        sup_text = m.group(2) or m.group(4)
+        if sub_text is not None:
+            run = paragraph.add_run(sub_text)
+            run.font.subscript = True
+        else:
+            run = paragraph.add_run(sup_text)
+            run.font.superscript = True
+        run.font.name = font_name
+        run.font.size = font_size
+        run.bold = bold
+        run.italic = italic
+        pos = m.end()
+    if pos < len(text):
+        _emit_plain(paragraph, text[pos:],
+                    font_name, font_size, bold, italic)
+
+
+def add_text_with_subsup(paragraph, text, font_name="Times New Roman",
+                         font_size=Pt(10), bold=False, italic=False):
+    """Add `text` to `paragraph`. Subscript/superscript handling is scoped to
+    math regions delimited by MATH_START / MATH_END sentinels (set up by
+    `latex_to_unicode`). Outside math regions, underscores and carets are
+    preserved literally so identifiers like `signature_analysis` and
+    `paper_a_results_v3.md` survive intact.
+    """
+    if MATH_START not in text:
+        _emit_math(paragraph, text, font_name, font_size, bold, italic) \
+            if False else \
+            _emit_plain(paragraph, text, font_name, font_size, bold, italic)
+        return
+
+    pos = 0
+    while pos < len(text):
+        s = text.find(MATH_START, pos)
+        if s == -1:
+            _emit_plain(paragraph, text[pos:],
+                        font_name, font_size, bold, italic)
+            break
+        if s > pos:
+            _emit_plain(paragraph, text[pos:s],
+                        font_name, font_size, bold, italic)
+        e = text.find(MATH_END, s + 1)
+        if e == -1:
+            # Unterminated math region — emit rest as plain.
+            _emit_plain(paragraph, text[s + 1:],
+                        font_name, font_size, bold, italic)
+            break
+        math_body = text[s + 1:e]
+        _emit_math(paragraph, math_body, font_name, font_size, bold, italic)
+        pos = e + 1
+
+
+# ---------------------------------------------------------------------------
+# Display-equation rendering (matplotlib mathtext → PNG → embedded image)
+# ---------------------------------------------------------------------------
+
+# matplotlib mathtext is a subset of LaTeX. A few common TeX-only macros need
+# to be substituted with mathtext-supported equivalents before parsing.
+_MATHTEXT_SUBS = [
+    (re.compile(r"\\tfrac\b"), r"\\frac"),       # text-frac → frac
+    (re.compile(r"\\dfrac\b"), r"\\frac"),       # display-frac → frac
+    (re.compile(r"\\operatorname\{([^{}]+)\}"),
+     lambda m: r"\mathrm{" + m.group(1) + "}"),  # operatorname → mathrm
+    (re.compile(r"\\,"), " "),                   # thin space
+    (re.compile(r"\\;"), " "),
+    (re.compile(r"\\!"), ""),
+]
+
+
+def _sanitise_for_mathtext(latex: str) -> str:
+    out = latex
+    for pat, repl in _MATHTEXT_SUBS:
+        out = pat.sub(repl, out)
+    return out
+
+
+def render_equation_png(latex: str, fontsize: int = 14) -> Path:
+    """Render a LaTeX math expression to a tightly-cropped PNG using
+    matplotlib mathtext, with content-addressed caching so a re-build only
+    re-renders changed equations. Returns the cached PNG path."""
+    sanitised = _sanitise_for_mathtext(latex.strip())
+    digest = hashlib.sha1(
+        (sanitised + f"|fs{fontsize}").encode("utf-8")).hexdigest()[:16]
+    out_path = EQUATION_CACHE_DIR / f"eq_{digest}.png"
+    if out_path.exists():
+        return out_path
+    fig = plt.figure(figsize=(8, 1.6))
+    fig.text(0.5, 0.5, f"${sanitised}$",
+             fontsize=fontsize, ha="center", va="center")
+    fig.savefig(str(out_path), dpi=220, bbox_inches="tight",
+                pad_inches=0.05)
+    plt.close(fig)
+    return out_path
+
+
+def add_equation_block(doc, latex: str, equation_number: int,
+                       width_inches: float = 4.5):
+    """Insert a centered display equation (rendered as PNG) followed by
+    a right-aligned equation number `(N)`. Width keeps the equation
+    visually proportional within the IEEE Access body column."""
+    img_path = render_equation_png(latex)
+    p = doc.add_paragraph()
+    p.alignment = WD_ALIGN_PARAGRAPH.CENTER
+    p.paragraph_format.space_before = Pt(6)
+    p.paragraph_format.space_after = Pt(6)
+    run = p.add_run()
+    run.add_picture(str(img_path), width=Inches(width_inches))
+    # Equation number on the same paragraph, tab-aligned to the right.
+    num_run = p.add_run(f"\t({equation_number})")
+    num_run.font.name = "Times New Roman"
+    num_run.font.size = Pt(10)
 
 
 def add_md_table(doc, table_lines):
@@ -79,14 +400,23 @@ def add_md_table(doc, table_lines):
     for r_idx, row in enumerate(rows_data):
         for c_idx in range(min(len(row), ncols)):
             cell = table.rows[r_idx].cells[c_idx]
-            cell.text = row[c_idx]
+            raw = row[c_idx]
-            for p in cell.paragraphs:
+            # Strip markdown emphasis markers; convert LaTeX before rendering.
-                p.alignment = WD_ALIGN_PARAGRAPH.CENTER
+            raw = re.sub(r"\*\*\*(.+?)\*\*\*", r"\1", raw)
-                for run in p.runs:
+            raw = re.sub(r"\*\*(.+?)\*\*", r"\1", raw)
-                    run.font.size = Pt(8)
+            raw = re.sub(r"\*(.+?)\*", r"\1", raw)
-                    run.font.name = "Times New Roman"
+            raw = re.sub(r"`(.+?)`", r"\1", raw)
-                    if r_idx == 0:
+            cell_text = latex_to_unicode(raw)
-                        run.bold = True
+            # Replace the default empty paragraph with one we control.
+            cell.text = ""
+            cp = cell.paragraphs[0]
+            cp.alignment = WD_ALIGN_PARAGRAPH.CENTER
+            add_text_with_subsup(
+                cp, cell_text,
+                font_name="Times New Roman",
+                font_size=Pt(8),
+                bold=(r_idx == 0),
+            )
     doc.add_paragraph()
 
 
@@ -105,10 +435,27 @@ def _insert_figures(doc, para_text):
             cr.italic = True
 
 
-def process_section(doc, filepath):
+def process_section(doc, filepath, equation_counter=None):
+    """Process one v3 markdown section. `equation_counter` is a single-element
+    list (used as a mutable counter shared across sections) tracking the
+    running display-equation number."""
+    if equation_counter is None:
+        equation_counter = [0]
     text = filepath.read_text(encoding="utf-8")
     text = strip_comments(text)
     lines = text.split("\n")
+    # Defensive blockquote handling: markdown blockquote lines (`> body`) are
+    # not rendered as Word callout blocks here, but stripping the leading
+    # `> ` keeps the body text from leaking the literal `>` and the empty
+    # `>` separator lines into the DOCX.
+    cleaned = []
+    for ln in lines:
+        s = ln.lstrip()
+        if s == ">" or s.startswith("> "):
+            cleaned.append(ln[ln.index(">") + 1:].lstrip() if "> " in ln else "")
+        else:
+            cleaned.append(ln)
+    lines = cleaned
     i = 0
     while i < len(lines):
         line = lines[i]
@@ -117,23 +464,44 @@ def process_section(doc, filepath):
             i += 1
             continue
         if stripped.startswith("# "):
-            h = doc.add_heading(stripped[2:], level=1)
+            h = doc.add_heading(
+                latex_to_unicode(stripped[2:]).replace(MATH_START, "").replace(MATH_END, ""),
+                level=1)
             for run in h.runs:
                 run.font.color.rgb = RGBColor(0, 0, 0)
             i += 1
             continue
         if stripped.startswith("## "):
-            h = doc.add_heading(stripped[3:], level=2)
+            h = doc.add_heading(
+                latex_to_unicode(stripped[3:]).replace(MATH_START, "").replace(MATH_END, ""),
+                level=2)
             for run in h.runs:
                 run.font.color.rgb = RGBColor(0, 0, 0)
             i += 1
             continue
         if stripped.startswith("### "):
-            h = doc.add_heading(stripped[4:], level=3)
+            h = doc.add_heading(
+                latex_to_unicode(stripped[4:]).replace(MATH_START, "").replace(MATH_END, ""),
+                level=3)
             for run in h.runs:
                 run.font.color.rgb = RGBColor(0, 0, 0)
             i += 1
             continue
+        if stripped.startswith("__TABLE_CAPTION__:"):
+            caption_text = stripped[len("__TABLE_CAPTION__:"):].strip()
+            caption_text = latex_to_unicode(caption_text)
+            cp = doc.add_paragraph()
+            cp.alignment = WD_ALIGN_PARAGRAPH.CENTER
+            cp.paragraph_format.space_before = Pt(6)
+            cp.paragraph_format.space_after = Pt(2)
+            add_text_with_subsup(
+                cp, caption_text,
+                font_name="Times New Roman",
+                font_size=Pt(9),
+                bold=True,
+            )
+            i += 1
+            continue
         if "|" in stripped and i + 1 < len(lines) and re.match(r"\s*\|[-|: ]+\|", lines[i + 1]):
             table_lines = []
             while i < len(lines) and "|" in lines[i]:
@@ -141,22 +509,74 @@ def process_section(doc, filepath):
                 i += 1
             add_md_table(doc, table_lines)
             continue
+        # Display math: a line starting with `$$` is treated as a single-line
+        # equation block and rendered as an embedded mathtext PNG with an
+        # auto-incrementing equation number.
+        if stripped.startswith("$$"):
+            # Accumulate until a closing $$ is found (single line in our
+            # corpus, but defensively support multi-line just in case).
+            buf = [stripped]
+            if not (stripped.count("$$") >= 2 and stripped.endswith("$$")):
+                while i + 1 < len(lines):
+                    i += 1
+                    buf.append(lines[i])
+                    if "$$" in lines[i]:
+                        break
+            joined = "\n".join(buf).strip()
+            # Strip the leading and trailing $$ delimiters and any trailing
+            # punctuation (e.g. the `,` that some equation lines end with).
+            inner = joined
+            if inner.startswith("$$"):
+                inner = inner[2:]
+            if inner.endswith("$$"):
+                inner = inner[:-2]
+            inner = inner.rstrip(", ")
+            equation_counter[0] += 1
+            try:
+                add_equation_block(doc, inner, equation_counter[0])
+            except Exception as exc:
+                # Fallback: render as plain centered Times-Roman line so the
+                # build doesn't fail on a single un-renderable equation.
+                p = doc.add_paragraph()
+                p.alignment = WD_ALIGN_PARAGRAPH.CENTER
+                run = p.add_run(f"[equation render failed: {exc}] {inner}")
+                run.font.name = "Times New Roman"
+                run.font.size = Pt(10)
+                run.italic = True
+            i += 1
+            continue
         if re.match(r"^\d+\.\s", stripped):
-            p = doc.add_paragraph(style="List Number")
+            # Manual numbering: keep the number from the markdown source and
-            content = re.sub(r"^\d+\.\s", "", stripped)
+            # apply a hanging-indent paragraph format. Avoids python-docx's
+            # `style='List Number'` which depends on a properly-set-up
+            # numbering definition that the default Document() lacks.
+            m = re.match(r"^(\d+)\.\s+(.*)$", stripped)
+            num, content = m.group(1), m.group(2)
+            p = doc.add_paragraph()
+            p.paragraph_format.left_indent = Inches(0.4)
+            p.paragraph_format.first_line_indent = Inches(-0.25)
+            p.paragraph_format.space_after = Pt(4)
+            content = re.sub(r"\*\*\*(.+?)\*\*\*", r"\1", content)
             content = re.sub(r"\*\*(.+?)\*\*", r"\1", content)
-            run = p.add_run(content)
+            content = re.sub(r"\*(.+?)\*", r"\1", content)
-            run.font.size = Pt(10)
+            content = re.sub(r"`(.+?)`", r"\1", content)
-            run.font.name = "Times New Roman"
+            content = latex_to_unicode(content)
+            add_text_with_subsup(p, f"{num}. {content}")
             i += 1
             continue
         if stripped.startswith("- "):
-            p = doc.add_paragraph(style="List Bullet")
+            # Manual bullets with hanging indent (same rationale as numbered).
+            p = doc.add_paragraph()
+            p.paragraph_format.left_indent = Inches(0.4)
+            p.paragraph_format.first_line_indent = Inches(-0.25)
+            p.paragraph_format.space_after = Pt(4)
             content = stripped[2:]
+            content = re.sub(r"\*\*\*(.+?)\*\*\*", r"\1", content)
             content = re.sub(r"\*\*(.+?)\*\*", r"\1", content)
-            run = p.add_run(content)
+            content = re.sub(r"\*(.+?)\*", r"\1", content)
-            run.font.size = Pt(10)
+            content = re.sub(r"`(.+?)`", r"\1", content)
-            run.font.name = "Times New Roman"
+            content = latex_to_unicode(content)
+            add_text_with_subsup(p, f"•  {content}")
             i += 1
             continue
         # Regular paragraph
@@ -179,14 +599,12 @@ def process_section(doc, filepath):
         para_text = re.sub(r"\*\*(.+?)\*\*", r"\1", para_text)
         para_text = re.sub(r"\*(.+?)\*", r"\1", para_text)
         para_text = re.sub(r"`(.+?)`", r"\1", para_text)
-        para_text = para_text.replace("$$", "")
         para_text = para_text.replace("---", "\u2014")
+        para_text = latex_to_unicode(para_text)
 
         p = doc.add_paragraph()
         p.paragraph_format.space_after = Pt(6)
-        run = p.add_run(para_text)
+        add_text_with_subsup(p, para_text)
-        run.font.size = Pt(10)
-        run.font.name = "Times New Roman"
 
         _insert_figures(doc, para_text)
 
@@ -234,15 +652,38 @@ def main():
     run.font.size = Pt(10)
     run.italic = True
 
+    equation_counter = [0]
     for section_file in SECTIONS:
         filepath = PAPER_DIR / section_file
         if filepath.exists():
-            process_section(doc, filepath)
+            process_section(doc, filepath, equation_counter=equation_counter)
         else:
             print(f"WARNING: missing section file: {filepath}")
 
     doc.save(str(OUTPUT))
     print(f"Saved: {OUTPUT}")
+    _run_linter()
+
+
+def _run_linter():
+    """Run the leak linter on the freshly built DOCX. Non-fatal: prints a
+    summary line. For full output run `python3 paper/lint_paper_v3.py`."""
+    try:
+        import lint_paper_v3  # local module
+    except Exception as exc:  # pragma: no cover
+        print(f"(lint skipped: {exc})")
+        return
+    findings = lint_paper_v3.lint_docx(OUTPUT)
+    errors = sum(1 for f in findings if f.severity == "ERROR")
+    warns = sum(1 for f in findings if f.severity == "WARN")
+    infos = sum(1 for f in findings if f.severity == "INFO")
+    if errors:
+        print(f"\n[lint] {errors} ERROR finding(s) in DOCX — run "
+              f"`python3 paper/lint_paper_v3.py --docx` for details.")
+    elif warns or infos:
+        print(f"[lint] DOCX clean of ERRORs ({warns} WARN, {infos} INFO).")
+    else:
+        print("[lint] DOCX clean.")
 
 
 if __name__ == "__main__":

paper/gemini_partner_redpen_audit_v3_19_0.md

@@ -0,0 +1,45 @@
+# Partner Red-Pen Regression Audit (v3.19.0) - Gemini 3.1 Pro
+
+
+### Overall Summary
+The authors have taken a highly rigorous and defensive route to addressing the partner's concerns. The most confusing and convoluted analytical constructs—specifically the accountant-level GMM and accountant-level BD/McCrary tests—have simply been **deleted entirely**. The surviving text has been rewritten to be direct, transparent about limitations, and free of AI-sounding filler. 
+
+Of the 11 specific lettered items (a–k) raised by the partner:
+- **8 are RESOLVED** (rewritten for clarity and precision)
+- **3 are N/A** (the underlying text/analysis was completely removed)
+- **0 are UNRESOLVED, PARTIAL, or IMPROVED**
+
+Additionally, the two overarching thematic items (Citation reality and ZH/EN alignment) are fully RESOLVED or N/A. The smallest residual set of polish required before the partner re-read is **empty**. The manuscript is clean and ready for review.
+
+---
+
+### Detailed Item-by-Item Audit
+
+#### Theme 1: Citation reality (suspected AI hallucinations)
+* **Item**: '輸入?', '有些幻覺像是研究方法', 'BD/McCrary 沒?', '引用?' (Are these hallucinated?)
+* **Status**: **RESOLVED**
+* **Citation**: `@paper/reference_verification_v3.md`, `@paper/paper_a_references_v3.md`
+* **Notes**: The authors conducted a comprehensive `WebFetch` audit of all 41 references. All statistical methods references ([37]-[41]: Hartigan, BD, McCrary, Dempster-Laird-Rubin, White) are 100% real and bibliographically accurate. The audit did catch one genuine error at ref [5] (wrong authors: "I. Hadjadj et al.") which the authors successfully fixed to "H.-H. Kao and C.-Y. Wen" in the current `paper_a_references_v3.md`.
+
+#### Theme 3: ZH/EN alignment gap
+* **Item**: '沒有跟英文嗎?比較' (no English alongside? compare) at end of III-H
+* **Status**: **N/A**
+* **Citation**: Entire manuscript
+* **Notes**: The v3.19.0 draft is now a finalized, monolingual English manuscript prepared for IEEE submission. The dual-language translation scaffolding that caused this misalignment has been removed, rendering the issue moot.
+
+#### Theme 2 & 4: Specific Prose and Numbers (The 11 Lettered Items)
+
+| Item | Partner's Red-Pen Mark | Status | Where it is addressed | Notes / Justification |
+| :--- | :--- | :--- | :--- | :--- |
+| **(a)** & **(h)** | **A1 stipulation, p.16** ('不太懂你的敘述' / entire paragraph red-circled) | **RESOLVED** | Sec III-G (`paper_a_methodology_v3.md`) | The paragraph was completely rewritten. It is no longer roundabout. It explicitly defines A1 as a "cross-year pair-existence property" and clearly lists three concrete conditions where it is *not* guaranteed (e.g., multiple template variants simultaneously, scan-stage noise). |
+| **(b)** | **Conservative structural-similarity, p.16** ('有點繞嗎?' / is it a bit roundabout?) | **RESOLVED** | Sec III-G (`paper_a_methodology_v3.md`) | Reduced to a single, highly literal sentence: "The independent minimum is unconditional on the cosine-nearest pair and is therefore the conservative structural-similarity statistic..." Extremely clear. |
+| **(c)** | **IV-G validation lead-in, p.18** ('不太懂為何陳述?' / don't follow why you say this) | **RESOLVED** | Sec IV-G (`paper_a_results_v3.md`) | The text now explicitly motivates the section: it explains that the prior capture rates are a circular "internal consistency check," so these three new analyses are needed because their "informative quantity does not depend on the threshold's absolute value." |
+| **(d)** & **(k)** | **BD/McCrary at accountant level, p.20** ('看不懂!' / '為何 accountant level 合計, 因為 component?') | **N/A** | *Removed entirely* | The authors deleted the entire accountant-level mixture analysis and accountant-level BD/McCrary test from the paper. Thresholding is now strictly signature-level, completely sidestepping this confusing narrative. |
+| **(e)** | **92.6% match rate, p.13** ('不太懂改善線' / don't follow the improvement angle) | **RESOLVED** | Sec III-D (`paper_a_methodology_v3.md`) | The "improvement angle" has been deleted. The 92.6% is now presented purely descriptively as a data processing metric, explaining that the 7.4% unmatched are "excluded for definitional reasons rather than discarded as noise." |
+| **(f)** | **0.95 cosine cut-off, p.18** ('Cut-off 對應!' / correspondence to what?) | **RESOLVED** | Sec III-K (`paper_a_methodology_v3.md`) | The text directly answers this now: "the cosine cutoff 0.95 corresponds to approximately the whole-sample Firm A P7.5 of the per-signature best-match cosine distribution..." |
+| **(g)** | **139/32 split in C1/C2 clusters, p.18** ('可能太倚加權因子!?' / too reliant on weighting factor?) | **N/A** | *Removed entirely* | Along with the rest of the accountant-level GMM (see item d/k), the C1/C2 cluster analysis and the 139/32 split have been entirely removed from the current draft. |
+| **(i)** | **Hartigan rejection-as-bimodality, p.19** ('?所以為何?' / so why?) | **RESOLVED** | Sec III-I.1 (`paper_a_methodology_v3.md`) | The text no longer falsely equates a dip-test rejection with bimodality. It correctly explains that a significant p-value simply means "more than one peak" and explains it is used only to "decide whether a KDE antimode is well-defined." |
+| **(j)** | **BIC strict-3-component upper-bound framing, p.20** (red-circled paragraph) | **RESOLVED** | Sec IV-D.3 (`paper_a_results_v3.md`) | The text abandons the tortured "upper-bound" framing and bluntly titles the subsection "A Forced Fit." It clearly states that because BIC strongly prefers 3 components, the 2-component parametric structure "is not supported by the data." |
+
+### Smallest Residual Set
+**None.** The authors did not just patch the confusing paragraphs; they systematically dropped the weakest, most complicated statistical claims (accountant-level mixtures) and grounded the remaining text in literal, descriptive language. The paper is safe, highly defensible, and ready to be sent back to the partner.

paper/gemini_review_v3_18_4.md

@@ -0,0 +1,68 @@
+# Independent Peer Review (Round 19) - Paper A v3.18.4
+
+## 1. Overall Verdict: Major Revision
+
+I recommend **Major Revision**. While v3.18.4 resolves the fabricated Appendix B paths and the cross-firm dual-descriptor arithmetic discrepancy, my independent audit found several profound new discrepancies, fabricated rationalizations, and a critical methodological flaw that survived the previous 18 review rounds.
+
+The most severe issues are:
+1. **Fabricated Rationalization for Excluded Documents:** Section IV-H claims 656 documents were excluded because they "carry only a single detected signature, for which no same-CPA pairwise comparison and therefore no best-match cosine / min dHash statistic is available." This fundamentally contradicts the pipeline's core logic (which computes maximum pairwise similarity across the *entire corpus* per CPA, not intra-document) and Section IV-D.1 (which correctly states only 15 signatures belong to singleton CPAs). The 656 documents were actually excluded because they had no CPA-matched signatures at all (`assigned_accountant IS NULL`).
+2. **Fabricated Provenance for Table XIII:** Appendix B claims Table XIII (Firm A per-year cosine distribution) is derived from `reports/accountant_similarity_analysis.json`. However, the generating script (`08_accountant_similarity_analysis.py`) neither extracts nor groups by the `year_month` field. The table's temporal data has no supporting script in the provided pipeline.
+3. **Fabricated Rationalization for Firm A Partners:** Section IV-F.2 claims "two [CPAs were] excluded for disambiguation ties" to explain the 178 vs. 180 Firm A partner split. The actual script `24_validation_recalibration.py` contains no disambiguation logic; it simply takes the set of unique CPAs successfully assigned to Firm A in the database, which happens to be 178.
+4. **Methodological Flaw in Inter-CPA Negative Anchor:** Script `21_expanded_validation.py` claims to generate ~50,000 random inter-CPA pairs for validation. However, the script artificially draws these pairs from a tiny pool of just `n=3,000` randomly selected signatures, rather than the full 168,755 corpus. This severely constrains diversity (reusing the same signatures ~33 times each) and artificially tightens the confidence intervals reported in Table X.
+
+These issues represent severe provenance, narrative, and statistical failures. The paper must undergo a major revision to correct these fabricated rationalizations and ensure the reported numbers and methodologies match the actual execution.
+
+## 2. Empirical-Claim Audit Table
+
+| Claim | Status | Audit basis / notes |
+|---|---|---|
+| 656 single-signature documents excluded because "no same-CPA pairwise comparison" is available | **FABRICATED** | Contradicts cross-document comparison logic and IV-D.1 (only 15 singleton CPAs lack comparison). The real reason is they failed CPA matching entirely. |
+| 178 Firm A CPAs in split vs 180 registry; "two excluded for disambiguation ties" | **FABRICATED** | `24_validation_recalibration.py` simply takes unique accountants with `firm=FIRM_A`. There is no disambiguation logic in the script. |
+| Table XIII (Firm A per-year cosine distribution) | **FABRICATED PROVENANCE** | App. B claims it's derived from `accountant_similarity_analysis.json`, but `08_accountant_similarity_analysis.py` doesn't extract or group by year. |
+| 50,000 inter-CPA negative pairs | **METHODOLOGICALLY FLAWED** | `21_expanded_validation.py` draws 50,000 pairs from a tiny pool of `n=3000` signatures, artificially constraining diversity. |
+| 145/50/180/35 byte-identity decomp | **VERIFIED-AGAINST-ARTIFACT** | Matches `28_byte_identity_decomposition.py`. |
+| Cross-firm convergence 42.12% vs 88.32% | **VERIFIED-AGAINST-ARTIFACT** | Denominators (65,514 and 55,922) reconcile correctly with the updated `accountants.firm` logic. |
+| 90,282 PDFs, 2013-2023, Taiwan | **VERIFIED-IN-TEXT** | Consistent across manuscript. |
+| 86,072 VLM-positive documents; 12 corrupted PDFs; final 86,071 | **VERIFIED-IN-TEXT** | Internally consistent in III-C. |
+| 182,328 extracted signatures; 168,755 CPA-matched; 13,573 unmatched | **VERIFIED-IN-TEXT** | Matches manuscript counts. |
+| 758 CPAs, 15 document types, 86.4% standard audit reports | **UNVERIFIABLE** | Plausible, but no direct packaged JSON verifies the 15/86.4% split. |
+| Qwen2.5-VL 32B, 180 DPI, first-quartile scan, temperature 0 | **UNVERIFIABLE** | No prompt/config/log artifact inspected. |
+| YOLO metrics (precision, recall, mAP) and 43.1 docs/sec throughput | **UNVERIFIABLE** | No training-results or runtime artifact in `signature_analysis/`. |
+| Same-CPA best-match N = 168,740, 15 fewer than matched due to singleton CPAs | **VERIFIED-AGAINST-ARTIFACT** | Matches dip-test report and script logic. |
+| ResNet-50 ImageNet-1K V2, 2048-d, L2 normalized | **VERIFIED-AGAINST-ARTIFACT** | Consistent with methods and ablation script. |
+| All-pairs intra/inter distribution N = 41,352,824 / 500,000; KDE crossover 0.837 | **VERIFIED-AGAINST-ARTIFACT** | Supported by formal-statistical script. |
+| Firm A dip result N=60,448, dip=0.0019, p=0.169 | **VERIFIED-AGAINST-ARTIFACT** | `15_hartigan_dip_test.py`. |
+| Beta mixture Delta BIC = 381 for Firm A; forced crossings 0.977/0.999 | **VERIFIED-AGAINST-ARTIFACT** | `17_beta_mixture_em.py`. |
+
+## 3. Methodological Soundness
+
+While the dual-descriptor design and replication-dominated anchor are fundamentally sound, there is a severe flaw in the inter-CPA negative anchor construction that must be corrected.
+**Flawed Inter-CPA Anchor Generation:** `21_expanded_validation.py` randomly selects just 3,000 feature vectors out of the 168,755 available signatures (via `load_feature_vectors_sample`), and then randomly pairs them to generate 50,000 negative samples. This means that each of the 3,000 signatures is reused in approximately 33 different pairs, artificially deflating the variance and diversity of the negative population. This compromises the tight Wilson 95% confidence intervals on FAR reported in Table X. The script should sample pairs uniformly across the entire 168,755 corpus.
+
+## 4. Narrative Discipline
+
+The manuscript's narrative discipline has improved regarding the removal of the "known-majority-positive" residue. However, the authors have resorted to fabricating rationalizations to explain simple arithmetic gaps:
+- **The 656 Document Exclusion:** Inventing a false methodological limitation ("single signature ... no same-CPA pairwise comparison") to explain a drop in document counts is unacceptable and undermines the paper's credibility, especially when the core methodology explicitly relies on cross-document matching.
+- **The 2 CPAs Exclusion:** Inventing "disambiguation ties" to explain why 178 CPAs are in the Firm A split instead of the registered 180 is similarly dishonest. If the database only successfully matched signatures to 178 Firm A CPAs, the text should state exactly that.
+
+## 5. IEEE Access Fit
+
+The work remains a strong fit for IEEE Access due to its scale and real-world application, provided the provenance and methodological issues are rectified. The journal emphasizes reproducibility, making the fabricated provenance for Table XIII and the statistical flaw in the FAR validation critical blockers for publication.
+
+## 6. Specific Actionable Revisions
+
+1. **Rewrite the 656-document exclusion explanation (Section IV-H):** State that 656 documents were excluded from the per-document classification because none of their extracted signatures could be successfully matched to a registered CPA name, not because single signatures lack cross-document comparison.
+2. **Remove the fabricated "disambiguation ties" claim (Section IV-F.2):** State simply that the 70/30 split was performed over the 178 Firm A CPAs who had successfully matched signatures in the corpus (compared to the 180 in the registry).
+3. **Provide actual script provenance for Table XIII:** Either supply the script that generates the year-by-year left-tail distribution, or remove Table XIII from the manuscript. Do not falsely attribute it to `08_accountant_similarity_analysis.py` (which does not group by year).
+4. **Fix the Inter-CPA Negative Anchor Script:** Modify `21_expanded_validation.py` to sample 50,000 pairs uniformly from the entire 168,755 matched-signature corpus, rather than from a pre-sampled subset of 3,000. Re-run and update Table X.
+5. **(Optional but recommended) Include Unverifiable Logs:** Add YOLO training logs, VLM configuration details, and the 15-document-type breakdown table to the supplementary materials so that claims in Section III-B, III-C, and III-D become verifiable.
+
+## 7. Disagreements with Codex Round-18
+
+I strongly disagree with the Round-18 Codex reviewer's conclusion that the manuscript only required a "Minor Revision." 
+- Codex completely missed that the "656 single-signature documents" explanation in Section IV-H is a fabricated rationalization that fundamentally contradicts the cross-document matching methodology correctly established elsewhere in the paper.
+- Codex blindly accepted the provenance of Table XIII (claiming it was derived from `accountant_similarity_analysis.json`) without checking that the generating script (`08_accountant_similarity_analysis.py`) contains absolutely no temporal (`year_month`) extraction or aggregation logic.
+- Codex missed the completely invented "two CPAs excluded for disambiguation ties" rationalization.
+- Codex missed the statistical flaw in `21_expanded_validation.py` where 50,000 negative pairs are artificially drawn from an overly restricted pool of only 3,000 signatures.
+
+These are significant issues involving empirical honesty and statistical validity that 18 rounds of AI review failed to catch. A Major Revision is strictly required before submission.

paper/gemini_review_v3_19_0.md

@@ -0,0 +1,45 @@
+# Independent Peer Review (Round 20) - Paper A v3.19.0
+
+## 1. Overall Verdict
+**Accept.** The authors have systematically and thoroughly resolved the four major blockers identified in the Round 19 review. The fabricated rationalizations have been entirely stripped out and replaced with honest, database-grounded explanations. The methodological flaw in the inter-CPA negative anchor has been corrected, resulting in statistically valid estimates. The manuscript now exhibits high empirical integrity and is ready for publication.
+
+## 2. Re-audit of Round-19 Findings
+
+| Round-19 finding | v3.19.0 status | Re-audit notes |
+|---|---|---|
+| Fabricated rationalization for 656-document exclusion | **RESOLVED** | The text now correctly explains that these 656 documents were excluded because none of their extracted signatures could be matched to a registered CPA name (`assigned_accountant IS NULL`), directly reflecting the filtering logic observed in `09_pdf_signature_verdict.py` (L44). |
+| Fabricated Table XIII provenance | **RESOLVED** | A new dedicated script (`29_firm_a_yearly_distribution.py`) has been introduced. It extracts and groups by the `year_month` field natively and reproduces the Table XIII data accurately. Appendix B has been updated accordingly. |
+| Fabricated 2-CPA disambiguation ties | **RESOLVED** | The text correctly identifies that the 2 missing Firm A CPAs are singletons (only one signature each). Because their `max_similarity_to_same_accountant` is undefined (NULL), they naturally drop out of the database view queried by `24_validation_recalibration.py` (L75). |
+| Methodological flaw in inter-CPA negative anchor | **RESOLVED** | `21_expanded_validation.py` was rewritten to uniformly sample 50,000 i.i.d. cross-CPA pairs from the full 168,755 matched corpus. The resulting FAR estimates and Wilson CIs in Table X are now statistically valid and methodologically sound. |
+
+## 3. Empirical-Claim Audit Table
+
+| Claim | Status | Audit basis / notes |
+|---|---|---|
+| 656 single-signature documents excluded because `assigned_accountant IS NULL` | **VERIFIED-AGAINST-ARTIFACT** | Matches `09_pdf_signature_verdict.py` filtering logic and accounts precisely for the 85,042 vs 84,386 PDF classification count difference. |
+| 178 Firm A CPAs in fold due to 2 singletons missing best-match statistics | **VERIFIED-AGAINST-ARTIFACT** | Matches SQL logic in `24_validation_recalibration.py` which explicitly requires `max_similarity_to_same_accountant IS NOT NULL`. |
+| Table XIII (Firm A per-year cosine distribution) | **VERIFIED-AGAINST-ARTIFACT** | Generated deterministically by the newly added `29_firm_a_yearly_distribution.py`. |
+| 50,000 inter-CPA negative pairs | **VERIFIED-AGAINST-ARTIFACT** | `21_expanded_validation.py` now explicitly samples uniformly from the `168k` matched corpus rather than a 3,000-row subset. |
+| Inter-CPA cosine stats (mean 0.763, P95 0.886, P99 0.915, max 0.992) | **VERIFIED-AGAINST-ARTIFACT** | Matches updated output logic generated by `21_expanded_validation.py` and cleanly reported in text. |
+| Table X FAR values (e.g. 0.0008 at 0.945, 0.0005 at 0.950) | **VERIFIED-IN-TEXT** | Plausible and updated correctly to reflect the new, unrestricted 50,000-pair draw. |
+| 145/50/180/35 byte-identity decomp | **VERIFIED-IN-TEXT** | Confirmed stable from prior artifact evaluations. |
+| Cross-firm convergence 42.12% vs 88.32% | **VERIFIED-IN-TEXT** | Confirmed stable; denominator math (55,922 Firm A signatures) reconciles natively. |
+| 90,282 PDFs, 2013-2023, Taiwan | **VERIFIED-IN-TEXT** | Consistent across the full manuscript. |
+| 86,072 VLM-positive documents; 12 corrupted PDFs; final 86,071 | **VERIFIED-IN-TEXT** | Consistent across the full manuscript. |
+| 182,328 extracted signatures; 168,755 CPA-matched; 13,573 unmatched | **VERIFIED-IN-TEXT** | Consistent across the full manuscript. |
+| 758 CPAs, 15 document types, 86.4% standard audit reports | **UNVERIFIABLE** | Plausible but no direct structured artifact evaluated. Acceptable as non-critical context. |
+| Qwen2.5-VL 32B, 180 DPI, first-quartile scan, temperature 0 | **UNVERIFIABLE** | Plausible operational config claim; acceptable for main-paper context. |
+| YOLO metrics (precision, recall, mAP) and 43.1 docs/sec throughput | **UNVERIFIABLE** | Plausible claims; acceptable for main-paper text. |
+| Same-CPA best-match N = 168,740, 15 fewer than matched due to singleton CPAs | **VERIFIED-AGAINST-ARTIFACT** | Matches SQL logic correctly excluding NULL best-match statistics. |
+
+## 4. Methodological Soundness
+Outstanding. The authors completely resolved the severe statistical flaw in the negative anchor generation. The new sampling procedure guarantees that the 50,000 negative pairs reflect the true inter-class variance of the full corpus rather than a repetitive subset, properly grounding the FAR Wilson CIs. The dual-descriptor approach, the empirical anchor choice, and the threshold characterization are solid.
+
+## 5. Narrative Discipline
+Excellent. The authors have purged the fabricated rationalizations that undermined previous versions. By plainly stating the mechanical, database-level realities (e.g., singleton records with `max_similarity_to_same_accountant IS NULL` dropping out of SQL views), the narrative is now both empirically honest and technically coherent. 
+
+## 6. IEEE Access Fit
+The manuscript is an excellent fit for IEEE Access. It presents a novel application of deep learning to a large-scale real-world problem, features strong empirical methodologies, and now possesses the rigorous provenance tracking expected of high-quality systems papers. 
+
+## 7. Specific Actionable Revisions
+None required. The manuscript is methodologically sound, narratively disciplined, and ready for publication as-is.

paper/gemini_review_v4_round1.md

@@ -0,0 +1,75 @@
+# Paper A Phase 5 Round 1 — Gemini 3.1 Pro independent review
+
+Reviewer: Gemini 3.1 Pro
+Date: 2026-05-14
+Target: paper/v4/paper_a_prose_v4_phase4.md + paper/v4/paper_a_methodology_v4_section_iii.md + paper/v4/paper_a_results_v4_section_iv.md
+Prior reviewer artifact: paper/codex_review_gpt55_v4_round7.md (codex GPT-5.5, Minor Revision)
+
+## Verdict
+Minor Revision. I corroborate codex's overall conclusion of Minor Revision, as the central empirical narrative (inter-CPA coincidence-rate calibration, K=3 descriptive demotion, and composition decomposition) is robust and strongly supported by the methodology and results sections. However, I explicitly dissent from codex on several critical findings. Most importantly, codex missed that references [42]-[44] *are* already present in the reference list, and codex did not flag the partner's dangerous suggestion to frame firm heterogeneity as "statistically insignificant." These gaps are addressed below.
+
+## Codex round-7 closure cross-check
+
+| # | Spot-checked finding | Verdict | Evidence |
+|---|---|---|---|
+| 1 | Replace "candidate classifiers" with "candidate checks" | CLOSED | `paper_a_prose_v4_phase4.md` explicitly uses "candidate checks" (e.g., "all three candidate checks — the inherited box rule..."); `paper_a_methodology_v4_section_iii.md` §III-K.4 uses "candidate check's positive-anchor miss rate". |
+| 2 | §II LOOO paragraph with refs [42]-[44] | CLOSED | `paper_a_prose_v4_phase4.md` §II contains a fully drafted addition citing Stone [42], Geisser [43], and Vehtari et al. [44]. |
+| 3 | Restore inherited v3.20.0 limitations in §V-G | CLOSED | `paper_a_prose_v4_phase4.md` §V-G lists "The last five are inherited from v3.20.0 §V-G..." and explicitly covers ImageNet features, red-stamp HSV, longitudinal confounds, source-exemplar misattribution, and legal interpretation. |
+| 4 | Limit full-dataset claims to K=3 + Spearman re-run | CLOSED | `paper_a_prose_v4_phase4.md` §V-G clarifies that full-dataset claims are limited: "We did not perform the full per-signature pool-normalised ICCR analysis at the full n = 686 scope; the §IV-K full-dataset Spearman re-run shows the K=3 + box-rule rank-convergence is preserved". |
+
+## Major findings
+
+1. **[Partner Query / Framing Risk] "Statistically insignificant" firm heterogeneity framing is unsupported.** (Codex missed) The partner queried whether firm heterogeneity could be framed as "statistically insignificant." This is completely unsupported and must be explicitly rejected. In `paper_a_results_v4_section_iv.md` §IV-M.5 (Table XXIII) and `paper_a_methodology_v4_section_iii.md` §III-L.4, logistic regression odds ratios for Firms B/C/D versus Firm A are 0.053, 0.010, 0.027. This indicates that Firms B/C/D have 19x to 100x *lower* odds than Firm A of firing the HC hit indicator even after controlling for pool size. This is an order-of-magnitude difference and highly statistically/practically significant.
+2. **[Codex Disagree] Codex falsely claimed refs [42]-[44] were absent.** (Codex disagree) Codex's round-7 review claimed that references [42]-[44] remained placeholders and were absent from `paper_a_references_v3.md`. This is incorrect; the reference file contains these three citations at the end of the list. The only residual issue is the draft note in the phase 4 close-out section (line ~145) stating `[add citation]`, which simply needs to be deleted.
+3. **[Disclaimer Adequacy] Unsupervised limits effectively disclosed.** The text properly disclaims the limits of its unsupervised setting. The "FAR" to "ICCR" terminology replacement reflects the structural fact that inter-CPA collision acts as a specificity proxy, fully acknowledging the "within-firm template-like collision" caveat in §III-L.4.
+4. **[K=3 Demotion Language] Consistent descriptive framing.** The language properly frames the K=2 and K=3 mixtures as firm-compositional partitions rather than inferential evidence for discrete mechanisms, correctly demoting K=3's operational standing based on the dip-test decomposition.
+5. **[Feature-Derived Scores] Caveat phrasing.** §III-K.1 and §V-E clearly caveat the high Spearman correlations ($\rho \ge 0.879$) as "not statistically independent measurements" since they are deterministic functions of the same descriptor pair, successfully framing it as internal consistency rather than external validation.
+
+## Minor findings
+
+1. **[m1] Stale internal draft notes and checklists.** The Phase 4 prose (`paper_a_prose_v4_phase4.md`) still contains internal draft notes at the top and the "Notes for Phase 4 close-out" at the bottom. Section III and IV files also contain similar `internal — remove before submission` blocks. These must be stripped.
+2. **[m2] Table numbering clash (XV-B vs XIX).** In `paper_a_results_v4_section_iv.md` §IV-J, a note acknowledges Table XV-B might need to be renumbered to Table XIX depending on journal style. This should be finalized to ensure sequential integer numbering (preferring Table XIX to avoid "B" suffixes).
+3. **[m3] Word count note.** The abstract word count note in the close-out checklist should be removed, as the abstract is independently verified at 243 words, satisfying the $\le 250$ requirement.
+
+## Provenance spot-checks
+
+1. **Spearman $\rho \ge 0.879$ floor.**
+   - *Claim Text:* "Three feature-derived scores agree on the per-CPA descriptor-position ranking at Spearman $\rho \ge 0.879$"
+   - *Location:* `paper_a_prose_v4_phase4.md` (Abstract & §V-E) and `paper_a_methodology_v4_section_iii.md` (§III-K.1).
+   - *Cited Script:* Script 38.
+   - *Verdict:* VERIFIED.
+2. **Firm A per-doc HC+MC alarm 0.62.**
+   - *Claim Text:* "Firm A's per-document HC+MC alarm rate is 0.62 versus 0.09–0.16 at Firms B/C/D"
+   - *Location:* `paper_a_prose_v4_phase4.md` (Abstract & §V-C).
+   - *Cited Script:* Script 45.
+   - *Verdict:* VERIFIED.
+3. **145/8/107/2 byte-identical split.**
+   - *Claim Text:* "262 byte-identical signatures in the Big-4 subset (Firm A 145, Firm B 8, Firm C 107, Firm D 2)"
+   - *Location:* `paper_a_methodology_v4_section_iii.md` (§III-K.4).
+   - *Cited Script:* Script 40.
+   - *Verdict:* VERIFIED.
+4. **Dip test $p_{\text{median}} = 0.35$ under joint centring + jitter.**
+   - *Claim Text:* "Once both confounds are removed (firm-mean centring plus uniform integer jitter), the Big-4 pooled dHash dip test yields $p_{\text{median}} = 0.35$"
+   - *Location:* `paper_a_methodology_v4_section_iii.md` (§III-I.4).
+   - *Cited Script:* Script 39e.
+   - *Verdict:* VERIFIED.
+5. **Logistic OR 0.053 / 0.010 / 0.027.**
+   - *Claim Text:* "logistic regression... yields odds ratios of 0.053 (Firm B), 0.010 (Firm C), and 0.027 (Firm D)"
+   - *Location:* `paper_a_methodology_v4_section_iii.md` (§III-L.4) and `paper_a_results_v4_section_iv.md` (Table XXIII).
+   - *Cited Script:* Script 44.
+   - *Verdict:* VERIFIED.
+
+## Newly introduced issues
+
+1. **Sample size nuance between text and tables:** §IV-J Table XV correctly sums to 150,442 signatures across Firms A-D, consistent with the descriptor-complete count. However, the vector-complete count of 150,453 used in the ICCR analyses (§III-L.2) could confuse readers comparing numbers across tables. A small footnote in Table XV directing readers back to the sample-size reconciliation in §III-G is recommended.
+
+## Phase 5 readiness
+
+Partial. The technical framing is solid and the methodological pivot is successfully integrated. Phase 5 readiness requires stripping the remaining internal draft notes and firmly rejecting any attempt to describe firm heterogeneity as "statistically insignificant."
+
+## Recommended next-step actions
+
+1. **[Empirical Blocker]** Explicitly reject the "statistically insignificant" framing of firm heterogeneity. The odds ratios derived from Script 44 confirm a massive, statistically significant difference between Firm A and Firms B/C/D.
+2. **[Copy-Edit Blocker]** Strip all internal draft notes, metadata tags, and the Phase 4 and Phase 3 close-out checklists from the prose, methodology, and results files.
+3. **[Copy-Edit Blocker]** Finalize Table XV-B's numbering (to Table XIX) to comply with sequential integer numbering formats typically preferred by IEEE Access.
+4. **[Copy-Edit Blocker]** Remove the `[add citation]` placeholder string in the notes; references [42]-[44] are fully integrated and listed in `paper_a_references_v3.md`.

paper/gemini_review_v4_round2.md

@@ -0,0 +1,42 @@
+# Paper A Phase 5 Round 2 — Gemini 3.1 Pro independent review
+
+Reviewer: Gemini 3.1 Pro
+Date: 2026-05-14
+Target: paper/v4/paper_a_prose_v4_phase4.md + paper/v4/paper_a_methodology_v4_section_iii.md + paper/v4/paper_a_results_v4_section_iv.md (post round-2 + round-3, commit 4a6f9c5)
+Prior reviewer artifacts: paper/codex_review_gpt55_v4_round7.md; paper/codex_review_gpt55_v4_round8.md; paper/gemini_review_v4_round1.md; paper/opus_review_v4_round1.md
+
+## Verdict
+Accept (Phase 5 Splice-Ready). The round-2 and round-3 changes have masterfully resolved the empirical and framing blockers surfaced by the multi-agent panel in round 1. No new empirical work is required. The manuscript is ready for Phase 5 master-file splice.
+
+## Round-1 / round-2 panel closure cross-check
+
+| Source | Finding | Current Status | Evidence / Note |
+|---|---|---|---|
+| Opus M1 | §IV K=3 mechanism-label reversion | CLOSED | Tables VIII, IX, XVI, XVII, and XVIII in `paper_a_results_v4_section_iv.md` now correctly use "low-cos / high-dHash" and "less-replication-dominated rate". The "hand-leaning" mechanistic framing has been successfully eradicated. |
+| Opus M3 | "98-100% within source firm" conflation | CLOSED | The Abstract in `paper_a_prose_v4_phase4.md` now accurately states "$77$–$99\%$ of inter-CPA collisions concentrate within the source firm" for the deployed any-pair rule, fixing the overclaim. |
+| Opus M4 | Duplicate §V-G heading | CLOSED | `paper_a_prose_v4_phase4.md` correctly sequence the sections as "G. Pixel-Identity..." and "H. Limitations". |
+| Codex r8 blocker | Abstract word count over 250 limit | CLOSED | The Abstract has been elegantly trimmed and now stands at approximately 235 words, well within the IEEE Access 250-word limit. |
+| Codex r8 blocker | §IV-I stale "Table XVI" cross-reference | CLOSED | The reference in `paper_a_results_v4_section_iv.md` now accurately points to "§IV-M Tables XXI–XXVI" for the ICCR calibration. |
+| Codex r8 blocker | §IV-J Table XV sample-size footnote | CLOSED | The footnote accurately reconciles the $150,442$ descriptor-complete versus $150,453$ vector-complete sub-samples in `paper_a_results_v4_section_iv.md`. |
+
+## Net-new findings
+1. **Abstract Trim:** The abstract trimming successfully reduced the word count without dropping any essential empirical substance. The retention of the $77$–$99\%$ any-pair collision stat over the $97$–$100\%$ same-pair stat is the right scientific choice, representing the actual deployed rule accurately.
+2. **"Replication-dominated" terminology:** The pivot to "less-replication-dominated" reads cleanly throughout §IV and maintains perfect consistency with the §III-J descriptive demotion.
+3. **Internal-note items:** The draft notes, close-out checklists, and the "Open questions remaining" in the files are tagged explicitly as `internal — remove before submission`. They are perfectly acceptable to defer to manuscript-splice time and are not empirical or structural blockers.
+
+## Provenance spot-checks
+I selected numerical claims not previously verified by Codex or Opus in their reviews:
+1. **Bootstrap CI half-width for marginal crossings:** Table VII in §IV-E reports a K=2 cosine crossing 95% CI of $[0.9742, 0.9772]$ and states a CI half-width of $0.0015$. $(0.9772 - 0.9742) / 2 = 0.0015$. The dHash CI of $[3.476, 3.969]$ yields a half-width of $(3.969 - 3.476) / 2 = 0.2465$, matching the reported $0.246$. VERIFIED.
+2. **Nine-tool validation table structure:** §III-M describes a "nine-tool unsupervised-validation collection." I verified the §III-M table counts exactly 9 diagnostics (from Per-comparison ICCR down to LOOO firm-level reproducibility) mapped to their untested assumptions. VERIFIED.
+3. **Table XVI K=3 Firm A Component Weights:** Table XVI in §IV-J reports Firm A has $0.00\%$ in C1 and $82.46\%$ in C3. This perfectly matches the prose claims in §V-C regarding Firm A's concentration in the templated end. VERIFIED.
+
+## Firm-heterogeneity framing audit
+The partner's suggestion to frame the firm heterogeneity as "statistically insignificant" remains correctly and decisively rejected in these post-round-3 drafts. The prose in §III-L.4 and the Abstract explicitly leverages the logistic regression odds ratios ($0.053, 0.010, 0.027$) to establish that Firms B/C/D have an order-of-magnitude lower HC alarm rate even after pool-size adjustment. Furthermore, the corrected any-pair $77$–$99\%$ / same-pair $97$–$100\%$ within-firm collision concentration explicitly *strengthens* the heterogeneity argument by showing that even false alarms cluster structurally within source firms. The framing is robust, decisive, and scientifically accurate.
+
+## Phase 5 readiness
+**Ready for Phase 5 Splice (Accept).** There are no remaining empirical, structural, or framing blockers.
+
+## Recommended next-step actions
+1. Execute the final master-file manuscript splice.
+2. During the splice, mechanically strip all markdown blocks tagged `> **Draft note... internal — remove before submission**`, as well as the close-out checklists and the open questions block at the end of §III.
+3. Finalize the `Table XV-B` versus `Table XIX` numbering decision based on the specific journal template requirements during typesetting.

paper/lint_paper_v3.py

@@ -0,0 +1,399 @@
+#!/usr/bin/env python3
+"""Paper A v3 markdown / DOCX leak linter.
+
+Runs two pass:
+
+  Source pass — scans the v3 markdown sources for syntax patterns that the
+  python-docx export pipeline does NOT render natively. Each finding is a
+  file:line:severity:message tuple. Severity is ERROR (will leak literal
+  syntax into Word), WARN (sometimes leaks), or INFO (style nits).
+
+  DOCX pass — opens the rendered DOCX and scans every paragraph and table
+  cell for known leak signatures. This is the authoritative check: even
+  if the source pass is clean, the DOCX pass tells you what your partner
+  will actually see. The DOCX pass currently checks for:
+
+    - leftover LaTeX commands (`\\cmd`)
+    - unstripped `$` math delimiters
+    - pandoc footnote markers (`[^name]`)
+    - markdown blockquote markers (lines starting with `> `)
+    - TeX brace tricks (`{=}`, `{,}`)
+    - PUA sentinels (`\\uE000`, `\\uE001`) leaking from the math-region
+      run-splitter
+    - the synthetic table-caption marker `__TABLE_CAPTION__:` if it ever
+      survives processing
+
+Exit code:
+  0  clean
+  1  WARN-level findings only (ship-able after review)
+  2  ERROR-level findings (do NOT ship)
+
+Usage:
+  python3 paper/lint_paper_v3.py           # both passes
+  python3 paper/lint_paper_v3.py --source  # source-side only
+  python3 paper/lint_paper_v3.py --docx    # DOCX-side only
+
+Designed to be run after `python3 export_v3.py` and before copying the
+DOCX to ~/Downloads.
+"""
+
+from __future__ import annotations
+
+import argparse
+import re
+import sys
+from dataclasses import dataclass
+from pathlib import Path
+
+PAPER_DIR = Path(__file__).resolve().parent
+DOCX_PATH = PAPER_DIR / "Paper_A_IEEE_Access_Draft_v3.docx"
+
+V3_SOURCES = [
+    "paper_a_abstract_v3.md",
+    "paper_a_introduction_v3.md",
+    "paper_a_related_work_v3.md",
+    "paper_a_methodology_v3.md",
+    "paper_a_results_v3.md",
+    "paper_a_discussion_v3.md",
+    "paper_a_conclusion_v3.md",
+    "paper_a_appendix_v3.md",
+    "paper_a_declarations_v3.md",
+    "paper_a_references_v3.md",
+]
+
+
+# ---------------------------------------------------------------------------
+# Finding model + ANSI colour helpers
+# ---------------------------------------------------------------------------
+
+SEVERITY_RANK = {"ERROR": 2, "WARN": 1, "INFO": 0}
+COLOR = {
+    "ERROR": "\033[31m",  # red
+    "WARN":  "\033[33m",  # yellow
+    "INFO":  "\033[36m",  # cyan
+    "RESET": "\033[0m",
+    "BOLD":  "\033[1m",
+}
+
+
+@dataclass
+class Finding:
+    severity: str
+    rule: str
+    location: str  # "file:line" or "DOCX:para 42" / "DOCX:table 6 row 3 col 2"
+    message: str
+    snippet: str = ""
+
+    def render(self, use_color: bool = True) -> str:
+        col = COLOR[self.severity] if use_color else ""
+        rst = COLOR["RESET"] if use_color else ""
+        bold = COLOR["BOLD"] if use_color else ""
+        head = f"{col}[{self.severity}]{rst} {bold}{self.rule}{rst} @ {self.location}"
+        body = f"\n    {self.message}"
+        snip = f"\n    > {self.snippet}" if self.snippet else ""
+        return head + body + snip
+
+
+# ---------------------------------------------------------------------------
+# Source-side rules
+# ---------------------------------------------------------------------------
+
+# Each rule: (pattern, severity, rule_id, message, predicate)
+# predicate(match, line) → bool: returns True to keep the finding (lets us
+# suppress matches that are inside HTML comments or fenced code blocks).
+
+def _outside_table_comment(match: re.Match, line: str, in_comment: bool, in_table: bool) -> bool:
+    """Suppress findings inside HTML comments (where they're allowed) or
+    inside markdown table rows (where they survive intact via add_md_table)."""
+    return not in_comment and not in_table
+
+
+def _always(match: re.Match, line: str, in_comment: bool, in_table: bool) -> bool:
+    return True
+
+
+SOURCE_RULES = [
+    # Pandoc footnote markers — leak as raw text in the DOCX.
+    (re.compile(r"\[\^[A-Za-z0-9_-]+\]"),
+     "ERROR", "pandoc-footnote",
+     "Pandoc-style footnote `[^name]` does not render in DOCX. "
+     "Inline the explanation as a parenthetical instead.",
+     _outside_table_comment),
+
+    # Markdown blockquote `> body` lines — exporter strips them defensively
+    # now, but flag for awareness so authors don't rely on them rendering.
+    (re.compile(r"^>\s"),
+     "WARN", "blockquote",
+     "Markdown blockquote `> ...` is stripped to plain paragraph in DOCX "
+     "(no quote-block formatting). If you intended a callout, use bold "
+     "lead-in instead.",
+     _always),
+
+    # Display-math fences `$$...$$` (only when the line itself starts with
+    # `$$`) — exporter does best-effort linearisation, but the result is
+    # ugly. Inline the equation as plain prose where possible.
+    (re.compile(r"^\$\$.+?\$\$\s*$|^\$\$\s*$"),
+     "WARN", "display-math",
+     "Display math `$$...$$` renders as a best-effort plain-text "
+     "linearisation in DOCX (no MathType/equation rendering). Consider "
+     "replacing with a numbered equation image or inline prose.",
+     _always),
+
+    # Inline math containing `\frac{...{...}...}` — nested braces in a
+    # frac argument are not handled by the exporter's regex.
+    (re.compile(r"\\t?frac\{[^{}]*\{[^{}]*\}[^{}]*\}\{|\\t?frac\{[^{}]+\}\{[^{}]*\{"),
+     "WARN", "nested-frac",
+     "Nested-brace `\\frac{...}{...}` may not linearise cleanly. Verify "
+     "the rendered DOCX paragraph or rewrite the math inline.",
+     _outside_table_comment),
+
+    # Setext-style headers (=== / ---) under a line of text — not handled.
+    (re.compile(r"^=+\s*$|^-{3,}\s*$"),
+     "INFO", "setext-header",
+     "Setext-style header (=== / ---) is not handled by the exporter; "
+     "use ATX (#, ##, ###) instead.",
+     _always),
+
+    # Pandoc fenced div `:::` — not handled.
+    (re.compile(r"^:::"),
+     "ERROR", "pandoc-fenced-div",
+     "Pandoc fenced div `:::` is not handled by the exporter and would "
+     "leak into the DOCX as plain text.",
+     _always),
+
+    # Pandoc bracketed-attribute spans `[text]{.class}` — not handled.
+    (re.compile(r"\][\{][^}]*[\}]"),
+     "WARN", "pandoc-attribute-span",
+     "Pandoc attribute span `[text]{.class}` is not parsed by the exporter "
+     "and the brace block will leak.",
+     _outside_table_comment),
+
+    # File paths in body text — Appendix B is the canonical home for
+    # script→artifact references.
+    (re.compile(r"`signature_analysis/\d+_[a-z_]+\.py`"),
+     "INFO", "script-path-in-body",
+     "Verbose script path in body text. Consider replacing with "
+     "'(reproduction artifact in Appendix B)' for body-prose tightness.",
+     _outside_table_comment),
+
+    # `reports/...json` paths in body text — same rationale.
+    (re.compile(r"`reports/[a-z_]+/[a-z_]+\.(?:json|md)`"),
+     "INFO", "report-path-in-body",
+     "Verbose report-artifact path in body text. Consider replacing with "
+     "'(see Appendix B provenance map)'.",
+     _outside_table_comment),
+
+    # Bare HTML comments that are NOT TABLE/FIGURE markers may indicate
+    # editorial residue. Stripped wholesale by exporter, so harmless, but
+    # worth visibility.
+    (re.compile(r"^<!--\s*$|^<!-- (?!TABLE |FIGURE )"),
+     "INFO", "html-comment",
+     "HTML comment block (non-TABLE) — stripped from DOCX. Keep for "
+     "editorial notes or remove for tidiness.",
+     _always),
+]
+
+
+def lint_sources() -> list[Finding]:
+    findings: list[Finding] = []
+    for src in V3_SOURCES:
+        path = PAPER_DIR / src
+        if not path.exists():
+            continue
+        in_comment = False
+        in_table = False
+        for line_no, line in enumerate(path.read_text(encoding="utf-8").splitlines(), 1):
+            # Track HTML-comment context (multi-line aware).
+            if "<!--" in line:
+                in_comment = True
+            stripped = line.strip()
+            if stripped.startswith("|") and stripped.endswith("|"):
+                in_table = True
+            else:
+                in_table = False
+            for pat, sev, rule, msg, predicate in SOURCE_RULES:
+                for m in pat.finditer(line):
+                    if not predicate(m, line, in_comment, in_table):
+                        continue
+                    findings.append(Finding(
+                        severity=sev,
+                        rule=rule,
+                        location=f"{src}:{line_no}",
+                        message=msg,
+                        snippet=line.rstrip()[:120],
+                    ))
+            if "-->" in line:
+                in_comment = False
+    return findings
+
+
+# ---------------------------------------------------------------------------
+# DOCX-side rules
+# ---------------------------------------------------------------------------
+
+DOCX_LEAK_PATTERNS = [
+    # (pattern, severity, rule_id, message)
+    (re.compile(r"\\[a-zA-Z]+(?:\{[^{}]*\})?"),
+     "ERROR", "leftover-latex-cmd",
+     "LaTeX command `\\cmd` leaked into DOCX. Either add a token rule to "
+     "`latex_to_unicode` in `export_v3.py` or rewrite the source as plain text."),
+
+    (re.compile(r"(?<!\\)\$[^$\s][^$]*\$"),
+     "ERROR", "unstripped-dollar-math",
+     "Inline math `$...$` was not stripped. The math-context handler in "
+     "`latex_to_unicode` should have wrapped the content with PUA sentinels."),
+
+    (re.compile(r"\[\^[A-Za-z0-9_-]+\]"),
+     "ERROR", "pandoc-footnote-leak",
+     "Pandoc footnote marker leaked into DOCX. Inline the footnote body "
+     "as a parenthetical at the source."),
+
+    (re.compile(r"^>\s"),
+     "ERROR", "blockquote-leak",
+     "Markdown blockquote `> ...` leaked literal `>` into DOCX. The "
+     "exporter pre-pass should strip these — check `process_section`."),
+
+    (re.compile(r"\{[,=<>+\-]\}"),
+     "ERROR", "tex-brace-trick",
+     "TeX brace-trick `{=}` / `{,}` leaked. Should be stripped by "
+     "`latex_to_unicode`."),
+
+    (re.compile(r"[]"),
+     "ERROR", "pua-sentinel-leak",
+     "Math-region PUA sentinel (\\uE000 / \\uE001) leaked. A render path "
+     "is bypassing `add_text_with_subsup`; check headings / list items / "
+     "title-page paragraphs."),
+
+    (re.compile(r"__TABLE_CAPTION__"),
+     "ERROR", "table-caption-marker-leak",
+     "Synthetic `__TABLE_CAPTION__:` marker leaked. The marker is meant "
+     "to be consumed by `process_section` and rendered as a centered "
+     "bold caption paragraph."),
+
+    (re.compile(r"signature[a-z]+analysis/\d+[a-z_]+\.py"),
+     "ERROR", "underscore-eaten-path",
+     "Underscores eaten from a script path (e.g., "
+     "`signatureanalysis/28byteidentitydecomposition.py`). The "
+     "math-context-scoped subscript handler in `add_text_with_subsup` "
+     "should leave underscores intact in plain text."),
+
+    (re.compile(r"\b(\w+_\w+)+\b", flags=re.UNICODE),
+     "INFO", "underscore-identifier",
+     "Underscored identifier in body text (e.g., a code symbol or path). "
+     "Verify it renders with underscores intact, not as subscripts."),
+]
+
+
+def lint_docx(docx_path: Path = DOCX_PATH) -> list[Finding]:
+    try:
+        from docx import Document
+    except ImportError:
+        return [Finding("ERROR", "missing-dep",
+                        "lint:docx",
+                        "python-docx is not installed; cannot run DOCX pass.")]
+
+    if not docx_path.exists():
+        return [Finding("ERROR", "missing-docx",
+                        str(docx_path),
+                        "Built DOCX not found. Run `python3 export_v3.py` first.")]
+
+    doc = Document(str(docx_path))
+    findings: list[Finding] = []
+    seen_signatures = set()  # dedupe identical leaks across paragraphs
+
+    def scan(text: str, location: str):
+        for pat, sev, rule, msg in DOCX_LEAK_PATTERNS:
+            for m in pat.finditer(text):
+                # Skip the INFO-level identifier rule unless it looks like
+                # an obvious math residue (e.g., dHash_indep or N_a).
+                if rule == "underscore-identifier":
+                    sample = m.group(0)
+                    # Only complain about identifiers that look like math
+                    # residue: short, underscore-separated single-char tokens.
+                    parts = sample.split("_")
+                    if not all(len(p) <= 4 for p in parts):
+                        continue
+                    if not all(p.isalnum() and not p.isdigit() for p in parts):
+                        continue
+                key = (rule, m.group(0))
+                if key in seen_signatures:
+                    continue
+                seen_signatures.add(key)
+                findings.append(Finding(
+                    severity=sev,
+                    rule=rule,
+                    location=location,
+                    message=msg,
+                    snippet=text[max(0, m.start() - 30):m.end() + 30].replace("\n", " ")[:140],
+                ))
+
+    for i, p in enumerate(doc.paragraphs):
+        if p.text:
+            scan(p.text, f"DOCX:para {i}")
+    for ti, t in enumerate(doc.tables):
+        for ri, row in enumerate(t.rows):
+            for ci, cell in enumerate(row.cells):
+                if cell.text:
+                    scan(cell.text, f"DOCX:table {ti + 1} row {ri} col {ci}")
+
+    return findings
+
+
+# ---------------------------------------------------------------------------
+# Reporter
+# ---------------------------------------------------------------------------
+
+def summarise(findings: list[Finding], use_color: bool = True) -> int:
+    def c(key: str) -> str:
+        return COLOR[key] if use_color else ""
+
+    if not findings:
+        print(f"{c('BOLD')}{c('INFO')}clean — no leaks detected{c('RESET')}")
+        return 0
+    counts = {"ERROR": 0, "WARN": 0, "INFO": 0}
+    findings.sort(key=lambda f: (-SEVERITY_RANK[f.severity], f.location))
+    for f in findings:
+        counts[f.severity] += 1
+        print(f.render(use_color))
+        print()
+    print(f"{c('BOLD')}summary{c('RESET')}: "
+          f"{c('ERROR')}{counts['ERROR']} ERROR{c('RESET')}  "
+          f"{c('WARN')}{counts['WARN']} WARN{c('RESET')}  "
+          f"{c('INFO')}{counts['INFO']} INFO{c('RESET')}")
+    if counts["ERROR"]:
+        return 2
+    if counts["WARN"]:
+        return 1
+    return 0
+
+
+def main():
+    ap = argparse.ArgumentParser(
+        description="Lint Paper A v3 markdown sources and rendered DOCX for "
+                    "syntax-leak issues.",
+    )
+    ap.add_argument("--source", action="store_true",
+                    help="run only the markdown source pass")
+    ap.add_argument("--docx", action="store_true",
+                    help="run only the rendered DOCX pass")
+    ap.add_argument("--no-color", action="store_true",
+                    help="disable ANSI colour output")
+    args = ap.parse_args()
+
+    use_color = sys.stdout.isatty() and not args.no_color
+    findings: list[Finding] = []
+    if args.source or not (args.source or args.docx):
+        print(f"{COLOR['BOLD'] if use_color else ''}--- source pass "
+              f"({len(V3_SOURCES)} files) ---{COLOR['RESET'] if use_color else ''}")
+        findings.extend(lint_sources())
+    if args.docx or not (args.source or args.docx):
+        print(f"{COLOR['BOLD'] if use_color else ''}\n--- docx pass "
+              f"({DOCX_PATH.name}) ---{COLOR['RESET'] if use_color else ''}")
+        findings.extend(lint_docx())
+
+    print()
+    sys.exit(summarise(findings, use_color))
+
+
+if __name__ == "__main__":
+    main()

paper/narrative_audit_v4.md

@@ -0,0 +1,220 @@
+# Paper A v4.0 — Narrative-Thread Audit
+
+Auditor: Claude Opus 4.7 (1M context)
+Date: 2026-05-14
+Target: paper/v4/paper_a_prose_v4_phase4.md + paper/v4/paper_a_methodology_v4_section_iii.md + paper/v4/paper_a_results_v4_section_iv.md (post round-5, commit 128a914)
+Purpose: Coherence check across Abstract / §I / §III / §IV / §V / §VI as a single argument, after Phase 5 AI peer-review panel convergence (3/3 in Accept/Minor band).
+
+## Headline assessment
+
+**Mostly Coherent — submission-ready after 2-3 small narrative-consistency patches.**
+
+The v4 story arc — *"v3.20.0 distributional path turns out to be composition + integer artefact; v4.0 replaces it with anchor-based ICCR + decisive firm heterogeneity; positioning is anchor-calibrated specificity-only screening, not validated detector"* — reads cleanly from Abstract through §VI. The 5 fix rounds + 3 reviewer panels have substantially closed the major framing, terminology, and provenance risks. What remains is narrow narrative-consistency residue between Phase 4 §I/§V prose and §III source-of-truth (3 specific items, all small), plus 1 interpretive caveat that would strengthen §V-H limitation 2.
+
+No empirical reruns required. No structural rewrites required. Submission-readiness gate: **conditionally pass** — recommend a 15-minute round-6 prose-consistency patch before manuscript-splice, then the manuscript is splice-ready.
+
+## 1. Abstract → body mirror audit
+
+The Abstract (Phase 4 line 11, 247 words) makes ~12 distinct claims. Each maps cleanly to a body location:
+
+| # | Abstract claim | §I location | §III/§IV body location | Status |
+|---:|---|---|---|---|
+| 1 | Non-hand-signed detection problem (regulation + digitization) | §I paras 1–3 (lines 19–21) | — (problem framing) | **Aligned** |
+| 2 | Pipeline: VLM + YOLOv11 + ResNet-50 + dual-descriptor | §I para 6 (line 29) + contribution 2 | §III-A..F (inherited) + §III-F dual-descriptor | **Aligned** |
+| 3 | 90,282 reports / 182,328 sigs / 758 CPAs | §I para 8 (line 39) | §IV-A..C (inherited) | **Aligned** |
+| 4 | Big-4 sub-corpus: 437 CPAs / 150,442 sigs | §I para 8 (line 39) | §III-G (line 19), §IV-D (line 9, line 15) | **Aligned** |
+| 5 | Composition decomposition $p_{\text{median}} = 0.35$ | §I para 5 (line 31) + contribution 4 | §III-I.4 (lines 55–73), §IV-M.1 Table XX (line 266) | **Aligned** |
+| 6 | Per-comparison ICCRs 0.0006 / 0.0013 / 0.00014 | §I para 6 (line 33) + contribution 5 | §III-L.1 (line 196), §IV-M.2 Table XXI (line 280) | **Aligned** |
+| 7 | Per-signature ICCR 0.11 | §I para 6 (line 33) | §III-L.2 (line 208), §IV-M.3 Table XXII (line 300) | **Aligned** |
+| 8 | Per-document ICCR 0.34 (HC+MC) | §I para 6 (line 33) | §III-L.3 (line 233), §IV-M.4 Table XXIII (line 317) | **Aligned** |
+| 9 | Firm heterogeneity: Firm A 0.62 vs B/C/D 0.09–0.16 | §I para 7 (line 35) + contribution 6 | §III-L.4 (line 259), §IV-M.4 (line 325) | **Aligned** |
+| 10 | Within-firm 77–99% (any-pair) | §I para 7 (line 35) + contribution 6 | §III-L.4 (line 283), §IV-M.5 Table XXV (line 340) | **Aligned** |
+| 11 | "Specificity-proxy-anchored screening + HITL, not validated detector" positioning | §I contribution 8 (line 57) | §III-M Table XXVII (line 316) + §V-G/H + §VI item 8 | **Aligned** |
+| 12 | "No calibrated error rates without ground truth" disclaimer | §I para 5 item (v) (line 25) | §III-M (line 312), §V-H limit 1 (line 113) | **Aligned** |
+
+Observation: Abstract does NOT mention the three-score Spearman convergence ($\rho \geq 0.879$). This is by design — the v4 pivot demoted three-score from a headline finding to "internal consistency" because the scores share inputs. §I contribution 7 and §V-E retain it with the demoted caveat. **No action needed.**
+
+## 2. §I contributions (8) → body implementation map
+
+| # | §I contribution | §III/§IV implementation | §V/§VI loop-back | Status |
+|---:|---|---|---|---|
+| 1 | Problem formulation | — (§V-A discusses) | §V-A (line 73), §VI implicit | **Aligned** |
+| 2 | End-to-end pipeline | §III-A..F (inherited) | §VI line 145 | **Aligned** |
+| 3 | Dual-descriptor verification | §III-F + §IV-L backbone ablation | §V-A/B implicit | **Aligned** |
+| 4 | Composition decomposition | §III-I.4 + §IV-M.1 Table XX | §V-B (line 81), §VI item 1 (line 147) | **Aligned** |
+| 5 | Anchor-based multi-level ICCR | §III-L + §IV-M.2/M.3/M.4 Tables XXI/XXII/XXIII | §V-F (line 99), §VI item 2 | **Aligned** |
+| 6 | Firm heterogeneity + within-firm collision | §III-L.4 + §IV-M.4/M.5 Tables XXIV/XXV | §V-C (line 83), §VI items 3+4 | **Aligned** |
+| 7 | K=3 descriptive + three-score convergence | §III-J + §III-K.1 + §IV-E/F/G | §V-D/E (lines 89–97), §VI items 5+6 | **Aligned** |
+| 8 | Annotation-free positive-anchor + ten-tool ceiling | §III-K.4 + §III-M Table XXVII + §IV-H Table XIV | §V-G (line 105), §VI items 7+8 | **Aligned** |
+
+All 8 contributions trace cleanly through §III/§IV implementation and §V/§VI loop-back. **No action needed.**
+
+## 3. v3→v4 pivot rhetoric thread
+
+The v4 pivot has four narrative nodes; each must reinforce the others:
+
+| Node | Location | Says |
+|---|---|---|
+| **Setup**: v3.x distributional path | §I para 5 line 31 (Phase 4 prose) | "Earlier work...adopted a distributional path...v4.0 reports a composition decomposition diagnostic that overturns this reading" |
+| **Proof**: 2×2 factorial composition decomposition | §III-I.4 Scripts 39b–39e (lines 55–73) | Joint firm-mean centring + integer-tie jitter eliminates rejection ($p_{\text{median}} = 0.35$) |
+| **Alternative**: anchor-based ICCR | §III-L (line 173+) | Replaces distributional thresholds with inter-CPA coincidence-rate calibration at 3 units |
+| **Discussion**: K=3 stays descriptive | §V-B + §V-D (lines 77–93) | Mixture fits are firm-compositional partitions, not mechanism modes |
+| **Conclusion**: pivot summary | §VI items 1+5 (line 147) | Demotes K=3 mechanism reading; positions ICCR as the operational calibration |
+
+All five nodes use consistent language ("composition + integer artefact"; "descriptive firm-compositional partition"; "no within-population bimodal antimode"). **No action needed.**
+
+## 4. K=3 demotion consistency
+
+Verified across 4 locations using consistent descriptor-position language (post round-2 M1 fix):
+
+- §III-J line 90 (source of truth): "The 'descriptive position' column replaces v3.x's 'hand-leaning / mixed / replicated' mechanism labels"
+- §I contribution 7 (line 55): "K=3 mixture demoted from 'three mechanism clusters' to a descriptive firm-compositional partition"
+- §V-D (line 93): "the K=3 stability supports a descriptive reading...*not* a three-mechanism latent-class structure"
+- §VI item 5 (line 147): same demotion language
+- §IV Tables XVI/XVII column headers: "C1 (low-cos / high-dHash) | C2 (central) | C3 (high-cos / low-dHash)" — descriptor-position labels throughout
+
+`grep -n "hand-leaning"` in v4 public prose: 0 hits (only internal-strip text). **Closed.**
+
+## 5. ICCR vs FAR terminology consistency
+
+Verified across all rate-reporting locations (post round-2 + round-5):
+
+- Abstract: "inter-CPA coincidence-rate (ICCR)" ✓
+- §I contribution 5 (line 51): explicit terminology adoption and FAR disclaimer ✓
+- §III-L.1 (line 185): "Terminological note on 'FAR'" with full disclaimer ✓
+- §IV-I (line 159): historical "FAR" cited only with the "v3.x terminology" caveat ✓
+- §V-G heading (line 105): "Inherited Inter-CPA Negative Anchor Reframed as Coincidence Rate" ✓
+- §V-H limitations: specificity-proxy framing under partially-violated assumption ✓
+- §VI item 2 (line 147): "explicit terminological replacement of 'FAR' by 'ICCR' given the unsupervised setting" ✓
+
+No public-prose "FAR" leak outside the historical-context caveats. **Closed.**
+
+## 6. Numbers consistency audit (cross-section)
+
+Headline numbers cross-referenced between Abstract / §I / §III / §IV:
+
+| Claim | Abstract | §I body | §III source | §IV table | Status |
+|---|---|---|---|---|---|
+| Per-comparison ICCR cos $>0.95$ | 0.0006 | 0.0006 | 0.00060 | 0.00060 Table XXI | **Match** (rounding consistent) |
+| Per-comparison ICCR dHash $\leq 5$ | 0.0013 | 0.0013 | 0.00129 | 0.00129 Table XXI | **Match** |
+| Per-comparison joint | 0.00014 | 0.00014 | 0.00014 | 0.00014 Table XXI | **Match** |
+| Per-signature ICCR | 0.11 | 0.11 | 0.1102 (Wilson) | 0.1102 Table XXII | **Match** |
+| Per-document ICCR (HC+MC) | 0.34 | 0.34 | 0.3375 | 0.3375 Table XXIII | **Match** |
+| Firm A doc HC+MC | 0.62 | 0.62 | 0.6201 | 0.6201 §IV-M.4 line 325 | **Match** |
+| Firms B/C/D doc HC+MC | 0.09–0.16 | 0.09–0.16 | 0.1600 / 0.1635 / 0.0863 | same | **Match** |
+| Within-firm any-pair | 77–99% (rounded) | 76.7–83.7% / 98.8% | same | Table XXV | **Match** |
+| Same-pair within-firm | — | 97.0–99.96% | 99.96 / 97.7 / 98.2 / 97.0 | line 349 | **Match** |
+| Composition $p_{\text{median}}$ | 0.35 | 0.35 | 0.35 | 0.35 Table XX | **Match** |
+| Logistic OR | — | 0.053 / 0.010 / 0.027 | same | Table XXIV | **Match** |
+| Spearman ρ floor | — | 0.879 | 0.879 | 0.8794 Table IX | **Match** (Spearman precision §III/§IV differ at 4th decimal — see §8) |
+
+All headline numbers reconcile across sections. **Closed.**
+
+## 7. Limitations vs §III-M Table XXVII coverage
+
+§V-H lists 14 limitations (9 v4-specific + 5 inherited from v3.20.0). Each Table XXVII assumption should be covered by an explicit §V-H item OR be self-evidently descriptive:
+
+| Table XXVII tool | Assumption | §V-H coverage |
+|---|---|---|
+| Composition decomposition | Jitter unbiased; Big-4 jittered + centred + jittered evidence | Implicit — covered by general "no signature-level ground truth" frame |
+| Per-comparison ICCR | Inter-CPA pairs are negative anchor (partially violated) | §V-H limit 2 (explicit) |
+| Per-signature ICCR | Same + pool replacement preserves negative-anchor property | §V-H limit 2 (implicit via "specificity-proxy rates under partially-violated assumption") |
+| Per-document ICCR | Same | Same |
+| Firm-heterogeneity logistic | Cluster-robust SE not run | **Gap** — no §V-H item explicitly flags the naive-SE caveat |
+| Cross-firm hit matrix | Deployed-rule semantics + mode-of-firms tie-break | §V-H limit 2 |
+| Alert-rate sensitivity | Descriptive gradient, not formal plateau | §V-H limit 5 (line 121, "alert-rate sensitivity analysis characterises only the HC threshold") |
+| Three-score Spearman | Scores share inputs | §V-H limit 6 (line 123, deployed-rate-excess interpretation) — partial; not the score-independence caveat directly |
+| Pixel-identical positive capture | Tautological (byte-identical ⇒ in HC region) | §V-H limit 4 (line 119, "pixel-identity is a conservative subset") |
+| LOOO firm-level reproducibility | Stability ≠ classification validity; K=3 membership ±12.8 pp | §V-H limit 8 (line 127, "K=3 hard-posterior membership is composition-sensitive") |
+
+**Gap 1**: §V-H does not explicitly flag the logistic-regression naive-SE caveat that Table XXVII row 5 discloses. Worth adding a half-sentence to §V-H or letting Table XXVII carry the disclosure since it's already in print.
+
+**Gap 2** (Opus N5 from round-2 audit): §V-H limit 2 discloses the firm-dependent within-firm violation numerically (98.8% at A; 76.7–83.7% at B/C/D) but does not interpret what this means for proxy reliability — namely that Firm A's per-firm ICCR is MORE contaminated by within-firm sharing than B/C/D's, so the per-firm B/C/D rates are closer to clean specificity. This nuance affects interpretation of the headline "firm heterogeneity is decisive" framing.
+
+## 8. Net-new narrative concerns (audit-surfaced)
+
+### Concern A — Phase 4 §I body line 31 cites "Script 39c" for jittered-dHash claim
+
+**Issue.** Phase 4 prose line 31 says:
+> "Within-firm signature-level cosine and jittered-dHash dip tests fail to reject in every individual Big-4 firm and in every individual mid/small firm with $\geq 500$ signatures (10 firms tested in Script 39c)."
+
+Codex round-9 verified that Script 39c on RAW dHash actually REJECTS unimodality in all 10 firms; only the JITTERED variant (codex's read-only spike on the Script 39c substrate) fails to reject. Round-5 corrected §III line 59 + provenance table line 382 to cite the spike attribution, but Phase 4 §I line 31's bare "Script 39c" citation now reads less precisely than the source-of-truth at §III line 59.
+
+**Severity.** Low. The qualitative claim ("fail to reject in 10 mid/small firms") is correct per codex's own rerun. The provenance attribution is what's slightly off.
+
+**Recommended fix.** Update Phase 4 line 31 to match §III line 59: "...in every individual mid/small firm with $\geq 500$ signatures (10 firms tested; cosine: Script 39c per-firm; jittered-dHash: codex-verified read-only spike on Script 39c substrate)." Or simpler: drop the parenthetical "in Script 39c" and let §III carry the precise provenance.
+
+### Concern B — Phase 4 §V-B line 81 carries the same jittered-dHash claim without provenance
+
+**Issue.** §V-B (line 81) says:
+> "Within-firm signature-level cosine and jittered-dHash dip tests fail to reject in every individual Big-4 firm and in every individual non-Big-4 firm with $\geq 500$ signatures (10 firms tested)."
+
+No script citation here; the bare "10 firms tested" is technically OK but less precise than §III line 59 after round-5.
+
+**Severity.** Low.
+
+**Recommended fix.** Add a §III cross-reference: "...10 firms tested; see §III-I.4 / §III provenance table line for the codex-verified read-only spike." Or leave as-is and let §III line 59 carry the detailed provenance.
+
+### Concern C — §III-K.4 line 149 stale cross-reference to v3.x §IV-I
+
+**Issue.** §III-K item 4 line 149 says:
+> "The corresponding signature-level inter-CPA negative-anchor ICCR evidence is developed in §III-L.1 (Big-4 sample) and the v3.x §IV-I corpus-wide version (reported under prior 'FAR' terminology)"
+
+After v4 §IV-I was shrunk to a 3-paragraph reframing stub (post round-3), the phrase "v3.x §IV-I corpus-wide version" is misleading — v4 §IV-I now exists, just as a pointer. Opus round-2 N6 flagged this; codex round-9 did not.
+
+**Severity.** Cosmetic.
+
+**Recommended fix.** Update line 149 to "§III-L.1 (Big-4 v4 sample) and the inherited corpus-wide v3.x version cited at §IV-I (reported under prior 'FAR' terminology)".
+
+### Concern D — Spearman precision mismatch §III vs §IV
+
+**Issue.** §III-K.1 line 123–127 reports Spearman ρ as 0.963 / 0.889 / 0.879 (3 decimal places); §IV-F Table IX line 81–87 reports 0.9627 / 0.8890 / 0.8794 (4 decimal places). Codex round-8 flagged this as OPEN / COPY-EDIT.
+
+**Severity.** Cosmetic.
+
+**Recommended fix.** Standardise on 4 decimal places (matches Script 38 reported precision) in §III + §IV + §V-E + §VI.
+
+## 9. Splice-readiness gate
+
+| Item | Status | Notes |
+|---|---|---|
+| Abstract word count | ✓ | 247 / 250 |
+| §I contributions count | ✓ | 8 contributions, all map to body |
+| §II LOOO addition with refs [42]-[44] | ✓ | Present (post round-1) |
+| §III sub-sections G..M complete | ✓ | Including Table XXVII numbered |
+| §IV table sequence V–XXVI sequential | ✓ | Post round-2 cascade |
+| §V sub-sections A..H complete | ✓ | Post round-2 M4 fix (G→H) |
+| §VI items 1..8 map to §I 1..8 | ✓ | 1:1 mapping verified |
+| References [1]–[44] present | ✓ | 44 entries; [42]-[44] = Stone 1974 / Geisser 1975 / Vehtari 2017 |
+| Internal draft notes stripped | ✗ | **Splice-time mechanical** — Phase 4 line 3 + lines 153-162; §III line 3 + lines 434-447; §IV line 3 + line 365+ |
+| "Nine-tool" / "Table XV-B" residue | ✗ | **Splice-time mechanical** — only in internal-strip text |
+| Cross-section number consistency | ✓ | All headline numbers match across Abstract / §I / §III / §IV |
+| Terminology consistency (ICCR / K=3 / less-replication-dominated) | ✓ | No public-prose leaks |
+| IEEE Access format | ✓ | Abstract single-paragraph ≤250 words; numbered references; numbered tables |
+
+## 10. Recommended round-6 narrative-consistency patch (15 min, optional)
+
+Before manuscript-splice, three small text patches would close the audit-surfaced concerns:
+
+1. **Concern A**: Phase 4 line 31 — narrow the "Script 39c" provenance attribution for the jittered-dHash claim to match §III line 59.
+2. **Concern C**: §III line 149 — update the "v3.x §IV-I corpus-wide version" wording to reflect v4 §IV-I's reframing-stub status.
+3. **Concern D**: Standardise Spearman precision to 4 decimal places across §III/§IV/§V.
+
+Optional:
+- **Gap 2 / Opus N5**: Add a half-sentence to §V-H limit 2 interpreting the firm-dependent within-firm violation as "Firm A's per-firm ICCR is more contaminated by within-firm sharing than Firms B/C/D's, so the per-firm B/C/D rates are closer to clean specificity than the pooled rate."
+
+None of these is empirical or structural. They are prose-level consistency polishings. **Submission can proceed without them**, but they would strengthen reviewer-pass robustness.
+
+## 11. Submission-readiness verdict
+
+**Conditionally ready.**
+
+The empirical core is sound and reproducible. The Phase 5 panel converged 3/3 in Accept/Minor band. Five fix rounds have closed every reviewer-flagged finding. Numbers are consistent across sections. Terminology is consistent. The v4 pivot's narrative thread reads as a coherent argument from Abstract through Conclusion.
+
+**Path A (ship now)**: proceed directly to manuscript-splice → DOCX export → partner Jimmy review → submission. Risk: the four audit-surfaced concerns above may be flagged by external reviewers as small cosmetic issues. Cost: zero pre-submission work.
+
+**Path B (15-min polish first)**: apply round-6 patches for the four concerns above → re-verify with `rg`-based grep → manuscript-splice → DOCX → partner. Risk: zero. Cost: 15 min.
+
+**Recommendation**: Path B. The four concerns are small, narrative-consistency only, and fixing them avoids putting cross-section attribution inconsistencies in front of partner Jimmy / IEEE Access reviewers.
+
+After Path B (or A): proceed to manuscript-splice as the next mechanical step.

paper/opus_review_v4_round1.md

@@ -0,0 +1,235 @@
+# Paper A Phase 5 Round 1 — Opus 4.7 max-effort independent review
+
+Reviewer: Claude Opus 4.7
+Date: 2026-05-14
+Target: paper/v4/paper_a_prose_v4_phase4.md + paper/v4/paper_a_methodology_v4_section_iii.md + paper/v4/paper_a_results_v4_section_iv.md
+Prior reviewer artifacts: paper/codex_review_gpt55_v4_round7.md (codex, Minor Revision); paper/gemini_review_v4_round1.md (Gemini 3.1 Pro, Minor Revision)
+
+## Verdict
+
+**Minor Revision (corroborates codex + Gemini on overall disposition), but I dissent on readiness.** The central empirical narrative — anchor-based multi-level ICCR calibration, the composition-decomposition demolition of distributional thresholds, the K=3-as-firm-compositional demotion, and the disclosed unsupervised-validation ceiling — is methodologically sound and survives my spot-checks. However, my independent pass surfaces three substantive blockers that both prior reviewers missed: (1) the headline "98–100% of inter-CPA collisions concentrated within the source firm" claim, repeated verbatim in the Abstract, §I contribution 6, §V-C, §V-G limitation 2, and §VI conclusion item 4, is **factually wrong for the deployed any-pair rule** at three of the four firms (Firms B/C/D within-firm rates are 77%/84%/77%, not 98–100%; the 98–100% range applies only to the stricter same-pair joint event); (2) §IV retains the demoted mechanism labels ("hand-leaning / mixed / replicated" in Table XVI columns, "hand-leaning rate" in Tables IX, X, XVII, XVIII, and prose at lines 102, 215, 226, 232, 254) that §III-J line 90 explicitly says are "replaced"; (3) §V has two sub-sections labelled "G" (line 105 "Pixel-Identity ..." and line 109 "Limitations"). I therefore corroborate Minor Revision on the empirical core but treat these three findings as additional copy-edit-or-rewrite blockers that must close before Phase 5 splice — finding (1) verges on empirical depending on whether the author chooses to rewrite or restrict scope.
+
+## Cross-reviewer agreement matrix
+
+| Theme | codex round-7 | Gemini round-1 | Opus round-1 |
+|---|---|---|---|
+| Overall disposition | Minor Revision | Minor Revision | Minor Revision |
+| Refs [42]-[44] present in `paper_a_references_v3.md` | Missed — claimed they were placeholders | Caught codex error | **Agree with Gemini.** Refs are at lines 87–91 of `paper/paper_a_references_v3.md` |
+| Partner "statistically insignificant" framing risk | Not flagged | Flagged as Major | **Agree with Gemini.** OR magnitudes 0.05/0.01/0.03 are decisive heterogeneity, not absence-of-effect |
+| Table XV-B → XIX renumbering | Not flagged | Flagged as Minor | **Agree with Gemini AND extend:** renumbering cascades — see Major M2 below |
+| Sample-size 150,442 vs 150,453 | Not flagged | Flagged as newly-introduced issue | **Partial agree:** the §IV-J line 177 footnote already reconciles this inline; the missing piece is a Table XV cross-pointer to §III-G |
+| Internal draft notes / checklists | Flagged | Flagged | **Agree.** Three draft-note blocks remain (Phase 4 prose, §III v7, §IV v3.3) plus three close-out checklists |
+| Three-score "feature-derived" caveat coverage | Closed | Closed | **Partial dissent.** Caveat is present at Abstract / §I / §III-K / §V-E, BUT §IV Tables IX/X/XVII/XVIII reintroduce mechanistic "hand-leaning" framing without the §III-K caveat |
+| K=3 demotion language consistency | Closed | Closed (line 25 says "Consistent descriptive framing") | **Dissent.** See Major M1 — §IV retains mechanistic labels throughout |
+| Cross-firm collision concentration "98–100%" | Closed implicitly | Verified VERIFIED in §IV-M.5 / §III-L.4 spot-check, but did not check whether prose claim matches the any-pair table | **Strong dissent.** See Major M3 |
+| Duplicate §V-G heading | Not flagged | Not flagged | **Both missed.** See Major M4 |
+
+## Major findings
+
+1. **§IV retains "hand-leaning / mixed / replicated" mechanism labels that §III-J line 90 explicitly demotes (both missed).**
+
+   - *Issue.* §III-I.4 establishes that the descriptor distributions contain no within-population bimodality; §III-J line 90 states: "The 'descriptive position' column **replaces** v3.x's 'hand-leaning / mixed / replicated' mechanism labels." §V-D, §I item 7, and §VI item 5 propagate the descriptive framing. **§IV does not.**
+   - *Where.*
+     - `paper_a_results_v4_section_iv.md` line 219 (Table XVI column headers): `C1 (hand-leaning) | C2 (mixed) | C3 (replicated)` — verbatim mechanism labels
+     - line 85 (Table IX): "K=3 P(C1) vs Paper A box-rule **hand-leaning** rate"
+     - line 86 (Table IX): "Reverse-anchor cosine percentile vs Paper A box-rule **hand-leaning** rate"
+     - line 93 (Table X): "mean Paper A **hand-leaning** rate"
+     - line 100: "more **hand-leaning** relative to the non-Big-4 reference"
+     - line 102: "the most-**hand-leaning** end of Big-4"; "more **hand-leaning**"; "ranks Firm D fractionally above Firm C"
+     - line 147 (Table XIV column): "Misclassified as **hand-leaning**"
+     - line 215 (§IV-J): "the per-CPA **hand-leaning** ranking"
+     - line 226 (Table XVI reading): "C3 **replicated** component"; "highest **hand-leaning** concentration of the Big-4"
+     - line 232 (§IV-K): "Paper A operational **hand-leaning** rate"
+     - line 238 (Table XVII): "C1 **hand-leaning**" / "C3 **replicated**"
+     - line 244 (Table XVIII title): "Paper A operational **hand-leaning** rate"
+     - line 246 (Table XVIII): "P(C1) vs Paper A **hand-leaning** rate"
+     - line 254 (§IV-K reading): "more non-templated CPAs"; "more mid/small-firm **hand-leaning** CPAs"
+   - *Reasoning.* §III-K explicitly renames Score 3 to "**inherited binary high-confidence box rule rate**" / "less-replication-dominated rate" (lines 119, 129), and §V-E uses "less-replication-dominated rate" (line 97). The §IV body silently retains v3.x mechanistic naming. A reader reaching §IV after §III-J's demotion will perceive that §IV has reverted to causal/mechanistic K=3 framing. This is the exact regression the v4 pivot is supposed to prevent.
+   - *Fix.* Global s/hand-leaning/less-replication-dominated/ in §IV; rename Table XVI columns to `C1 (low-cos / high-dHash)` / `C2 (central)` / `C3 (high-cos / low-dHash)` matching the §III-J Table 8 "descriptive position" column; rename Table XVII C1/C3 row labels and Table XVIII row labels likewise; rewrite the §IV-K Reading prose to use descriptor-position language ("CPAs whose descriptor mean sits further from the templated end of the descriptor plane").
+   - *Tag.* **Both missed.**
+
+2. **Table-numbering cascade if XV-B → XIX is finalised (Gemini missed the cascade; codex did not flag).**
+
+   - *Issue.* Gemini correctly recommended renumbering Table XV-B → Table XIX. But the §IV-M tables (§IV-M.1 through §IV-M.6) currently occupy Tables XIX, XX, XXI, XXII, XXIII, XXIV, XXV. Bumping XV-B → XIX forces a cascade renumber of all seven §IV-M tables to XX–XXVI. There is no acknowledgement of this cascade in either §IV-J close-out item 2 (line 370) or in Gemini's recommendation. Without addressing the cascade, fixing XV-B alone produces a duplicate Table XIX in the manuscript.
+   - *Where.* `paper_a_results_v4_section_iv.md` lines 192 (XV-B), 266 (XIX), 280 (XX), 300 (XXI), 317 (XXII), 329 (XXIII), 340 (XXIV), 353 (XXV).
+   - *Reasoning.* Sequential integer numbering is the IEEE Access norm. Either Table XV-B stays (and §IV-M tables stay XIX–XXV) or the rename cascades.
+   - *Fix.* If XV-B → XIX is preferred, also renumber §IV-M tables XIX→XX, XX→XXI, ..., XXV→XXVI; update all in-text references (§III provenance table at line 387 onward; §IV-J line 188 and 217 referencing "Table XVI"; §I and §V cross-references). Run a manuscript-wide grep on "Table XVI"…"Table XXVI" after the renumbering to catch stale internal pointers.
+   - *Tag.* **Gemini missed (cascade implication); codex missed entirely.**
+
+3. **The "98–100% of inter-CPA collisions concentrated within the source firm" claim is factually wrong for the deployed any-pair rule at three of four firms (both missed).**
+
+   - *Issue.* The Abstract (line 11), §I item 6 (line 53), §V-C (line 87), §V-G limitation 2 (line 115), and §VI conclusion item 4 (line 147) report "98–100% of inter-CPA collisions concentrated within the source firm". This range applies only to the **same-pair joint event** (a single inter-CPA candidate satisfying both cos>0.95 AND dHash≤5), which is the **stricter alternative** classifier explicitly contrasted to the deployed any-pair rule in §III-L.0 (line 183) and §III-L.4 (line 281).
+   - *Verification.* §IV-M Table XXIV (line 342) reports any-pair cross-firm hit counts. Computing within-firm fractions from Table XXIV:
+     - Firm A: 14,447 / 14,622 = 98.8%
+     - Firm B: 371 / 484 = 76.7%
+     - Firm C: 149 / 178 = 83.7%
+     - Firm D: 106 / 137 = 77.4%
+     - **Any-pair within-firm range: [76.7%, 98.8%]**, not [98%, 100%]. The 98–100% range corresponds to the same-pair rates (99.96/97.7/98.2/97.0%) stated separately at §III-L.4 line 281 and §IV-M.5 line 349.
+   - *Reasoning.* §III-L.0 makes a careful any-pair vs same-pair distinction and emphasises the deployed rule is **any-pair**. The Abstract / §I / §VI summarise the within-firm concentration using the same-pair number while attributing it to the deployed rule. §V-C line 87 is even worse: "98–100% of inter-CPA collisions originate from candidates within the source firm, **regardless of which Big-4 firm is the source**" — explicitly asserting the strong claim across all four firms, contradicted by Firms B/C/D any-pair rates of 77/84/77%.
+   - *Fix options.*
+     - **Option A (preferred):** Restate as "98% within-firm at Firm A, 77–84% at Firms B/C/D for the deployed any-pair rule; 97–100% within-firm under the stricter same-pair joint event across all four firms" in Abstract / §I / §V / §VI. This preserves the headline finding (within-firm dominance) while accurately characterising the gap between any-pair and same-pair.
+     - **Option B:** If the manuscript prefers the cleaner same-pair number, every occurrence must be re-attributed to "the same-pair joint event under the stricter alternative classifier (§III-L.4)", not to "the deployed rule" / "inter-CPA collisions". The current Abstract / §I framing ties the number to "inter-CPA collisions" without qualifier, which reads as applying to the deployed rule.
+   - *Severity.* This is the headline forensic finding in the Abstract and §I. Mis-stating it across five locations is more than a copy-edit — partners and reviewers form their understanding of "what the paper shows" from the Abstract.
+   - *Tag.* **Both missed.** Gemini's provenance spot-check #3 verified the byte-identical split (145/8/107/2) but did not verify whether the 98–100% claim in the Abstract matches the deployed-rule numbers in Table XXIV.
+
+4. **Duplicate §V-G heading (both missed).**
+
+   - *Issue.* `paper_a_prose_v4_phase4.md` has two §V-G headings: line 105 "G. Pixel-Identity as a Hard Positive Anchor; Inherited Inter-CPA Negative Anchor Reframed as Coincidence Rate" and line 109 "G. Limitations". The second should be "H. Limitations".
+   - *Where.* `paper_a_prose_v4_phase4.md` lines 105, 109. Also: line 37 (§I) cross-references "§V-G" for the conservative-subset caveat — ambiguous given two G sections; the content actually lives in BOTH (line 105 has "we caution that this result is necessary but not sufficient: for the box rule it is close to tautological", and line 119 has the "Pixel-identity is a conservative subset" bullet).
+   - *Reasoning.* §V originally went A through F in the v3 draft. The Phase 4 prose added a new §V-G ("Pixel-Identity ...") between §V-F and the existing §V-G Limitations without renaming the latter. Both prior reviewers spot-checked §V-G Limitations content for completeness (codex M9, Gemini codex-closure check #3) and confirmed the inherited v3.20.0 limitations are restored — but neither noticed the duplicate letter.
+   - *Fix.* Rename the second to §V-H ("H. Limitations"). Update every internal §V-G cross-reference: line 37, line 111 ("inherited from v3.20.0 §V-G"), line 160 (close-out checklist), and any §III or §IV pointer.
+   - *Tag.* **Both missed.**
+
+5. **Stale "seven limitations" assertion in close-out checklist (both missed).**
+
+   - *Issue.* Phase 4 close-out item 4 (line 160) says "The seven limitations are listed flat". Actual count in the §V-G Limitations section is **14** (9 v4.0-specific + 5 inherited from v3.20.0). §V-G line 111 correctly says "The first nine are v4.0-specific; the last five are inherited". The checklist item is stale from a previous version where the v4.0-specific count was 2.
+   - *Fix.* Either update to "The fourteen limitations are listed flat" or remove the checklist on Phase 5 splice (already on Gemini's list under m1 and codex's list under A8).
+   - *Tag.* **Both missed (a small instance of the broader internal-notes cleanup).**
+
+6. **§I contribution 4 and §IV-D both refer to a "2×2 factorial diagnostic" but only §IV-M Table XIX summarises it; §IV-D's body does not reproduce the 2×2 factorial table from §III-I.4 (codex implicitly covered; Gemini partly covered).**
+
+   - *Issue.* §IV-D line 23 says "the v4-new composition-decomposition diagnostics that establish this finding are tabulated in §IV-M below". The reader expects to find the §III-I.4 factorial table (4 rows: raw / centred-only / jittered-only / both) reproduced in §IV-M. But §IV-M Table XIX only summarises three of the four rows (collapses the "centred-only" and "jittered-only" cells into descriptive prose) and omits the "raw" baseline row's $p$-value comparison. Readers comparing §III-I.4 line 67 to §IV-M Table XIX will see a partial reproduction.
+   - *Severity.* Cosmetic — the same content is in §III-I.4 in full. But the §IV reader is implicitly told §IV-M is the destination for the §III-I.4 evidence, and §IV-M is not as complete as §III-I.4.
+   - *Fix.* Either reproduce the full 4-row 2×2 factorial table in §IV-M (currently Table XIX), or change §IV-D line 23 to "see §III-I.4 Table for the full factorial; §IV-M Table XIX summarises the key diagnostic outcomes".
+   - *Tag.* Both implicitly missed.
+
+7. **Inconsistent precision in three-score Spearman across §III-K and §IV-F (Gemini implicitly missed).**
+
+   - *Issue.* §III-K Table at line 125–127 reports three-score pairwise Spearman as +0.963, +0.889, +0.879 (3 decimal places). §IV-F Table IX at lines 85–87 reports the same correlations as +0.9627, +0.8890, +0.8794 (4 decimal places). §III provenance table line 365 uses 3 decimals.
+   - *Severity.* Low — the values are consistent at 3-dp rounding. But mixed precision across §III table / §IV table / Abstract is a copy-edit signal.
+   - *Fix.* Standardise on 4 decimal places everywhere (the §IV precision matches Script 38's output more faithfully) and update §III-K Table + §III provenance row + the Phase 4 abstract / §I body to match. Alternatively, standardise on the lower-bound floor "ρ ≥ 0.879" that the Abstract uses.
+   - *Tag.* Both missed (low priority).
+
+## Minor findings
+
+1. **Stale "approximately 235 words" / "243-244 words" abstract counts in two checklists.** Phase 4 close-out item 1 (line 157) says "Current draft is 243–244 words"; codex round-7 verified `wc -w` returns 243. Both are within the IEEE Access 250-word budget. Just remove the abstract-word-count notes before splice. (Gemini m3.)
+
+2. **Three "internal — remove before submission" draft notes still present.** Phase 4 line 3, §III v7 line 3 (lines 1–5), §IV v3.3 line 3. Plus two close-out checklists (§IV lines 365–375, Phase 4 lines 153–162) and §III's cross-reference index (lines 431–448). All flagged by codex (A8) and Gemini (m1). I corroborate.
+
+3. **§IV-J Table XV-B header pointer.** §IV-J line 228 "Document-level worst-case aggregation outputs are reported in Table XV-B above." — fine prose but does not specify which table number is the document-level table when XV-B → XIX renumber occurs. Once Major M2 is resolved, update accordingly.
+
+4. **Mixed pp vs decimal vs percentage notation across Abstract / §I / §III-L / §IV-M for the same numerical claims.** Examples:
+   - Abstract line 11 uses "0.34 for the operational HC+MC alarm"
+   - §I line 33 uses "33.75% of Big-4 documents"
+   - §III-L.3 Table reports "0.3375 [Wilson 95% [0.3342, 0.3409]]"
+   - §IV-M Table XXII reports "0.3375"
+   - Phase 4 conclusion line 147 uses "(0.34 for the operational HC+MC alarm)"
+   - Standardise on either decimal or percentage throughout numeric quotations; the lossy "0.34" in the Abstract drops the 95% CI which is reported elsewhere. The IEEE Access norm in this corpus is decimal-with-CI in tables, percentage-without-CI in prose summary.
+
+5. **"v3.x §IV-F.1" cross-reference is not validated against the v3 file's actual section letter.** §III-H line 37 and §V-C line 85 both cite "v3.x §IV-F.1" for the 145 pixel-identical signatures across ~50 distinct Firm A partners. If `paper_a_results_v3.md` uses a different sub-section letter (e.g., §IV-G or §IV-F-1), the citation will be stale at splice time. I did not verify this but flag it as a splice-time check.
+
+6. **"~50 distinct partners of 180" vs §III-H "50 distinct Firm A partners of 180 registered" — Firm A registered partner count of 180 is inherited from v3.20.0 / Script 28 with no in-paper provenance verification.** This is flagged in §III-H line 37 ("inherited from v3 §IV-F.1 / Script 28 / Appendix B byte-decomposition output and were not regenerated in v4.0 spike scripts") — adequately disclosed. Just confirm at splice that v3.20.0's count of 180 is reproducible.
+
+7. **§I.5 contribution 5 line 51 claim "We adopt 'inter-CPA coincidence rate' as the metric name throughout and reserve 'False Acceptance Rate' for terminology that requires ground-truth negative labels".** §IV-I line 159 prose uses "False Acceptance Rate (FAR)" in the historical-context phrase "previously reported as 'False Acceptance Rate' in v3.x" — this is intentional historical reference, but it does mean "FAR" appears once in §IV body, which contradicts the §I claim of "throughout". Either weaken §I.5 to "throughout v4.0 framing, with one historical-reference exception in §IV-I" or strip "FAR" from §IV-I line 159 and reword to "previously reported using biometric-verification 'False Acceptance Rate' terminology".
+
+8. **MC band per-firm proportions in §IV-J line 215.** "10.76% / 35.88% / 41.44% / 29.33% across Firms A through D". Cross-checking against Table XV per-firm breakdown line 183–186: Firm A MC = 10.76%, Firm B MC = 35.88%, Firm C MC = 41.44%, Firm D MC = 29.33%. Consistent ✓.
+
+9. **§V-G limitation 1 list says "first nine are v4.0-specific" but Phase 4 close-out item 4 calls them "seven limitations".** Already covered under Major M5.
+
+10. **Provenance-table cross-references inside the §III provenance table.** Line 364: "K=3 LOOO held-out C1 absolute differences 1.8–12.8 pp | direct | Script 37 held-out prediction check". Cross-checked against §IV-G Table XIII (line 134–137): the four held-out absolute differences are 4.68, 1.76, 12.77, 5.81 → range 1.76 to 12.77, which rounds to "1.8–12.8" ✓.
+
+11. **Abstract line 11 phrasing "98–100% of inter-CPA collisions concentrated within the source firm — consistent with firm-level template-like reuse"** —  this is the most-public statement of Major M3. The Abstract is the only thing many readers will read; the imprecision is more costly here than in §V-G.
+
+## Provenance spot-checks
+
+I deliberately chose 5 claims NOT covered by Gemini's spot-checks.
+
+1. **Within-source-firm any-pair collision rates (the "98–100%" claim).**
+   - *Claim text.* "with $98$–$100\%$ of inter-CPA collisions concentrated within the source firm" (Abstract line 11, §I line 53, §V-C line 87, §V-G line 115, §VI line 147).
+   - *Manuscript location of evidence.* Table XXIV in §IV-M.5 (line 342); §III-L.4 cross-firm hit matrix (line 274–281).
+   - *Cited script.* Script 44.
+   - *Verdict.* **INCONSISTENT.** Computed from Table XXIV: any-pair within-firm fractions are 98.8% / 76.7% / 83.7% / 77.4% (range 76.7–98.8%, not 98–100%). The 98–100% range corresponds to the same-pair joint event explicitly reported as the **stricter alternative** at §III-L.4 line 281 and Table XXIV line 349 (99.96% / 97.7% / 98.2% / 97.0%). The narrative-level claim conflates two distinct rule semantics. The script logic (44_firm_matched_pool_regression.py lines 274–327) does compute both matrices; the manuscript reports both correctly inside §III-L.4 / §IV-M.5; the failure is in the Abstract / §I / §V-C / §V-G / §VI narrative-summary layer. See Major M3.
+
+2. **Per-pair conditional ICCR dHash≤5 given cos>0.95 = 0.234 (Wilson [0.190, 0.285], 70 of 299 pairs).**
+   - *Claim text.* §III-L.1 line 208; Phase 4 §V-F line 103.
+   - *Cited script.* Script 40b.
+   - *Verdict.* **VERIFIED for logic; numerical value not independently verifiable from worktree.** Script 40b at lines 22–23 explicitly computes "Conditional FAR(dh<=k | cos>0.95)" for k∈[0, 20] and prints "P(dh<=k | cos>0.95)" for each k (lines 226–229 of the script). The 0.234 / 70 / 299 numbers cannot be checked without access to `/Volumes/NV2/PDF-Processing/signature-analysis/reports/v4_big4/inter_cpa_far_sweep/` (not in the worktree). Wilson [0.190, 0.285] is a plausible 95% CI for $\hat{p} = 70/299 = 0.2341$ (manual check: SE ≈ $\sqrt{0.234 \cdot 0.766/299} = 0.0245$, normal-approx 95% CI [0.186, 0.282] — Wilson CIs are slightly wider on each side, so [0.190, 0.285] is consistent). Logic + analytic plausibility verified.
+
+3. **Pooled Big-4 per-signature any-pair ICCR 0.1102 with Wilson [0.1086, 0.1118].**
+   - *Claim text.* §III-L.2 Table at line 220; Phase 4 §V-F line 101.
+   - *Cited script.* Script 43.
+   - *Verdict.* **VERIFIED analytically.** For $\hat{p} = 0.1102$ on $n_{\text{sig}} = 150{,}453$, the Wilson-approximate 95% CI half-width is $\approx 1.96 \cdot \sqrt{0.1102 \cdot 0.8898 / 150453} = 0.00159$, giving [0.1086, 0.1118] — exactly the reported interval. CPA-block bootstrap [0.0908, 0.1330] is much wider, as expected once CPA-level clustering is recognised (CIs widen by factor of $\sim 9\times$ here), consistent with the strong within-CPA correlation that the manuscript discusses.
+
+4. **Logistic OR pool-size effect = 4.01.**
+   - *Claim text.* §III-L.4 Table line 266; §IV-M.5 Table XXIII line 336.
+   - *Cited script.* Script 44.
+   - *Verdict.* **VERIFIED for logic; magnitude is intuitive.** Script 44 at lines 219–227 fits `logistic(hit) ~ intercept + FirmB + FirmC + FirmD + log(pool_size_centered)` and reports `OR = exp(beta)`. A 4× per-log-unit pool-size effect means each $e\times$ pool-size increase quadruples the per-signature odds — consistent with the §III-L.2 decile trend (decile 1 = 0.0249, decile 10 = 0.1905, ratio 7.65×, pool ratio approximately $1115/201 \approx 5.5$, log-ratio $\approx 1.71$, predicted OR $\approx 4.01^{1.71} = 10.7$ vs observed 0.1905/0.0249 = 7.65; the gap is small enough to be consistent with monotonicity + finite-sample). No red flag.
+
+5. **Alert-rate sensitivity local-gradient ratios (cos=0.95: ≈25×; dHash=5: ≈3.8×; dHash=15: ≈0.08).**
+   - *Claim text.* §III-L.5 line 291; §IV-M.6 Table XXV line 357–359.
+   - *Cited script.* Script 46.
+   - *Verdict.* **PARTIALLY VERIFIABLE.** Script 46 at lines 236–276 explicitly computes the local-to-median gradient ratio at the v3-inherited threshold (cos=0.95 and dh=5) and emits `ratio_local_to_median` for both. **However**, the script does NOT emit a dh=15 ratio: `DH_GRID = np.arange(0, 21, 1)` covers 15 (so the swept rates ARE available in the saved JSON), but the script only computes the ratio at dh=5. The ≈0.08 ratio at dh=15 in Table XXV must therefore be a **derived quantity** computed by the author from the JSON output post-hoc, not a direct script output. This is not a fabrication risk (the author has access to the swept rates), but the manuscript should either (a) add a brief in-paper note that the dh=15 ratio is computed identically to the dh=5 ratio but post-hoc on the JSON, or (b) extend Script 46's plateau-detection block to emit ratios at both dh=5 and dh=15.
+
+   *Secondary note.* The cosine sweep prose at §III-L.5 line 291 quotes "cosine sweep at dHash ≤ 5 yields rates of 0.5091 at cos > 0.945 vs 0.4789 at cos > 0.955". From Script 46 line 56–57: `COS_FOR_2D = np.arange(0.85, 1.00, 0.01)` only covers integer hundredths, so 0.945 and 0.955 are NOT on the 2D grid. The 1D `COS_GRID` covers the sweep at which 0.945 / 0.955 may be in the grid; I cannot fully verify without seeing the saved JSON. The numerical specificity is suspiciously precise (four decimals on a derived rate) — flag for partner spot-check at splice.
+
+## Newly introduced issues
+
+1. **The "specificity-proxy-anchored screening framework with human-in-the-loop review" positioning is consistent across Abstract / §I / §III-M / §V-G / §VI — except that the term "human-in-the-loop" appears NOWHERE in §III or §IV, only in §I line 30 (item 5), §III-L.3 line 255, §III-M line 334, and §VI line 147.** §I item 5 promises "anchor-based threshold calibration at three units of analysis ... against an inter-CPA negative-anchor coincidence-rate proxy", and §III-M closes with "positioning as a specificity-proxy-anchored screening framework with human-in-the-loop review". But §IV's results discussion never operationalises what "human-in-the-loop review" means: the per-document HC+MC alarm rate of 0.34 is reported as a number; whether the human reviewer triages all 34% of flagged docs or only a sub-set is not specified. Newly introduced v4-positional claim that v3.20.0 did not make, lacking a concrete operationalisation. Either explicitly punt to future work or describe the intended triage workflow briefly in §V or §VI.
+
+2. **The "feature-derived" qualifier coverage IS reliable across §III-K, §V-E, Abstract, §I, §VI — but breaks down in §IV.** Major M1 above. Newly introduced in v4 because v3.20.0 used "three independent scores" without the caveat; v4 added the caveat in §III/§V/§I/§VI but not §IV.
+
+3. **The "9-tool nine-diagnostic" validation collection (§III-M Table at lines 318–329) and §VI item 8 nine-tool claim — count is internally consistent (9 rows in the §III-M table), but the table is currently unnumbered.** All other tables in §III/§IV have a Table N: header. The §III-M validation table should be Table XXVI (or whatever number after the §IV-M cascade) for IEEE Access. Otherwise it is an unnumbered table inside a discussion of "Table-numbering scheme" elsewhere — inconsistent.
+
+4. **The §I cross-reference list at line 29 ("(1) signature page identification...; (2) signature region detection...; ... (8) a multi-tool unsupervised validation strategy") promises 8 pipeline steps in numbered order, but the actual pipeline order in §III is steps 1–5 are the embedding pipeline and 6–8 are validation framework / decomposition / framework positioning, not strict pipeline stages.** Cosmetic; the v3.20.0 contribution list had 7 sequential pipeline-stages; the v4.0 list collapses pipeline + validation into 8 items, blurring stage vs framework. Either say "Eight elements" rather than imply 8 pipeline steps, or split into "Pipeline (1–4): ...; Calibration and validation framework (5–8): ...".
+
+## Disagreements with prior reviewers
+
+1. **I disagree with codex round-7 M6 (closed) "§II citation-number gap and placeholder contradiction."** codex round-7 says "[42]-[44] remain placeholders and absent from the reference list." Gemini correctly identified that codex was wrong: [42]–[44] **are** present at lines 87–91 of `paper_a_references_v3.md`. I corroborate Gemini's correction. The only residual is the `[add citation]` placeholder string in Phase 4 close-out note line 159 (just a stale comment in a checklist that will be stripped at splice). codex's claim was based on the file ending at [41] in some earlier snapshot; the current file ends at [44].
+
+2. **I partially disagree with Gemini's finding #4 "K=3 Demotion Language Consistency" (verdict: "CLOSED. Consistent descriptive framing.").** §III, §I, §V, §VI properly demote K=3, but §IV (which Gemini did not separately enumerate for this check) retains mechanism labels throughout (Major M1 above). The K=3 demotion is **partially closed**, not fully closed.
+
+3. **I corroborate Gemini's finding #1 on "statistically insignificant" framing risk** (Major in Gemini's review, missed by codex). The OR magnitudes are 0.05/0.01/0.03 — these are 19×/100×/37× effects after pool-size adjustment, with standard errors that the manuscript flags as needing cluster-robust treatment but which are nonetheless an order of magnitude below 1. There is no defensible reading under which this is "statistically insignificant"; any such partner framing must be rejected.
+
+4. **Neither codex nor Gemini examined the cross-firm hit matrix arithmetic** to verify that the Abstract's "98–100%" claim is consistent with Table XXIV. Major M3 is my net-new finding.
+
+5. **Neither codex nor Gemini flagged the duplicate §V-G heading.** Major M4 is my net-new finding.
+
+6. **Neither codex nor Gemini flagged the §IV vs §III terminology drift around "hand-leaning / replicated".** Major M1 is my net-new finding.
+
+7. **codex M1 (Abstract independent-score correction closed) — I corroborate the closure for the Abstract** (line 11 says "Three feature-derived scores"), but extend the finding: the same correction is NOT propagated to §IV Tables IX/X/XVII/XVIII where the "Paper A box-rule hand-leaning rate" label retains mechanistic framing. This is Major M1 again.
+
+## Phase 5 readiness
+
+**Partial — closer to "blocked on copy-edit pass" than to "ready".** Three major issues require manuscript-text rewrites before Phase 5 splice:
+
+- **Empirical-language blocker:** Major M3 (98–100% within-firm claim) requires Abstract / §I / §V-C / §V-G / §VI prose rewrite. This is more than copy-edit — the headline finding's numerical scope must be reconciled with Table XXIV.
+- **Terminology blocker:** Major M1 (§IV mechanism labels) requires global s/hand-leaning/less-replication-dominated/ across §IV tables and prose plus Table XVI column-header rename.
+- **Structural blocker:** Major M4 (duplicate §V-G heading) requires §V renumber.
+- **Numbering blocker:** Major M2 (table-numbering cascade) requires either keeping XV-B suffix or cascading XIX→XX→...→XXVI across §IV-M and updating every cross-reference.
+
+The empirical core (scripts 32–46, ICCR multi-level calibration, composition decomposition, three-score convergence, byte-identical anchor, logistic regression) is sound and reproducible from script inspection. No new statistical work is required for Phase 5.
+
+## Recommended next-step actions
+
+Ranked by severity, distinguishing prose-rewrite blockers from copy-edit-only items.
+
+1. **[Manuscript rewrite — prose blocker]** Reconcile the "98–100% within-firm" claim with Table XXIV. Adopt Major M3 Option A or Option B in Abstract / §I item 6 / §V-C / §V-G limitation 2 / §VI item 4. Preferred phrasing (Option A): "98% of inter-CPA collisions at Firm A and 77–84% at Firms B/C/D under the deployed any-pair rule, rising to 97–100% across all four firms under the stricter same-pair joint event".
+
+2. **[Manuscript rewrite — terminology blocker]** Global s/hand-leaning/less-replication-dominated/ across §IV (Table IX/X column labels, Table XIV column, Table XVI columns, Table XVII rows, Table XVIII title, §IV-J/K/M prose). Rename Table XVI columns to match §IV-E Table VIII's "descriptive position" column. Keep §I, §V, §VI as currently worded.
+
+3. **[Manuscript rewrite — structural blocker]** Rename §V's second "G. Limitations" to "H. Limitations". Update every §V-G cross-reference.
+
+4. **[Manuscript rewrite — partner framing]** Explicitly state at §V-C or §V-G that firm heterogeneity is highly statistically significant (OR magnitudes 19×, 100×, 37× after pool-size adjustment with $z$-equivalent ≥ 10 in absolute value on standard SE; cluster-robust SE is flagged as a robustness check, not as the reason heterogeneity might be insignificant). Reject any reframing as "statistically insignificant". (Gemini Major #1.)
+
+5. **[Copy-edit blocker — table renumbering]** Decide on the XV-B suffix. If renamed to XIX, cascade through §IV-M XIX→XX, XX→XXI, …, XXV→XXVI. Update §III provenance table cross-references. Update §IV-J line 188 / 217 / 228 references. Update §IV-L line 258 to disambiguate from v3.20.0 Table XVIII.
+
+6. **[Copy-edit blocker — strip internal artefacts]** Remove:
+   - Phase 4 prose draft note (line 3) and close-out checklist (lines 153–162)
+   - §III v7 draft note (lines 1–5) and cross-reference index + open-questions block (lines 431–448)
+   - §IV v3.3 draft note (line 3) and close-out checklist (lines 365–375)
+
+7. **[Copy-edit — terminology consistency]** Decide on decimal vs percentage notation for the headline ICCRs (0.34 vs 33.75% etc.) and apply uniformly across Abstract / §I / §V / §VI.
+
+8. **[Copy-edit — Spearman precision]** Standardise three-score correlations on either 3-dp (0.963/0.889/0.879) or 4-dp (0.9627/0.8890/0.8794) across §III-K / §IV-F / §III provenance.
+
+9. **[Copy-edit — §III-M validation table]** Assign a table number (Table XXVI after the §IV-M cascade) to the nine-tool validation table at §III-M lines 318–329.
+
+10. **[Copy-edit — §V-G item count]** Update Phase 4 close-out item 4 "seven limitations" to "fourteen limitations" (or strip the checklist).
+
+11. **[Splice-time check]** Verify v3.20.0 §IV-F.1 cross-reference letters resolve correctly when §III-H / §V-C are spliced into the master file. Verify v3.20.0 Table XVIII (backbone ablation) does not collide with v4 Table XVIII (Spearman drift) once the final renumbering is applied. Verify all "v3.x" pointers point to actual sections that survive in the master manuscript.
+
+12. **[Minor — script 46 dh=15 ratio]** Either add a brief note in §IV-M that the dh=15 plateau ratio is computed post-hoc from the swept JSON (the swept rates themselves are emitted by Script 46), or extend Script 46 to compute the ratio at dh=15 directly.
+
+13. **[Minor — pipeline-step vs framework-element framing]** Reword §I line 29's "(1)…(8)" enumeration to distinguish pipeline stages (1–4) from validation/framework elements (5–8).
+
+14. **[Minor — human-in-the-loop operationalisation]** Add one sentence in §V or §VI describing what "human-in-the-loop review" means operationally (e.g., manual inspection of the 0.34 flagged-document fraction; sampling strategy; reviewer-effort estimate).

paper/opus_review_v4_round2.md

@@ -0,0 +1,233 @@
+# Paper A Phase 5 Round 2 — Opus 4.7 max-effort independent review
+
+Reviewer: Claude Opus 4.7
+Date: 2026-05-14
+Target: paper/v4/paper_a_prose_v4_phase4.md + paper/v4/paper_a_methodology_v4_section_iii.md + paper/v4/paper_a_results_v4_section_iv.md (post round-2 + round-3, commit 4a6f9c5)
+Prior reviewer artifacts: paper/codex_review_gpt55_v4_round7.md; paper/codex_review_gpt55_v4_round8.md; paper/gemini_review_v4_round1.md; paper/opus_review_v4_round1.md
+
+## Verdict
+
+**Minor Revision (corroborates codex round-8).** The empirical core is sound and reproducible. Round-2 (b884d39) closed my round-1 M1–M4 cleanly, and round-3 (4a6f9c5) closed codex round-8's three concrete splice blockers (abstract trim 247 w; §IV-J n=150,442 vs 150,453 footnote correctly distinguishing descriptor-complete vs vector-complete; §IV-I "§IV-M Tables XXI-XXVI" replacing the stale "Table XVI"). I dissent on **readiness for splice**: a fresh round-2 pass surfaces three net-new findings that the panel collectively missed — a denominator inconsistency between §IV-J Table XIX and §IV-M.4 per-firm doc counts (379-doc mixed-firm-PDF mode-of-firms tie-break), the absence of the composition-decomposition diagnostic from the §III-M nine-tool validation table that anchors the v4 narrative, and the unnumbered status of the §III-M table itself. None of the three is empirical-blocker grade; they are substantive copy-edit / structural fixes that should be patched in a single round-4 pass before splice.
+
+I align with codex round-8 on the disposition and on M1-M4 closure judgments. I do NOT corroborate Gemini round-2 directly (parallel; not read).
+
+## Cross-reviewer convergence summary
+
+Post-round-3 (4a6f9c5):
+
+| Theme | codex r8 | Opus r2 (this) |
+|---|---|---|
+| Overall disposition | Minor Revision | Minor Revision (corroborate) |
+| Opus M1 K=3 mechanism-label reversion | CLOSED (Table XI residue → fixed in r3 to "replication-dominated vs less-replication-dominated") | **CLOSED.** Verified by grep: `hand-leaning` returns 0 matches in §IV body; the only 2 §III matches are internal-checklist open-question text (line 445) and an unrelated $\Delta$BIC line, both stripped at splice. |
+| Opus M2 Table XV-B cascade | CLOSED in public body | **CLOSED.** Tables XV→XIX, §IV-M cascades XX-XXVI correctly. Only "XV-B" residue is in internal draft notes (lines 3, 370) that strip at splice. |
+| Opus M3 within-firm any-pair vs same-pair | CLOSED in body; abstract uses rounded any-pair 77-99% | **CLOSED.** Abstract line 11 "77-99%" rounds the deployed-rule any-pair range (76.7-98.8%). §I item 6 (line 53), §V-C (line 87), §V-H limitation 2 (line 115), §VI item 4 (line 147), §VI future work (line 149) all give the correct any-pair 76.7-83.7% / 98.8% split plus same-pair 97.0-99.96% subrange. |
+| Opus M4 duplicate §V-G | CLOSED | **CLOSED.** §V headings now run A-H sequentially (lines 73/77/83/89/95/99/105/109). |
+| Gemini Table XV sample-size footnote | CLOSED in r3 | **CLOSED.** §IV-J line 177 footnote now correctly groups §IV-M.2/M.3/M.5 (Scripts 40b/43/44) as vector-complete 150,453. |
+| Codex r8 splice blockers | r8 status open; r3 fixed | **CLOSED.** Abstract 247 w (verified `wc -w` on line 11); §IV-I (line 161) now points to "§IV-M Tables XXI-XXVI"; binary-collapse label is "replication-dominated vs less-replication-dominated" (§III line 131, §IV Table XI line 104). |
+| Internal draft notes | OPEN (splice-strip pending) | **OPEN.** Three draft-note blocks + three close-out checklists + §III cross-reference index + §III open-questions block still present and must strip at splice. |
+
+## M1–M4 closure verification (full audit)
+
+### M1 — §IV K=3 mechanism-label reversion: CLOSED
+
+Provenance grep `hand-leaning` returns only 2 matches in §III, both non-substantive:
+- `paper_a_methodology_v4_section_iii.md:90` — false positive ("lower than $K{=}2$ by $3.48$", the term "leaning" never appears).
+- `paper_a_methodology_v4_section_iii.md:445` — Open-question item 2 in the §III internal checklist ("Firm C is the firm most concentrated in C1 hand-leaning at 23.5%"). This is in the **author working notes** at lines 441-447, scheduled for splice-strip per Gemini m1 / codex Opus minor 2.
+
+§IV body is entirely cleansed: Table IX (line 85-87) "less-replication-dominated rate"; Table X (line 93) "mean Paper A less-replication-dominated rate"; Table XI (line 104) "binary collapse, replication-dominated vs less-replication-dominated" (round-3 fix); Table XIV (line 147) "Misclassified as less-replication-dominated"; Table XVI (line 219) "C1 (low-cos / high-dHash) | C2 (central) | C3 (high-cos / low-dHash)"; Table XVII (line 238) "C1 (low-cos / high-dHash)" / "C3 (high-cos / low-dHash)"; Table XVIII (line 244-246) "Paper A operational less-replication-dominated rate"; §IV-F prose (line 102) "the most replication-dominated"; §IV-K reading (line 254) "non-templated CPAs". All match §III-J line 90's intent.
+
+**One residual concern (low priority):** the K=3 LOOO upstream Script 37 report file at `/Volumes/NV2/PDF-Processing/signature-analysis/reports/v4_big4/k3_loo_check/k3_loo_report.md` still uses "C1 hand-leaning / C2 mixed / C3 replicated" labels (lines 7-9). This is in the empirical pipeline output, not the manuscript; it doesn't affect the published paper but does mean a reviewer following the data trail will see legacy labels. Worth noting in author working notes that the script report file naming convention has not been updated.
+
+### M2 — Table-numbering cascade: CLOSED
+
+Verified the cascade: §IV-J Table XV (line 167, five-way per-sig), Table XVI (line 217, K=3 cross-tab), Table XVII (line 234, full-vs-Big-4 K=3 drift), Table XVIII (line 244, Spearman full-vs-Big-4), Table XIX (line 192, document-level worst-case), Table XX (line 266, composition decomposition), Table XXI (line 280, per-comparison ICCR), Table XXII (line 300, pool-normalised per-sig ICCR), Table XXIII (line 317, document-level ICCR), Table XXIV (line 329, logistic regression), Table XXV (line 340, cross-firm hit matrix), Table XXVI (line 353, alert-rate sensitivity).
+
+In-text §IV cross-references all consistent:
+- §IV-D line 23 "tabulated in §IV-M below" — non-specific, OK.
+- §IV-I line 161 (round-3 fix) "§IV-M Tables XXI-XXVI" — correct.
+- §IV-J line 188 "qualitatively aligns with the K=3 cluster cross-tab of Table XVI" — correct.
+- §IV-J line 228 "Document-level worst-case aggregation outputs are reported in Table XIX above" — correct.
+
+§III provenance table (lines 386-427) does not by-number reference §IV tables, only by §III subsection / Script — robust to the cascade.
+
+Phase 4 prose: no Table-XV-X references in §V or §VI; safe.
+
+Only stale "Table XV-B" residue lives in the §IV draft-note (line 3) and close-out checklist (line 370), strip-at-splice items.
+
+### M3 — Within-firm collision semantic conflation: CLOSED
+
+The Abstract (line 11) uses the rounded any-pair-only range "77-99% of inter-CPA collisions concentrate within the source firm — consistent with firm-level template-like reuse". 76.7-98.8% rounds to 77-99% (lossy at the boundary but defensible as a 2-significant-figure summary).
+
+I verified by grepping "98-100\\|98–100" — zero matches in §III/§IV/Phase 4. The legacy framing is fully removed. The corrected pattern (any-pair + same-pair subrange disclosure) appears at:
+- §III-J line 99: "within-firm collision concentration is $98.8\\%$ at Firm A and $76.7$–$83.7\\%$ at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–$99.96\\%$ within-firm across all four firms)"
+- §III-L.4 line 283: same body text in primary methodology location.
+- §V-C line 87 (Phase 4): same pattern + the v3.x byte-level evidence cross-reference.
+- §V-H limitation 2 line 115 (Phase 4): correctly attributes the partial-violation framing.
+- §VI item 4 line 147 (Phase 4): same pattern.
+- §VI future work line 149: "(any-pair $76.7$–$98.8\\%$ across Big-4; same-pair joint $97.0$–$99.96\\%$)" — full precision.
+
+§I contribution 6 (line 53) also gives the any-pair 98.8% / 76.7-83.7% + same-pair 97.0-99.96% split.
+
+### M4 — Duplicate §V-G heading: CLOSED
+
+`grep -n '^## [A-Z]\\.'` on Phase 4 prose shows §V sub-sections A-H in correct order: A (line 73), B (77), C (83), D (89), E (95), F (99), G (105 "Pixel-Identity..."), H (109 "Limitations"). One cross-reference to §V-G in the close-out checklist (line 160) is internal-strip-at-splice; §V-H line 111 "inherited from v3.20.0 §V-G" correctly cites the original v3.x letter.
+
+## Net-new findings (fresh look post-round-3)
+
+### N1. Denominator inconsistency between §IV-J Table XIX per-firm document counts and §IV-M.4 per-firm D2 counts — **discoverable in 30s by a careful reader**.
+
+*Issue.* §IV-J Table XIX (lines 207-211) reports per-firm document-level breakdown **"single-firm PDFs only; mixed-firm PDFs $n = 379$ excluded"**: Firm A 30,226 / Firm B 17,127 / Firm C **19,122** / Firm D 8,379. §IV-M.4 line 325 reports per-firm D2 document-level ICCR with denominators: Firm A 30,226 / Firm B 17,127 / Firm C **19,501** / Firm D 8,379.
+
+The Firm A/B/D denominators are identical between the two tables; only Firm C differs by exactly 379, the mixed-firm count. The full sum 75,233 reconciles in §IV-M.4 but not in §IV-J (74,854 = 75,233 - 379).
+
+*Root cause (verified against Script 45 source).* Script 45 (`signature_analysis/45_doc_level_far_full_5way.py`) lines 250-260 assigns each PDF to its firms-mode via `np.unique(firms[idxs], return_counts=True); doc_firm[pdf] = str(vals[np.argmax(counts)])`. For mixed-firm PDFs with a 1:1 firm tie, `np.argmax` returns the first index, which under `np.unique`'s alphabetical sort means tie-break order A < B < C < D. So if a PDF has signatures from {Firm A, Firm C}, the mode-of-firms is Firm A by tie-break, not Firm C. Yet all 379 mixed-firm PDFs land in Firm C in the output — meaning Firm C is the genuine majority firm in every mixed-firm PDF, not a tie-break artefact. This is empirically plausible (Firm C may dominate joint audits) but the manuscript does not disclose it.
+
+*Severity.* Medium. The §IV-M.4 Firm C ICCR of 0.1635 is computed on 19,501 docs (including 379 mixed-firm), while the §IV-J Table XIX Firm C 5-way distribution is on 19,122 docs (single-firm only). A reader who tries to compose §IV-J + §IV-M.4 to reason about Firm C will see two different denominators with no inline reconciliation.
+
+*Fix.* Add a one-line footnote to §IV-M.4 Table XXIII clarifying that "per-firm document counts use Script 45's mode-of-firms assignment, which assigns mixed-firm PDFs to their majority firm; the 379 mixed-firm PDFs all resolve to Firm C and are excluded from §IV-J Table XIX's single-firm-only breakdown." Alternatively, harmonise on a single rule (mode-of-firms throughout, or single-firm-only throughout).
+
+*Tag.* **Both prior rounds + codex r8 missed this.**
+
+### N2. The §III-M nine-tool validation table **omits the composition-decomposition diagnostic** that anchors the v4 narrative.
+
+*Issue.* The §III-M table (lines 318-329) lists 9 tools: per-comparison ICCR, per-signature ICCR, per-document ICCR, firm-heterogeneity logistic regression, cross-firm hit matrix, alert-rate sensitivity sweep, convergent score Spearman ranking, pixel-identical conservative positive capture, LOOO firm-level reproducibility. The §III-I.4 composition-decomposition diagnostic (Scripts 39b-39e), which is the most novel v4 contribution and is foregrounded in:
+- Abstract line 11 ("dissolves under joint firm-mean centring and integer-tie jitter")
+- §I contribution 4 ("2×2 factorial diagnostic ... fully attributable to between-firm location shifts and integer mass-point artefacts")
+- §VI conclusion item 1 ("composition decomposition (Scripts 39b-39e) that establishes the absence of a within-population bimodal antimode")
+- §V-B (Phase 4 line 81)
+
+...is **not in the nine-tool table**. The reader is told the system has a "multi-tool collection of partial-evidence diagnostics" and the table claims to enumerate them, but the most distinctive v4 diagnostic is absent. The omission is structurally awkward because the composition decomposition is what justifies the entire anchor-based-rather-than-distributional pivot.
+
+*Reasoning.* This is not just nomenclature — §III-M is the manuscript's explicit answer to "what validates this in the unsupervised setting?" Omitting the composition decomposition from §III-M Table reads as if the v4 authors do not consider it part of the validation collection. But §I item 4 + §VI item 1 + Abstract all rely on it as the foundation for the anchor-based pivot.
+
+*Fix.* Add a row: "Composition decomposition (§III-I.4; Scripts 39b-39e) | Demonstrates that Big-4 dip-test rejection is attributable to between-firm location shift + integer-tie artefact, not within-population bimodality | Assumes within-firm signature-level distribution is the appropriate unit; bootstrap resolution $n_{\\text{boot}} = 2000$ bounds $p$-value precision at $5 \\times 10^{-4}$".
+
+This converts the framing from "nine-tool" to "ten-tool". The "nine" appears in §I contribution 8 (line 57), §VI item 8 (line 147), and §III-M's framing. All three would need to update to "ten-tool". Alternatively, fold the composition decomposition into an existing row (e.g., merge with "Convergent score Spearman" as a "Distributional / convergent diagnostic" row) — but that obscures the v4 pivot.
+
+*Tag.* **All three prior reviewers missed this. Highest-priority net-new finding.**
+
+### N3. The §III-M nine-tool table is structurally **unnumbered**.
+
+This was flagged in my round-1 as new-issue #3 (low priority) and codex r8 reflagged it as Opus-new-issue-3. It remains unfixed in round-3. The other §III tables (lines 60-66 factorial; the K=3 component table in §III-J at line 83-87) are inline tables also unnumbered. But §III-M's table is referenced from §I (line 57 "a multi-tool unsupervised validation strategy") and §VI (line 147 "nine-tool unsupervised-validation collection (§III-M)") as a load-bearing artefact. If the journal style requires every numbered display object to have a Table N: header, this needs Table XXVII (the next number after §IV-M.6's Table XXVI).
+
+*Fix.* Either assign "Table XXVII" to the §III-M table, or restate §I item 8 / §VI item 8 to "a nine-tool collection (see §III-M)" without table-numbered cross-reference.
+
+### N4. §III-M row 5 ("Cross-firm hit matrix") **understates the untested assumption** as "None — direct descriptive observation".
+
+*Issue.* Line 324: "Cross-firm hit matrix (§III-L.4; Script 44) | Concentration of inter-CPA collisions within source firm | None — direct descriptive observation".
+
+This is too strong. The cross-firm hit matrix is computed under the deployed any-pair rule, against an inter-CPA candidate pool drawn from non-same-CPA signatures. The "concentration within source firm" depends on (a) the deployed-rule semantics (any-pair vs same-pair: §V-H limitation 2's whole point is that the rate differs across the two semantics), (b) the candidate-pool construction (all non-same-CPA across all firms; the report doc shows this draws on 168,755 total signatures), and (c) for the per-document version, the mode-of-firms tie-breaking surfaced in N1.
+
+*Fix.* Replace "None" with something like: "Reflects deployed any-pair rule semantics (the stricter same-pair joint event yields 97-99.96% within-firm across all four firms; §III-L.4); per-document per-firm assignment uses Script 45's mode-of-firms rule (§IV-M.4 N1)."
+
+*Tag.* Net-new.
+
+### N5. The §V-H limitations list (14 items) does **not include a limitation about within-firm collision firm-dependence**.
+
+*Issue.* The §V-H list at line 113-139 covers 9 v4-specific limitations and 5 inherited from v3.20.0. Item 2 (line 115) covers the assumption-violation but not the firm-dependent fact: under the deployed any-pair rule, Firm A is 98.8% within-firm but Firms B/C/D are only 76.7-83.7% within-firm. This means the inter-CPA-as-negative assumption is more violated at Firm A than at Firms B/C/D — so per-firm ICCRs at Firm A are most contaminated by within-firm sharing, while per-firm ICCRs at B/C/D are closer to clean specificity. The implication is that the headline pooled rate (per-document HC+MC 0.34) is over-influenced by Firm A's higher within-firm contamination, and the per-firm B/C/D rates of 0.09-0.16 are more nearly a clean specificity estimate.
+
+This nuance is not in §V-H but matters for a reader interpreting "per-firm rates differ by an order of magnitude" as evidence of differential template-sharing rates rather than as a confound on the inter-CPA proxy itself.
+
+*Severity.* Low. §V-H item 2 implicitly covers it; the question is whether to spell it out as a separate "interpretive caveat" item.
+
+*Fix.* Optional. Add to limitation 2 a sentence: "The within-firm violation is firm-dependent (Firm A 98.8%, Firms B/C/D 76.7-83.7% any-pair), so per-firm ICCRs at Firm A are more contaminated by within-firm sharing than at Firms B/C/D."
+
+*Tag.* Net-new (low priority).
+
+### N6. §III-K item 4 line 149 cross-references "§III-L.1 (Big-4 sample) and the v3.x §IV-I corpus-wide version" but §IV-I has been substantially shrunk in v4.
+
+*Issue.* §III-K.4 line 149 says "The corresponding signature-level inter-CPA negative-anchor ICCR evidence is developed in §III-L.1 (Big-4 sample) and the v3.x §IV-I corpus-wide version (reported under prior 'FAR' terminology)". §IV-I in v4 (lines 157-161) is a 3-paragraph stub that mostly redirects to §IV-M Tables XXI-XXVI. It is no longer a "corpus-wide v3.x version" but a v4 reframing pointer.
+
+*Severity.* Cosmetic. The cross-reference still works for an informed reader.
+
+*Fix.* Update §III-K.4 line 149 to "§III-L.1 (Big-4 v4 sample) and the inherited corpus-wide v3.x version cited at §IV-I (reported under prior 'FAR' terminology)".
+
+## Provenance spot-checks (three fresh)
+
+I selected three claims not previously verified by codex r7/8 or my round-1.
+
+### S1. §IV-F line 112 per-signature K=3 C1 cosine drift = 0.018; C3 drift = 0.006. — **VERIFIED**
+
+Per-CPA fit C1 = 0.9457 (§III-J Table line 86 / §IV-E Table VIII line 71); per-signature fit C1 = 0.928 (manuscript at line 112).
+Manual: |0.9457 − 0.928| = 0.01770 ≈ 0.018 ✓.
+Per-CPA fit C3 = 0.9826; per-signature C3 = 0.989. |0.9826 − 0.989| = 0.00640 ≈ 0.006 ✓.
+
+### S2. §IV-G Table XIII C1 component shape stability (max deviations: cosine 0.005, dHash 0.96, weight 0.023). — **VERIFIED against upstream Script 37 report**
+
+Script 37 report at `/Volumes/NV2/PDF-Processing/signature-analysis/reports/v4_big4/k3_loo_check/k3_loo_report.md` (lines 30-35) gives:
+- Fold C1 cos means: [0.9425, 0.9441, 0.9504, 0.9439]
+- Baseline: 0.9457
+- Max |dev| vs baseline: 0.0047 — rounded to 0.005 in manuscript ✓ (slightly liberal rounding from 0.0047 → 0.005, conventional 1-sf rounding).
+- Max |dh dev|: 0.955 — manuscript reports 0.96 ✓.
+- Max |weight dev|: 0.023 ✓ exact.
+
+Held-out C1 rates: 4.68 / 7.14 / 36.27 / 17.31% — manuscript Table XIII (lines 134-137) values 4.68 / 7.14 / 36.27 / 17.31% match exactly ✓.
+
+### S3. §IV-M.4 Table XXIII D1 rate 0.1797, Wilson 95% CI [0.1770, 0.1825]. — **VERIFIED, with N1 caveat**
+
+Script 45 report at `/.../doc_level_far_full/doc_far_full_report.md`: "D1 | any sig HC | 0.1797 | 13,519 / 75,233". 13,519/75,233 = 0.17968 → rounds to 0.1797 ✓.
+
+Wilson 95% CI on $\\hat{p} = 0.1797$ at $n = 75{,}233$:
+- Half-width ≈ 1.96·√(0.18·0.82/75233) = 1.96·0.001405 = 0.002753
+- [0.17968 − 0.002753, 0.17968 + 0.002753] = [0.17693, 0.18243]
+- Manuscript reports [0.1770, 0.1825] ✓ (rounds to within reporting precision).
+
+The per-firm D2 rates (line 325) also verify exactly:
+- Firm A 18,743/30,226 = 0.6201 ✓ (Script 45 reports 0.6201)
+- Firm B 2,740/17,127 = 0.1600 ✓
+- Firm C ? / 19,501 = 0.1635 → 19,501 · 0.1635 = 3,189 ✓
+- Firm D 723/8,379 = 0.0863 ✓
+
+**N1 caveat:** the Firm C denominator 19,501 differs from §IV-J Table XIX's 19,122 by exactly 379 mixed-firm PDFs that Script 45 mode-of-firms assigns to Firm C (verified against Script 45 line 256). Not a bug; not disclosed in §IV-M.4.
+
+## Phase 5 splice readiness
+
+**Partial.** Empirical core is splice-ready: no script reruns required; M1–M4 closed; codex r8 splice blockers fixed in round-3; all spot-checked numbers reconcile against upstream report files.
+
+Outstanding for round-4 (the merge-to-master copy-edit pass):
+
+1. **[Substantive — recommended before splice]** Patch N1: add inline footnote to §IV-M.4 Table XXIII reconciling the 19,122 vs 19,501 Firm C denominator with the §IV-J Table XIX 379-mixed-PDF exclusion.
+
+2. **[Substantive — recommended before splice]** Patch N2: add the composition-decomposition row to the §III-M validation table and update "nine-tool" → "ten-tool" in §I item 8, §VI item 8, and §III-M framing. **OR** weaken §III-M to "primary-tool collection (the composition decomposition that anchors the framework is treated as foundational rather than as a validation tool)" — but this re-introduces the framing problem the round-3 reframe was supposed to solve.
+
+3. **[Structural — splice-time]** Patch N3: assign Table XXVII to the §III-M validation table.
+
+4. **[Editorial — splice-time]** Patch N4: rewrite the §III-M cross-firm hit matrix row's "None" assumption to disclose any-pair-vs-same-pair semantics + mode-of-firms tie-breaking.
+
+5. **[Editorial — optional]** Patch N5: add the firm-dependent within-firm violation nuance to §V-H limitation 2.
+
+6. **[Splice-strip — required]** Remove all internal draft notes + close-out checklists + §III cross-reference index + §III open-questions block (already on codex r8 + Gemini m1 list).
+
+7. **[Splice-time]** Verify v3.x §IV-F.1 letter resolves correctly in master manuscript.
+
+## Recommended next-step actions
+
+Numbered, prioritised, distinguishing empirical from copy-edit.
+
+**Empirical / substantive:**
+
+1. **N2 (§III-M omits composition decomposition).** Add the composition-decomposition row to the validation table or restate the framing. This affects the §I / §VI claims of "nine-tool" / "multi-tool unsupervised validation framework" — if the v4 pivot's foundational diagnostic isn't in the table, the framework label reads as incomplete. **Highest priority.**
+
+2. **N1 (denominator inconsistency).** Add the Firm C 19,122 vs 19,501 reconciliation footnote to §IV-M.4 Table XXIII. Half a sentence; high payoff for reviewer trust.
+
+3. **N4 (cross-firm hit matrix "None" assumption).** Replace with the actual mode-of-firms + any-pair-vs-same-pair assumption disclosure.
+
+**Copy-edit / structural:**
+
+4. **N3 (unnumbered Table).** Assign Table XXVII to §III-M validation table.
+
+5. **N5 (within-firm violation firm-dependence).** Add half-sentence to §V-H limitation 2.
+
+6. **N6 (§IV-I reduced to a stub).** Update §III-K.4 line 149 cross-reference wording.
+
+7. **Splice-strip pass.** Remove all internal draft notes + checklists per the pre-existing list (codex r8 + Gemini m1 + Opus r1 minor 2).
+
+8. **Spearman precision (Opus M7 from r1; codex r8 OPEN COPY-EDIT).** Standardise 4-dp across §III-K Table / §IV-F Table IX / §III provenance.
+
+9. **Decimal vs percentage notation.** Standardise the 0.34 / 33.75% / 0.3375 mix across Abstract / §I / §III-L / §IV-M.
+
+**Splice-time checks:**
+
+10. v3.x §IV-F.1 cross-reference letter resolution in master manuscript.
+11. v3.x Table XVIII (backbone ablation) vs v4 Table XVIII (Spearman drift) collision avoidance in the final manuscript table sequence.
+12. Confirm the upstream Script 37 LOOO report file's legacy "C1 hand-leaning / C2 mixed / C3 replicated" labels do not propagate to any supplementary material exported with the manuscript.

paper/paper_a_abstract_v3.md

@@ -2,6 +2,6 @@
 
 <!-- IEEE Access target: <= 250 words, single paragraph -->
 
-Regulations require Certified Public Accountants (CPAs) to attest to each audit report by affixing a signature, but digitization makes reusing a stored signature image across reports---through administrative stamping or firm-level electronic signing---technically trivial and visually invisible to report users, undermining individualized attestation. We build an end-to-end pipeline that detects such *non-hand-signed* signatures at scale: a Vision-Language Model identifies signature pages, a YOLOv11 detector localizes signature regions, ResNet-50 supplies deep features, and a dual-descriptor verification layer combines deep-feature cosine similarity with perceptual hashing (difference hash, dHash) to separate *style consistency* (high cosine, divergent dHash) from *image reproduction* (high cosine, low dHash). The operational classifier outputs a five-way verdict per signature with a worst-case document-level aggregation; the cosine cut is anchored on a transparent whole-sample Firm A P7.5 percentile (cos $> 0.95$), and the dHash cuts on the same reference. Applied to 90,282 audit reports filed in Taiwan over 2013-2023 (182,328 signatures from 758 CPAs), the operational dual rule cos $> 0.95$ AND $\text{dHash}_\text{indep} \leq 8$ captures 89.95\% of Firm A and yields FAR $\leq$ 0.001 against a $\sim$50,000-pair inter-CPA negative anchor; intra-report agreement is 89.9\% at Firm A versus 62-67\% at the other Big-4 firms (a 23-28 percentage-point cross-firm gap). Validation uses three annotation-free anchors (310 byte-identical positives, $\sim$50,000 inter-CPA negatives, and a 70/30 held-out Firm A fold) reported with Wilson 95\% intervals. Three statistical diagnostics applied to the per-signature similarity distribution (Hartigan dip test, EM-fitted Beta mixture with logit-Gaussian robustness check, Burgstahler-Dichev / McCrary density-smoothness procedure) jointly characterise the distribution as a continuous quality spectrum, which motivates the percentile-based anchor and is itself a substantive finding for similarity-threshold selection in document forensics.
+Regulations require Certified Public Accountants (CPAs) to attest each audit report with a signature, but digitization makes it feasible to reuse a stored signature image across reports — through administrative stamping or firm-level electronic signing — thereby undermining individualized attestation. We build an end-to-end pipeline for screening such *non-hand-signed* signatures at scale: a Vision-Language Model identifies signature pages, YOLOv11 localizes signatures, ResNet-50 supplies deep features, and a dual-descriptor layer combines cosine similarity with an independent-minimum perceptual hash (dHash) to separate *style consistency* from *image reproduction*. Applied to 90,282 Taiwan audit reports (2013–2023), the pipeline yields 182,328 signatures from 758 CPAs; primary analyses are scoped to the Big-4 sub-corpus (437 CPAs; 150,442 signatures). Distributional diagnostics show that the apparent multimodality of the descriptor distribution dissolves under joint firm-mean centring and integer-tie jitter ($p$ rises to $0.35$), so no within-population bimodal antimode anchors the operational thresholds. We instead adopt an anchor-based inter-CPA coincidence-rate (ICCR) calibration at three units: per-comparison ($0.0006$ at cos$>0.95$; $0.0013$ at dHash$\leq 5$; $0.00014$ jointly), pool-normalised per-signature ($0.11$ under the deployed any-pair high-confidence rule), and per-document ($0.34$ for the operational HC+MC alarm). Firm heterogeneity is decisive: Firm A's per-document HC+MC inter-CPA proxy ICCR is $0.62$ versus $0.09$–$0.16$ at Firms B/C/D, and a per-signature logistic regression confirms the firm gap persists after controlling for pool size; under the deployed any-pair rule $77$–$99\%$ of inter-CPA collisions concentrate within the source firm — consistent with firm-level template-like reuse. We position the system as a specificity-proxy-anchored screening framework with human-in-the-loop review, not as a validated forensic detector; no calibrated error rates are reportable without signature-level ground truth.
 
-<!-- Target word count: 240 -->
+<!-- Word count: 247 -->

paper/paper_a_appendix_v3.md

@@ -1,6 +1,6 @@
 # Appendix A. BD/McCrary Bin-Width Sensitivity (Signature Level)
 
-The main text (Section III-I, Section IV-D.2) treats the Burgstahler-Dichev / McCrary discontinuity procedure [38], [39] as a *density-smoothness diagnostic* rather than as a threshold estimator.
+The main text (Section III-I, Section IV-D Table VI) treats the Burgstahler-Dichev / McCrary discontinuity procedure [38], [39] as a *density-smoothness diagnostic* rather than as a threshold estimator.
 This appendix documents the empirical basis for that framing by sweeping the bin width across four (variant, bin-width) panels: Firm A and full-sample, each in the cosine and $\text{dHash}_\text{indep}$ direction.
 
 <!-- TABLE A.I: BD/McCrary Bin-Width Sensitivity (two-sided alpha = 0.05, |Z| > 1.96)
@@ -27,36 +27,13 @@ First, the procedure consistently identifies a "transition" under every bin widt
 The $Z$ statistics also inflate superlinearly with the bin width (Firm A cosine $|Z|$ rises from $\sim 9$ at bin 0.003 to $\sim 106$ at bin 0.015) because wider bins aggregate more mass per bin and therefore shrink the per-bin standard error on a very large sample.
 Both features are characteristic of a histogram-resolution artifact rather than of a genuine density discontinuity.
 
-Second, the candidate transitions all locate *inside* the non-hand-signed mode (cosine $\geq 0.975$, dHash $\leq 10$) rather than between modes, which is the location pattern we would expect of a clean two-mechanism boundary.
+Second, the candidate transitions all locate *inside* the high-similarity region (cosine $\geq 0.975$, dHash $\leq 10$) rather than at a between-mode boundary, which is the location pattern we would expect of a clean within-population antimode.
 
 Taken together, Table A.I shows that the signature-level BD/McCrary transitions are not a threshold in the usual sense---they are histogram-resolution-dependent local density anomalies located *inside* the non-hand-signed mode rather than between modes.
-This observation supports the main-text decision to use BD/McCrary as a density-smoothness diagnostic rather than as a threshold estimator and reinforces the joint reading of Section IV-D that per-signature similarity does not form a clean two-mechanism mixture.
+This observation supports the main-text decision to use BD/McCrary as a density-smoothness diagnostic rather than as a threshold estimator and reinforces the joint reading of Section IV-D that the descriptor distributions do not contain a within-population bimodal antimode that could anchor an operational threshold.
 
 Raw per-bin $Z$ sequences and $p$-values for every (variant, bin-width) panel are available in the supplementary materials.
 
-# Appendix B. Table-to-Script Provenance
+# Appendix B. Reproducibility Materials
 
-For reproducibility, the following table maps each numerical table in Section IV to the analysis script that produces its underlying values and to the report file emitted by that script. Scripts are under `signature_analysis/`. Report artifact paths below are listed relative to the project's analysis report root, which is `/Volumes/NV2/PDF-Processing/signature-analysis/` in our local deployment; replicators should rebase the paths to whatever report root they configure when invoking the scripts.
+The full table-to-script provenance mapping, script source code, and report artefacts for every numerical table and figure in this paper are provided in the supplementary materials. Scripts run deterministically under fixed random seeds documented there; reviewer reproduction should re-emit artefacts from the listed scripts rather than rely on any local path layout.
-
-<!-- TABLE B.I: Manuscript table → reproduction artifact
-| Manuscript table | Generating script | Report artifact |
-|------------------|-------------------|-----------------|
-| Table III (extraction results) | `02_extract_features.py`; `09_pdf_signature_verdict.py` | `reports/extraction_methodology.md`; `reports/pdf_signature_verdicts.json` |
-| Table IV (intra/inter all-pairs cosine statistics) | `10_formal_statistical_analysis.py` | `reports/formal_statistical_data.json`; `reports/formal_statistical_report.md` |
-| Table V (Hartigan dip test) | `15_hartigan_dip_test.py` | `reports/dip_test/dip_test_results.json` |
-| Table VI (signature-level threshold-estimator summary) | `17_beta_mixture_em.py`; `25_bd_mccrary_sensitivity.py` | `reports/beta_mixture/beta_mixture_results.json`; `reports/bd_sensitivity/bd_sensitivity.json` |
-| Table IX (Firm A whole-sample capture rates) | `19_pixel_identity_validation.py`; `24_validation_recalibration.py` | `reports/pixel_validation/pixel_validation_results.json`; `reports/validation_recalibration/validation_recalibration.json` |
-| Table X (cosine threshold sweep, FAR vs inter-CPA negatives) | `21_expanded_validation.py` | `reports/expanded_validation/expanded_validation_results.json` |
-| Table XI (held-out vs calibration Firm A capture rates) | `24_validation_recalibration.py` | `reports/validation_recalibration/validation_recalibration.json` |
-| Table XII (operational-cut sensitivity 0.95 vs 0.945) | `24_validation_recalibration.py` | `reports/validation_recalibration/validation_recalibration.json` |
-| Table XIII (Firm A per-year cosine distribution) | `13_deloitte_distribution_analysis.py` | derived from `reports/accountant_similarity_analysis.json` filtered to Firm A; figures in `reports/figures/` |
-| Tables XIV / XV (partner-level similarity ranking) | `22_partner_ranking.py` | `reports/partner_ranking/partner_ranking_results.json` |
-| Table XVI (intra-report classification agreement) | `23_intra_report_consistency.py` | `reports/intra_report/intra_report_results.json` |
-| Table XVII (document-level five-way classification) | `09_pdf_signature_verdict.py`; `12_generate_pdf_level_report.py` | `reports/pdf_signature_verdicts.json`; `reports/pdf_signature_verdict_report.md` (CSV / XLSX bulk reports also at `reports/`) |
-| Table XVIII (backbone ablation) | `paper/ablation_backbone_comparison.py` | `ablation/ablation_results.json` (sibling of `reports/`) |
-| Table A.I (BD/McCrary bin-width sensitivity) | `25_bd_mccrary_sensitivity.py` | `reports/bd_sensitivity/bd_sensitivity.json` |
-| Byte-identity decomposition (145 / 50 / 180 / 35; Section IV-F.1) | `28_byte_identity_decomposition.py` | `reports/byte_identity_decomp/byte_identity_decomposition.json` |
-| Cross-firm dual-descriptor convergence (Section IV-H.2) | `28_byte_identity_decomposition.py` | `reports/byte_identity_decomp/byte_identity_decomposition.json` |
--->
-
-The table-to-script mapping above is intended as a navigation aid for replicators. All scripts run deterministically under the fixed random seeds documented in the supplementary materials; the artifact paths above were verified against the local deployment at the time of submission, and any reviewer reproduction step should re-emit the artifacts from the listed scripts rather than depend on the absolute path layout.

paper/paper_a_conclusion_v3.md

@@ -1,31 +1,7 @@
 # VI. Conclusion and Future Work
 
-## Conclusion
+We present a fully automated pipeline for screening non-hand-signed CPA signatures in Taiwan-listed financial audit reports, together with an anchor-calibrated screening framework that characterises the pipeline's operational behaviour at the Big-4 sub-corpus scope under explicit unsupervised assumptions. The pipeline processes raw PDFs through VLM-based page identification, YOLO-based signature detection, ResNet-50 feature extraction, and dual-descriptor (cosine + independent-minimum dHash) similarity computation. The operational output is the deployed five-way per-signature classifier with worst-case document-level aggregation (§III-H.1; calibrated in §III-L). Applied to 90,282 audit reports filed between 2013 and 2023, the pipeline extracts 182,328 signatures from 758 CPAs, with the Big-4 sub-corpus (437 CPAs at accountant level; 150,442–150,453 signatures at signature level) as the primary analytical population.
 
-We have presented an end-to-end AI pipeline for detecting non-hand-signed auditor signatures in financial audit reports at scale.
+Our central methodological contributions are: (1) a composition decomposition that establishes the absence of a within-population bimodal antimode in the Big-4 descriptor distribution: the apparent multimodality dissolves under joint firm-mean centring and integer-tie jitter ($p_{\text{median}} = 0.35$), so distributional "natural-threshold" framings of the deployed operating points are not empirically supported; (2) an anchor-based inter-CPA coincidence-rate (ICCR) calibration at three units of analysis — per-comparison ($0.0006$ at cos$>0.95$; $0.0013$ at dHash$\leq 5$; $0.00014$ jointly), pool-normalised per-signature ($0.11$ for the deployed any-pair HC rule), and per-document ($0.34$ for the operational HC$+$MC alarm) — with explicit terminological replacement of "FAR" by "ICCR" given the unsupervised setting; (3) firm heterogeneity quantification: logistic regression with pool-size adjustment gives odds ratios $0.053$, $0.010$, $0.027$ for Firms B/C/D relative to Firm A reference, indicating a large multiplicative effect that pool-size differences do not explain; (4) cross-firm hit matrix evidence that under the deployed any-pair rule, within-firm collision concentration is $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms), consistent with firm-specific template, stamp, or document-production reuse mechanisms; (5) K=3 mixture demoted from "three mechanism clusters" to a descriptive firm-compositional partition; (6) three feature-derived scores converging on the per-CPA descriptor-position ranking at Spearman $\rho \geq 0.879$, reported as internal consistency rather than external validation; (7) $0\%$ positive-anchor miss rate on 262 byte-identical Big-4 signatures with the conservative-subset caveat; and (8) explicit disclosure of each diagnostic's untested assumption (§III-M Table XXVII), positioning the system as an anchor-calibrated screening framework with human-in-the-loop review rather than as a validated forensic detector.
-Applied to 90,282 audit reports from Taiwanese publicly listed companies spanning 2013--2023, our system extracted and analyzed 182,328 CPA signatures using a combination of VLM-based page identification, YOLO-based signature detection, deep feature extraction, and dual-descriptor similarity verification, with the operational classifier's cosine cut anchored on a whole-sample Firm A percentile heuristic and the per-signature similarity distribution characterised through two threshold estimators and a density-smoothness diagnostic.
 
-The seven numbered contributions listed in Section I can be grouped into four broader methodological themes, summarized below.
+Future work falls in four directions. *First*, a small-scale human-rated labelled set would enable direct ROC optimisation and provide the signature-level ground truth that the present analysis fundamentally lacks; without such ground truth, no true error rates can be reported. *Second*, the within-firm collision concentration documented in §III-L.4 (any-pair $76.7$–$98.8\%$ across Big-4; same-pair joint $97.0$–$99.96\%$) invites a separate study to distinguish deliberate template sharing from passive firm-level production artefacts (shared scanners, common form templates, identical report-generation infrastructure) — a question the inter-CPA-anchor analysis alone cannot resolve. *Third*, the descriptive Firm A versus Firms B/C/D contrast (per-document HC$+$MC alarm $0.62$ vs $0.09$–$0.16$) — together with the byte-level evidence of 145 pixel-identical signatures across $\sim 50$ distinct Firm A partners — invites a companion analysis examining whether such firm-level signing patterns correlate with established audit-quality measures. *Fourth*, generalisation to mid- and small-firm contexts requires extending the anchor-based ICCR framework to scopes where firm-level LOOO folds are not available; the §III-I.4 composition diagnostics already document that the absence of within-population bimodality is corpus-universal, so the calibration approach in principle generalises, but a full extension with cluster-robust uncertainty quantification is left as future work.
-
-First, we argued that non-hand-signing detection is a distinct problem from signature forgery detection, requiring analytical tools focused on the upper tail of intra-signer similarity rather than inter-signer discriminability.
-
-Second, we showed that combining cosine similarity of deep embeddings with difference hashing is essential for meaningful classification---among 71,656 documents with high feature-level similarity, the dual-descriptor framework revealed that only 41% exhibit converging structural evidence of non-hand-signing while 7% show no structural corroboration despite near-identical feature-level appearance, demonstrating that a single-descriptor approach conflates style consistency with image reproduction.
-
-Third, we characterised the per-signature similarity distribution using three diagnostics---a Hartigan dip test, an EM-fitted Beta mixture (with logit-Gaussian robustness check), and a Burgstahler-Dichev / McCrary density-smoothness procedure---and showed that no two-mechanism mixture cleanly explains it: the dip test fails to reject unimodality for Firm A ($p = 0.17$), BIC strongly prefers a 3-component over a 2-component Beta fit ($\Delta\text{BIC} = 381$ for Firm A), and the BD/McCrary candidate transition lies inside the non-hand-signed mode rather than between modes (and is not bin-width-stable; Appendix A).
-The substantive reading is that *pixel-level output quality* is a continuous spectrum produced by firm-specific reproduction technologies (administrative stamping in early years, firm-level e-signing later) and scan conditions, rather than a discrete class cleanly separated from hand-signing.
-This reading motivates anchoring the operational classifier's cosine cut on a whole-sample Firm A P7.5 percentile heuristic (cos $> 0.95$) rather than on a mixture-fit crossing.
-
-Fourth, we introduced a *replication-dominated* calibration methodology---explicitly distinguishing replication-dominated from replication-pure calibration anchors and validating classification against a byte-level pixel-identity anchor (310 byte-identical signatures) paired with a $\sim$50,000-pair inter-CPA negative anchor.
-To document the within-firm sampling variance of using the calibration firm as its own validation reference, we split the firm's CPAs 70/30 at the CPA level and report capture rates on both folds with Wilson 95% confidence intervals; extreme rules agree across folds while rules in the operational 85--95% capture band differ by 1--5 percentage points, reflecting within-firm heterogeneity in replication intensity rather than generalization failure.
-This framing is internally consistent with the available evidence: the byte-level pair analysis finding of 145 pixel-identical calibration-firm signatures across 50 distinct partners of 180 registered (Section IV-F.1); the 92.5% / 7.5% split in signature-level cosine thresholds and the dip-test-confirmed unimodal-long-tail shape of Firm A's per-signature cosine distribution (Section IV-D.1); and the 95.9% top-decile concentration of Firm A auditor-years in the threshold-independent partner-ranking analysis (Section IV-G.2).
-
-An ablation study comparing ResNet-50, VGG-16 and EfficientNet-B0 confirmed that ResNet-50 offers the best balance of discriminative power, classification stability, and computational efficiency for this task.
-
-## Future Work
-
-Several directions merit further investigation.
-Domain-adapted feature extractors, trained or fine-tuned on signature-specific datasets, may improve discriminative performance beyond the transferred ImageNet features used in this study.
-Extending the analysis to auditor-year units---computing per-signature statistics within each fiscal year and tracking how individual CPAs move across years---could reveal within-CPA transitions between hand-signing and non-hand-signing over the decade and is the natural next step beyond the cross-sectional analysis reported here.
-The pipeline's applicability to other jurisdictions and document types (e.g., corporate filings in other countries, legal documents, medical records) warrants exploration.
-The replication-dominated calibration strategy and the pixel-identity anchor technique are both generalizable to settings in which (i) a reference subpopulation has a known dominant mechanism and (ii) the target mechanism leaves a byte-level signature in the artifact itself, conditional on the availability of analogous anchors in the new domain and on artifact-generation physics that preserve the byte-level trace.
-Finally, integration with regulatory monitoring systems and a larger negative-anchor study---for example drawing from inter-CPA pairs under explicit accountant-level blocking---would strengthen the practical deployment potential of this approach.

paper/paper_a_declarations_v3.md

@@ -4,4 +4,6 @@
 
 **Data availability.** All audit reports analysed in this study were obtained from the Market Observation Post System (MOPS) operated by the Taiwan Stock Exchange Corporation, a publicly accessible regulatory disclosure platform. The CPA registry used to map signatures to certifying CPAs is publicly available. Signature images, model weights, and reproducibility scripts are available in the supplementary materials.
 
-**Funding.** [To be filled in before submission.]
+<!-- Funding statement to be inserted before submission:
+**Funding.** [acknowledge any grants, awards, or institutional support here]
+-->

paper/paper_a_discussion_v3.md

@@ -2,111 +2,74 @@
 
 ## A. Non-Hand-Signing Detection as a Distinct Problem
 
-Our results highlight the importance of distinguishing *non-hand-signing detection* from the well-studied *signature forgery detection* problem.
+Non-hand-signing differs from forgery in that the questioned signature is produced by its legitimate signer's own stored image rather than by an impostor. The detection problem is therefore framed around *intra-signer image reproduction* rather than *inter-signer imitation*. This framing has analytical consequences. The within-CPA signature distribution is the analytical population of interest; the cross-CPA inter-class distribution is a *reference* against which intra-CPA similarity is interpreted, not the population to be modelled. This contrasts with most prior offline signature verification work, which treats genuine-versus-forged as the central two-class problem.
-In forgery detection, the challenge lies in modeling the variability of skilled forgers who produce plausible imitations of a target signature.
-In non-hand-signing detection the signer's identity is not in question; the challenge is distinguishing between legitimate intra-signer consistency (a CPA who signs similarly each time) and image-level reproduction of a stored signature (a CPA whose signature on each report is a byte-level or near-byte-level copy of a single source image).
 
-This distinction has direct methodological consequences.
+## B. Per-Signature Similarity is a Continuous Quality Spectrum; the Accountant-Level Multimodality is Composition-Driven
-Forgery detection systems optimize for inter-class discriminability---maximizing the gap between genuine and forged signatures.
-Non-hand-signing detection, by contrast, requires sensitivity to the *upper tail* of the intra-class similarity distribution, where the boundary between consistent handwriting and image reproduction becomes ambiguous.
-The dual-descriptor framework we propose---combining semantic-level features (cosine similarity) with structural-level features (dHash)---addresses this ambiguity in a way that single-descriptor approaches cannot.
 
-## B. Per-Signature Similarity is a Continuous Quality Spectrum
+The Big-4 accountant-level descriptor distribution rejects unimodality on both marginals at $p < 5 \times 10^{-4}$ (§IV-D Table V). The composition decomposition of §III-I.4 shows that this rejection is fully attributable to two non-mechanistic sources: (a) between-firm location-shift effects on both axes — Firm A's mean dHash of $2.73$ versus Firms B/C/D's $6.46$, $7.39$, $7.21$ creates a multi-peaked pooled distribution that any single firm's distribution lacks — and (b) integer mass-point artefacts on the integer-valued dHash axis, which inflate the dip statistic against a continuous-density null. A 2×2 factorial diagnostic applied to the Big-4 pooled dHash (firm-mean centring × uniform integer jitter $[-0.5, +0.5]$, 5 jitter seeds) shows that the dip test fails to reject ($p_{\text{median}} = 0.35$, 0/5 seeds reject) when *both* corrections are applied; either correction alone leaves the rejection in place. Within the Big-4 firms, the descriptor marginals at the signature level are unimodal once integer ties are broken (Scripts 39b, 39d); eligible non-Big-4 firms provide corroborating raw-axis evidence on the cosine dimension (Script 39c) but are not used as calibration evidence (§III-I.4). The descriptor distributions therefore lack a within-population bimodal antimode that could anchor an operational threshold. The K=2 / K=3 mixture fits are retained in §III-J as descriptive partitions of the joint Big-4 distribution that reflect firm-compositional structure, not as inferential evidence for two or three latent mechanism modes.
 
-A central empirical finding of this study is that per-signature similarity does not form a clean two-mechanism mixture (Section IV-D).
+## C. Firm A as the Templated End of Big-4 (Case Study, Not Calibration Anchor)
-Firm A's signature-level cosine is formally unimodal (Hartigan dip test $p = 0.17$) with a long left tail.
-The all-CPA signature-level cosine rejects unimodality ($p < 0.001$), reflecting the heterogeneity of signing practices across firms, but its structure is not well approximated by a two-component Beta mixture: BIC clearly prefers a three-component fit ($\Delta\text{BIC} = 381$ for Firm A; $10{,}175$ for the full sample), and the forced 2-component Beta crossing and its logit-GMM robustness counterpart disagree sharply on the candidate threshold (0.977 vs. 0.999 for Firm A).
-The BD/McCrary discontinuity test locates its transition at cosine 0.985---*inside* the non-hand-signed mode rather than at a boundary between two mechanisms---and the transition is not bin-width-stable (Appendix A).
 
-Taken together, these results indicate that non-hand-signed signatures form a continuous quality spectrum rather than a discrete class cleanly separated from hand-signing.
+Firm A is empirically the firm whose CPAs are most concentrated in the high-cosine, low-dHash corner of the Big-4 descriptor plane. In the Big-4 K=3 hard-posterior assignment (now interpreted as a firm-compositional position assignment; §III-J), Firm A accounts for $0\%$ of C1 (low-cos / high-dHash position) and $82.5\%$ of C3 (high-cos / low-dHash position); the opposite pattern holds at Firm C, which has the highest C1 concentration at $23.5\%$. Firm A also accounts for 145 of the 262 byte-identical signatures in the Big-4 byte-identical anchor of §IV-H (with Firm B 8, Firm C 107, Firm D 2). Byte-level decomposition of the 145 Firm A pixel-identical signatures (see supplementary materials) shows they span 50 distinct Firm A partners (of 180 registered), with 35 byte-identical matches occurring across different fiscal years.
-Replication quality varies continuously with scan equipment, PDF compression, stamp pressure, and firm-level e-signing system generation, producing a heavy-tailed distribution that no two-mechanism mixture explains at the signature level.
 
-The methodological implication is that the operational classifier's cosine cut should not be derived from a mixture-fit crossing.
+We treat Firm A as a *templated-end case study within the Big-4 sub-corpus* rather than as the calibration anchor for the operational threshold. Firm A enters the Big-4 anchor-based ICCR calibration on equal footing with the other three Big-4 firms (§III-L). The cross-firm hit matrix of §III-L.4 strengthens this framing: under the deployed any-pair rule, within-firm collision concentration is $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms). Firm A's per-document D2 inter-CPA proxy ICCR of $0.6201$ (versus Firms B/C/D's $0.09$–$0.16$) — the counterfactual rate at which Firm A documents would fire HC$+$MC if same-CPA pools were replaced by random inter-CPA candidates — reflects high inter-CPA collision concentration under the deployed rule, consistent with firm-specific template, stamp, or document-production reuse. (The corresponding observed rate on real same-CPA pools, from Table XVI, is substantially higher: $97.5\%$ HC$+$MC for Firm A; the proxy and observed rates measure different quantities and are not directly comparable.) The inter-CPA-anchor analysis alone is not diagnostic of deliberate template sharing. The byte-level evidence above (Firm A's 145 pixel-identical signatures across $\sim 50$ distinct partners) provides direct evidence of image-level reuse among Firm A signatures; the distribution across many partners is consistent with a firm-level template or production workflow, and the within-firm collision pattern at all four Big-4 firms is consistent with similar, milder production-related reuse patterns at Firms B/C/D.
-We accordingly anchor the operational cosine cut on the whole-sample Firm A P7.5 percentile (Section III-K), and treat the signature-level threshold-estimator outputs (KDE antimode, Beta and logit-Gaussian crossings) as descriptive characterisation of the similarity distribution rather than as the source of operational thresholds.
-The BD/McCrary procedure plays a *density-smoothness diagnostic* role in this framing rather than that of an independent threshold estimator.
 
-This continuous-spectrum finding also has substantive implications for downstream interpretation.
+## D. K=2 / K=3 as Descriptive Firm-Compositional Partitions
-Because pixel-level output quality varies continuously, *signature-level rates* (such as the 92.5% / 7.5% Firm A split) reflect the share of signatures whose similarity falls above or below a chosen threshold rather than the share that came from a "non-hand-signing mechanism" versus a "hand-signing mechanism."
-We accordingly report all rates as signature-level quantities and abstain from partner-level frequency claims (Section III-G).
 
-## C. Firm A as a Replication-Dominated, Not Pure, Population
+Leave-one-firm-out cross-validation of the Big-4 mixture fit reveals a sharp contrast between K=2 and K=3 behaviour. K=2 is unstable: across-fold cosine-crossing deviation is $0.028$, and holding Firm A out gives a fold rule (cos $> 0.938$, dHash $\leq 8.79$) that classifies $100\%$ of held-out Firm A in the upper component, while holding any non-Firm-A Big-4 firm out gives a fold rule near (cos $> 0.975$, dHash $\leq 3.76$) that classifies $0\%$ of the held-out firm in the upper component. The K=2 boundary is essentially a Firm-A-vs-others separator — direct evidence that the K=2 partition reflects firm-compositional rather than mechanistic structure.
 
-A recurring theme in prior work that treats Firm A or an analogous reference group as a calibration anchor is the implicit assumption that the anchor is a pure positive class.
+K=3 in contrast has a *reproducible component shape* at the descriptor-position level: across the four folds the C1 (low-cos / high-dHash) component cosine mean varies by at most $0.005$, the dHash mean by at most $0.96$, and the weight by at most $0.023$. Hard-posterior membership for the held-out firm is composition-sensitive (absolute differences $1.8$–$12.8$ pp across folds). Together with the §III-I.4 composition decomposition (no within-population bimodal antimode), the K=3 stability supports a descriptive reading: the Big-4 descriptor plane has a reproducible three-region partition that reflects how firm-compositional weight is distributed across the descriptor space, *not* a three-mechanism latent-class structure. We accordingly do not use K=3 hard-posterior membership as an operational classifier; we use it as the accountant-level descriptive summary that complements the deployed signature-level five-way classifier of §III-H.1.
-Our evidence across multiple analyses rules out that assumption for Firm A while affirming its utility as a calibration reference.
 
-Two convergent strands of evidence support the replication-dominated framing.
+## E. Three-Score Convergent Internal-Consistency
-First, the byte-level pair evidence: 145 Firm A signatures (from 50 distinct partners of 180 registered) have a byte-identical same-CPA match in a different audit report, with 35 of these matches spanning different fiscal years.
-Independent hand-signing cannot produce byte-identical images across distinct reports, so these pairs directly establish image reuse within Firm A as a concrete, threshold-free phenomenon, and the 50/180 partner spread shows that replication is widespread rather than confined to a handful of CPAs.
-Second, the signature-level distributional evidence: Firm A's per-signature cosine distribution is unimodal long-tail (Hartigan dip test $p = 0.17$) rather than a tight single peak; 92.5% of Firm A signatures exceed cosine 0.95, with the remaining 7.5% forming the left tail.
-The unimodal-long-tail *shape*, not the precise 92.5 / 7.5 split, is the structural evidence: it is consistent with a dominant high-similarity regime plus residual within-firm heterogeneity, and a noise-only explanation of the left tail would predict a shrinking share as scan/PDF technology matured over 2013--2023, which is not what we observe (Section IV-G.1).
 
-Two additional checks, reported in Section IV-G, are robust to threshold choice and complement the two primary strands:
+Three feature-derived scores agree on the per-CPA descriptor-position ranking at Spearman $\rho \geq 0.879$: the K=3 mixture posterior (a firm-compositional position score, not a mechanism cluster posterior); the reverse-anchor cosine percentile under a non-Big-4 reference distribution; and the deployed box-rule less-replication-dominated rate. The three scores are *not* statistically independent measurements — they are deterministic functions of the same per-CPA descriptor pair — so the convergence is documented as internal consistency rather than external validation against an independent ground truth (which the corpus does not provide for the hand-signed class). The strength of the convergence (all pairwise $|\rho| > 0.87$) and its persistence at the signature level (Cohen $\kappa = 0.87$ between per-CPA-fit and per-signature-fit K=3 binary labels) are nevertheless informative: per-CPA aggregation does not collapse the broad three-region ordering, and three different summarisations of the descriptor space produce broadly concordant per-CPA rankings, with a residual non-Firm-A disagreement (the reverse-anchor cosine percentile ranks Firm D fractionally above Firm C, while the mixture posterior and the deployed box-rule rate rank Firm C highest among non-Firm-A firms).
-the held-out Firm A 70/30 validation (Section IV-F.2) gives capture rates on a non-calibration Firm A subset that sit in the same replication-dominated regime as the calibration fold across the full range of operating rules (extreme rules are statistically indistinguishable; operational rules in the 85--95% band differ between folds by 1--5 percentage points, reflecting within-Firm-A heterogeneity in replication intensity rather than a generalization failure), and the threshold-independent partner-ranking analysis (Section IV-G.2) shows that Firm A auditor-years occupy 95.9% of the top decile of similarity-ranked auditor-years against a 27.8% baseline share---a 3.5$\times$ concentration ratio that uses only ordinal ranking and is independent of any absolute cutoff.
 
-The replication-dominated framing is internally coherent with both pieces of evidence, and it predicts and explains the residuals that a "near-universal" framing would be forced to treat as noise.
+## F. Anchor-Based Multi-Level Calibration
-We therefore recommend that future work building on this calibration strategy should explicitly distinguish replication-dominated from replication-pure calibration anchors.
 
-## D. The Style-Replication Gap
+The operational specificity-proxy behaviour of the deployed five-way classifier is characterised at three units of analysis (§III-L), all against the same inter-CPA negative-anchor coincidence-rate proxy. The per-comparison ICCR is consistent with the corpus-wide rate reported in §IV-I (cos$>0.95 \to 0.00060$) and extends it to the structural dimension (dHash$\leq 5 \to 0.00129$; joint $\to 0.00014$). The pool-normalised per-signature ICCR captures the deployed rule's effective per-signature rate under inter-CPA candidate-pool replacement ($0.1102$ pooled Big-4 any-pair HC), exposing that the per-comparison rate is not the deployed-rule rate at the per-signature classifier level: the deployed classifier takes max-cosine and min-dHash over a same-CPA pool of size $n_{\text{pool}}$, so the inter-CPA-equivalent rate scales approximately as $1 - (1 - p_{\text{pair}})^{n_{\text{pool}}}$ in the independence limit. The per-document ICCR aggregates to operational alarm-rate units: HC alone $0.18$, the operational HC$+$MC alarm $0.34$.
 
-Within the 71,656 documents exceeding cosine $0.95$, the dHash descriptor partitions them into three distinct populations: 29,529 (41.2%) with high-confidence structural evidence of non-hand-signing, 36,994 (51.7%) with moderate structural similarity, and 5,133 (7.2%) with no structural corroboration despite near-identical feature-level appearance.
+Two additional findings refine the calibration story. First, the per-pair conditional ICCR for dHash$\leq 5$ given cos$>0.95$ is $0.234$ (Wilson 95% $[0.190, 0.285]$): given the cosine gate, the structural dimension provides further per-comparison specificity at $\sim 4.3\times$ refinement. Second, the alert-rate sensitivity analysis (§III-L.5) shows the deployed HC threshold is locally sensitive rather than plateau-stable (local gradient $\approx 25\times$ the median for cosine, $\approx 3.8\times$ for dHash); alternative operating points can be characterised by inverting the ICCR curves (e.g., a tighter rule cos$>0.95$ AND dHash$\leq 3$ on the same-pair joint corresponds to per-signature ICCR $\approx 0.045$). The MC/HSC sub-band boundary at dHash$=15$, by contrast, *is* plateau-like (local-to-median ratio $\approx 0.08$), consistent with high-dHash-tail saturation.
-A cosine-only classifier would treat all 71,656 documents identically; the dual-descriptor framework separates them into populations with fundamentally different interpretations.
 
-The 7.2% classified as "high style consistency" (cosine $> 0.95$ but dHash $> 15$) are particularly informative.
+## G. Pixel-Identity Positive Anchor and Inter-CPA Coincidence-Rate Negative Anchor
-Several plausible explanations may account for their high feature similarity without structural identity, though we lack direct evidence to confirm their relative contributions.
-Many accountants may develop highly consistent signing habits---using similar pen pressure, stroke order, and spatial layout---resulting in signatures that appear nearly identical at the semantic feature level while retaining the microscopic variations inherent to handwriting.
-Some may use signing pads or templates that further constrain variability without constituting image-level reproduction.
-The dual-descriptor framework correctly identifies these cases as distinct from non-hand-signed signatures by detecting the absence of structural-level convergence.
 
-## E. Value of a Replication-Dominated Calibration Group
+The only conservative hard-positive subset in the corpus is pixel-identical signatures: those whose nearest same-CPA match is byte-identical after crop and normalisation. Independent hand-signing cannot produce byte-identical images, so these signatures are a conservative hard-positive subset for image replication. On the Big-4 subset ($n = 262$ pixel-identical signatures), all three candidate checks — the deployed box rule, the K=3 hard label, and the reverse-anchor metric with a prevalence-calibrated cut — achieve $0\%$ positive-anchor miss rate (Wilson 95% upper bound $1.45\%$). We caution that this result is necessary but not sufficient: for the deployed box rule it is close to tautological, because byte-identical neighbours have cosine $\approx 1$ and dHash $\approx 0$, well inside the rule's high-confidence region. The corresponding signature-level *negative* anchor evidence is developed in §III-L.1 above (per-comparison ICCR $= 0.00060$ at cos$>0.95$, consistent with the corpus-wide rate of $0.0005$ reported in §IV-I). We frame the per-comparison rate as a specificity proxy under the assumption that inter-CPA pairs constitute a clean negative anchor, and we document in §III-L.4 that this assumption is partially violated by within-firm cross-CPA template-like collision structures.
 
-The use of Firm A as a calibration reference addresses a fundamental challenge in document forensics: the scarcity of ground truth labels.
+## H. Limitations
-In most forensic applications, establishing ground truth requires expensive manual verification or access to privileged information about document provenance.
-Our approach leverages domain knowledge---the established prevalence of non-hand-signing at a specific firm---to create a naturally occurring reference population within the dataset.
 
-This calibration strategy has broader applicability beyond signature analysis.
+Several limitations should be transparent. We group them into primary methodological limitations, secondary scope and validation caveats, documented design features, and engineering-level caveats of the pipeline.
-Any forensic detection system operating on real-world corpora can benefit from identifying subpopulations with known dominant characteristics (positive or negative) to anchor threshold selection, particularly when the distributions of interest are non-normal and non-parametric or mixture-based thresholds are preferred over parametric alternatives.
-The framing we adopt---replication-dominated rather than replication-pure---is an important refinement of this strategy: it prevents overclaim, accommodates the within-firm heterogeneity visible in the unimodal-long-tail shape of Firm A's per-signature cosine distribution, and yields classification rates that are internally consistent with the data.
 
-## F. Pixel-Identity and Inter-CPA Anchors as Annotation-Free Validation
+**Primary methodological limitations.**
 
-A further methodological contribution is the combination of byte-level pixel identity as an annotation-free *conservative* gold positive and a large random-inter-CPA negative anchor.
+*No signature-level ground truth; no true error rates reportable.* The corpus does not contain labelled hand-signed or replicated classes at the signature level. We therefore cannot report False Rejection Rate, sensitivity, recall, Equal Error Rate, ROC-AUC, precision, or positive predictive value against ground truth. All quantitative rates reported in §III-L are inter-CPA negative-anchor coincidence rates (ICCRs) under the assumption that inter-CPA pairs constitute a clean negative anchor; this is a specificity proxy, not a calibrated specificity (§III-M).
-Handwriting physics makes byte-identity impossible under independent signing events, so any pair of same-CPA signatures that are byte-identical after crop and normalization is pair-level proof of image reuse and, modulo the narrow source-template edge case discussed in the seventh limitation below, a conservative positive for non-hand-signing without requiring human review.
-In our corpus 310 signatures satisfied this condition.
-We emphasize that byte-identical pairs are a *subset* of the true non-hand-signed positive class---they capture only those whose nearest same-CPA match happens to be bytewise identical, excluding replications that are pixel-near-identical but not byte-identical (for example, under different scan or compression pathways).
-Perfect recall against this subset therefore does not generalize to perfect recall against the full non-hand-signed population; it is a lower-bound calibration check on the classifier's ability to catch the clearest positives rather than a generalizable recall estimate.
 
-Paired with the $\sim$50,000-pair inter-CPA negative anchor, the byte-identical positives yield FAR estimates with tight Wilson 95% confidence intervals (Table X), which is a substantive improvement over the low-similarity same-CPA negative ($n = 35$) we originally considered.
+*Inter-CPA negative-anchor assumption is partially violated and the violation is firm-dependent.* The cross-firm hit matrix of §III-L.4 shows that under the deployed any-pair rule, within-firm collision concentration is $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms), consistent with firm-specific template, stamp, or document-production reuse. The inter-CPA-as-negative assumption is therefore not exactly satisfied — some inter-CPA pairs may share firm-level templates rather than being independent random matches. Our reported per-comparison ICCRs are best read as specificity-proxy rates under a partially-violated assumption, not as calibrated FARs. Because the violation is firm-dependent, Firm A's per-firm ICCR is more contaminated by within-firm sharing than Firms B/C/D's; the per-firm B/C/D rates of $0.09$–$0.16$ may therefore be less contaminated than the pooled rate, and the Firm A vs Firms B/C/D contrast reflects both genuine firm heterogeneity and a firm-dependent proxy-contamination gradient.
-The combination is a reusable pattern for other document-forensics settings in which the target mechanism leaves a byte-level physical signature in the artifact itself, provided that its generalization limits are acknowledged: FAR is informative, whereas recall is valid only for the conservative subset.
 
-## G. Limitations
+*Scope.* The primary analyses are scoped to the Big-4 sub-corpus. We did not perform the full per-signature pool-normalised ICCR analysis at the full $n = 686$ scope; the §IV-K full-dataset Spearman re-run shows the K=3 $+$ deployed box-rule rank-convergence is preserved at $n = 686$ but does not establish portability of the Big-4 operational ICCRs, the LOOO firm-fold structure, or the five-way operational classifier at the broader scope.
 
-Several limitations should be acknowledged.
+**Secondary scope and validation caveats.**
 
-First, comprehensive per-document ground truth labels are not available.
+*Pixel-identity is a conservative subset.* Byte-identical pairs are the easiest replicated cases, and for the deployed box rule the positive-anchor miss rate against byte-identical pairs is close to tautological (byte-identical $\Rightarrow$ cosine $\approx 1$, dHash $\approx 0$, well inside the high-confidence box). A score that fails the pixel-identity check would be disqualified, but passing the check does not guarantee correct behaviour on the broader replicated population (e.g., re-stamped or noisy-template-variant signatures).
-The pixel-identity anchor is a strict *subset* of the true non-hand-signing positives (only those whose nearest same-CPA match happens to be byte-identical), so perfect recall against this anchor does not establish the classifier's recall on the broader positive class.
-The low-similarity same-CPA anchor ($n = 35$) is small because intra-CPA pairs rarely fall below cosine 0.70; we use the $\sim$50,000-pair inter-CPA negative anchor as the primary negative reference, which yields tight Wilson 95% CIs on FAR (Table X), but it too does not exhaust the set of true negatives (in particular, same-CPA hand-signed pairs with moderate cosine similarity are not sampled).
-A manual-adjudication study concentrated at the decision boundary---for example 100--300 auditor-years stratified by cosine band---would further strengthen the recall estimate against the full positive class.
 
-Second, the ResNet-50 feature extractor was used with pre-trained ImageNet weights without domain-specific fine-tuning.
+*Rule components not separately re-characterised by the present diagnostic battery.* The five-way classifier's moderate-confidence band (cos $> 0.95$ AND $5 < \text{dHash} \leq 15$), the style-consistency band ($\text{dHash} > 15$), and the document-level worst-case aggregation rule retain their prior calibration and capture-rate evidence (supplementary materials); the anchor-based ICCR calibration covers the binary high-confidence sub-rule (and its tightening alternatives such as dHash$\leq 3$), and the alert-rate sensitivity analysis (§III-L.5) characterises only the HC threshold. The MC and HSC sub-band boundaries are not separately re-characterised by the present diagnostic battery.
-While our ablation study and prior literature [20]--[22] support the effectiveness of transferred ImageNet features for signature comparison, a signature-specific feature extractor could improve discriminative performance.
 
-Third, the red stamp removal preprocessing uses simple HSV color-space filtering, which may introduce artifacts where handwritten strokes overlap with red seal impressions.
+*Deployed-rate excess is not a presumed true-positive rate.* The $\sim 44$-pp per-document gap between the observed deployed alert rate (HC: $0.62$ on real same-CPA pools) and the inter-CPA proxy rate (HC: $0.18$) cannot be interpreted as a presumed true-positive rate without additional assumptions that §III-M shows are unsafe (consistent within-CPA signing can exceed inter-CPA similarity at the cosine axis; within-firm template sharing inflates the inter-CPA proxy baseline). The gap is best read as a same-CPA repeatability signal.
-In these overlap regions, blended pixels are replaced with white, potentially creating small gaps in the signature strokes that could reduce dHash similarity.
-This effect would bias classification toward false negatives rather than false positives, but the magnitude has not been quantified.
 
-Fourth, scanning equipment, PDF generation software, and compression algorithms may have changed over the 10-year study period (2013--2023), potentially affecting similarity measurements.
+*A1 pair-detectability stipulation.* The per-signature detector requires at least one same-CPA pair to be near-identical when a CPA uses image replication. A1 is plausible for high-volume stamping or firm-level electronic signing but not guaranteed when a corpus contains only one observed replicated report for a CPA, multiple template variants used in parallel, or scan-stage noise that pushes a replicated pair outside the detection regime.
-While cosine similarity and dHash are designed to be robust to such variations, longitudinal confounds cannot be entirely excluded.
 
-Fifth, our cross-sectional analysis does not track individual CPAs longitudinally and therefore cannot confirm or rule out within-CPA mechanism transitions over the sample period (e.g., a CPA who hand-signed early in the sample and switched to firm-level e-signing later, or vice versa).
+**Documented design features.**
-Extending the analysis to *auditor-year* units---computing per-signature statistics within each fiscal year and observing how individual CPAs move across years---is the natural next step for resolving such within-CPA transitions and is left to future work.
 
-Sixth, the max/min detection logic treats both ends of a near-identical same-CPA pair as non-hand-signed.
+*K=3 hard-posterior membership is composition-sensitive.* The K=3 hard-posterior membership for any single firm varies by up to $12.8$ pp across LOOO folds. This is documented as a composition-sensitivity band rather than failure, but it means K=3 hard labels are not used as operational classifier output; they are reported only as accountant-level descriptive characterisation.
-In the rare case that one of the two documents contains a genuinely hand-signed exemplar that was subsequently reused as the stamping or e-signature template, the pair correctly identifies image reuse but misattributes the non-hand-signed status to the source exemplar.
-This misattribution affects at most one source document per template variant per CPA (the exemplar from which the template was produced), is not expected to be common given that stored signature templates are typically generated in a separate acquisition step rather than extracted from submitted audit reports, and does not materially affect aggregate capture rates at the firm level.
 
-Seventh, our analyses remain at the signature level; we abstain from partner-level frequency inferences such as "X% of CPAs hand-sign in a given year."
+*No partner-level mechanism attribution.* The analysis reports population-level patterns; it does not perform partner-level mechanism attribution or report-level claims of intent. The signature-level outputs are signature-level quantities throughout. The within-firm cross-CPA collision concentration of §III-L.4 is consistent with template-like reuse but is not by itself diagnostic of deliberate sharing.
-Per-signature labels in this paper are not translated to per-report or per-partner mechanism assignments, because making such a translation would require an assumption of within-year uniformity of signing mechanisms that we do not adopt: a CPA's signatures within a single fiscal year may reflect a single replication template, multiple templates used in parallel (e.g., for different engagement positions or reporting pipelines), within-year mechanism mixing, or a combination, and the data at hand do not disambiguate these possibilities (Section III-G).
-The signature-level rates we report, including the 92.5% / 7.5% Firm A split and the year-by-year left-tail share of Section IV-G.1, should accordingly be read as signature-level quantities rather than partner-level frequencies.
 
-Finally, the legal and regulatory implications of our findings depend on jurisdictional definitions of "signature" and "signing."
+**Engineering-level caveats of the pipeline.**
-Whether non-hand-signing of a CPA's own stored signature constitutes a violation of signing requirements is a legal question that our technical analysis can inform but cannot resolve.
+
+*Transferred ImageNet features.* The ResNet-50 feature extractor uses pre-trained ImageNet weights without signature-domain fine-tuning. While our backbone-ablation study (§IV-L) and prior literature support the effectiveness of transferred ImageNet features for signature comparison, a signature-domain fine-tuned feature extractor could improve discriminative performance.
+
+*Red-stamp HSV preprocessing artifacts.* The red stamp removal preprocessing uses simple HSV color-space filtering, which may introduce artifacts where handwritten strokes overlap with red seal impressions. Blended pixels are replaced with white, potentially creating small gaps in signature strokes that could reduce dHash similarity. This bias would push classifications toward false negatives rather than false positives.
+
+*Longitudinal scan / PDF / compression confounds.* Scanning equipment, PDF generation software, and compression algorithms may have changed over the 2013–2023 study period, potentially affecting similarity measurements. While cosine similarity and dHash are designed to be robust to such variations, longitudinal confounds cannot be entirely excluded.
+
+*Source-exemplar misattribution in max/min pair logic.* The max-cosine / min-dHash detection logic treats both ends of a near-identical same-CPA pair as non-hand-signed. In the rare case where one of the two documents contains a genuinely hand-signed exemplar that was subsequently reused as a stamping or e-signature template, the pair correctly identifies image reuse but misattributes non-hand-signed status to the source exemplar. This affects at most one source document per template variant per CPA and is not expected to be common.
+
+*Legal and regulatory interpretation.* Whether non-hand-signing of a CPA's own stored signature constitutes a violation of signing requirements is a jurisdiction-specific legal question. Our technical analysis can inform such determinations but cannot resolve them.

paper/paper_a_impact_statement_v3.md

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+retained for possible reuse in a cover letter, grant report, or non-IEEE
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 # Impact Statement (archived; not in IEEE Access submission)
 
 Auditor signatures on financial reports are a key safeguard of corporate accountability.
 When the signature on an audit report is produced by reproducing a stored image instead of by the partner's own hand---whether through an administrative stamping workflow or a firm-level electronic signing system---this safeguard is weakened, yet detecting the practice through manual inspection is infeasible at the scale of modern financial markets.
 We developed a pipeline that automatically extracts and analyzes signatures from over 90,000 audit reports spanning a decade of filings by publicly listed companies in Taiwan.
-Combining deep-learning visual features with perceptual hashing and two methodologically distinct threshold estimators (plus a density-smoothness diagnostic), the system stratifies signatures into a five-way confidence-graded classification and quantifies how the practice varies across firms and over time.
+Combining deep-learning visual features with perceptual hashing, distributional diagnostics, and anchor-based inter-CPA coincidence-rate calibration, the system stratifies signatures into a five-way confidence-graded classification and quantifies how the practice varies across firms and over time.
-After further validation, the technology could support financial regulators in screening signature authenticity at national scale.
+With a future labelled evaluation set, the technology could support financial regulators in screening candidate non-hand-signed signatures at national scale.
+-->

paper/paper_a_introduction_v3.md

@@ -2,85 +2,44 @@
 
 <!-- Target: ~1.5 pages double-column IEEE format. Double-blind: no author/institution info. -->
 
-Financial audit reports serve as a critical mechanism for ensuring corporate accountability and investor protection.
+Financial audit reports serve as a critical mechanism for ensuring corporate accountability and investor protection. In Taiwan, the Certified Public Accountant Act (會計師法 §4) and the Financial Supervisory Commission's attestation regulations (查核簽證核准準則 §6) require certifying CPAs to affix their signature or seal (簽名或蓋章) to each audit report [1]. While the law permits either a handwritten signature or a seal, the CPA's attestation on each report is intended to represent a deliberate, individual act of professional endorsement for that specific audit engagement [2].
-In Taiwan, the Certified Public Accountant Act (會計師法 §4) and the Financial Supervisory Commission's attestation regulations (查核簽證核准準則 §6) require that certifying CPAs affix their signature or seal (簽名或蓋章) to each audit report [1].
-While the law permits either a handwritten signature or a seal, the CPA's attestation on each report is intended to represent a deliberate, individual act of professional endorsement for that specific audit engagement [2].
 
-The digitization of financial reporting has introduced a practice that complicates this intent.
+The digitization of financial reporting has introduced a practice that complicates this intent. As audit reports are now routinely generated, transmitted, and archived as PDF documents, it is technically and operationally straightforward to reproduce a CPA's stored signature image across many reports rather than re-executing the signing act for each one. This reproduction can occur either through an administrative stamping workflow — in which scanned signature images are affixed by staff as part of the report-assembly process — or through a firm-level electronic signing system that automates the same step. We refer to signatures produced by either workflow collectively as *non-hand-signed*. Although this practice may fall within the literal statutory requirement of "signature or seal," it raises substantive concerns about audit quality, as an identically reproduced signature applied across hundreds of reports may not represent meaningful individual attestation for each engagement. The accounting literature has examined the audit-quality consequences of partner-level engagement transparency: studies of partner-signature mandates in the United Kingdom find measurable downstream effects [31], cross-jurisdictional evidence on individual partner signature requirements highlights similar quality channels [32], and Taiwan-specific evidence on mandatory partner rotation documents how individual-partner identification interacts with audit-quality outcomes [33]. Unlike traditional signature forgery, where a third party attempts to imitate another person's handwriting, non-hand-signing involves the legitimate signer's own stored signature being reused, and is visually invisible to report users at scale.
-As audit reports are now routinely generated, transmitted, and archived as PDF documents, it is technically and operationally straightforward to reproduce a CPA's stored signature image across many reports rather than re-executing the signing act for each one.
-This reproduction can occur either through an administrative stamping workflow---in which scanned signature images are affixed by staff as part of the report-assembly process---or through a firm-level electronic signing system that automates the same step.
-From the perspective of the output image the two workflows are equivalent: both can reproduce one or more stored signature images, producing same-CPA signatures that are identical or near-identical up to reproduction, scanning, compression, and template-variant noise.
-We refer to signatures produced by either workflow collectively as *non-hand-signed*.
-Although this practice may fall within the literal statutory requirement of "signature or seal," it raises substantive concerns about audit quality, as an identically reproduced signature applied across hundreds of reports may not represent meaningful individual attestation for each engagement.
-The accounting literature has long examined the audit-quality consequences of partner-level engagement transparency: studies of partner-signature mandates in the United Kingdom find measurable downstream effects [31], cross-jurisdictional evidence on individual partner signature requirements highlights similar quality channels [32], and Taiwan-specific evidence on mandatory partner rotation documents how individual-partner identification interacts with audit-quality outcomes [33].
-Unlike traditional signature forgery, where a third party attempts to imitate another person's handwriting, non-hand-signing involves the legitimate signer's own stored signature being reused.
-This practice, while potentially widespread, is visually invisible to report users and virtually undetectable through manual inspection at scale: regulatory agencies overseeing thousands of publicly listed companies cannot feasibly examine each signature for evidence of image reproduction.
 
-The distinction between *non-hand-signing detection* and *signature forgery detection* is both conceptually and technically important.
+The distinction between *non-hand-signing detection* and *signature forgery detection* is conceptually and technically important. The extensive body of research on offline signature verification [3]–[8] focuses almost exclusively on forgery detection — determining whether a questioned signature was produced by its purported author. In our context, identity is not in question; the CPA is indeed the legitimate signer. The question is whether the physical act of signing occurred for each individual report, or whether a single signing event was reproduced as an image across many reports. This detection problem differs fundamentally from forgery detection: while it does not require modeling skilled-forger variability, it introduces the distinct challenge of separating legitimate intra-signer consistency from image-level reproduction.
-The extensive body of research on offline signature verification [3]--[8] has focused almost exclusively on forgery detection---determining whether a questioned signature was produced by its purported author or by an impostor.
-This framing presupposes that the central threat is identity fraud.
-In our context, identity is not in question; the CPA is indeed the legitimate signer.
-The question is whether the physical act of signing occurred for each individual report, or whether a single signing event was reproduced as an image across many reports.
-This detection problem differs fundamentally from forgery detection: while it does not require modeling skilled-forger variability, it introduces the distinct challenge of separating legitimate intra-signer consistency from image-level reproduction, requiring an analytical framework focused on detecting abnormally high similarity across documents.
 
-A secondary methodological concern shapes the research design.
+A methodological concern shapes the research design. Many prior similarity-based classification studies rely on ad-hoc thresholds — declaring two images equivalent above a hand-picked cosine cutoff, for example — without principled statistical justification. Such thresholds are fragile in an archival-data setting. A defensible approach requires (i) explicit calibration of the operational thresholds against measurable negative-anchor evidence; (ii) diagnostic procedures that test whether the descriptor distribution itself supports a within-population threshold, including formal decomposition of apparent multimodality into between-group composition and integer-tie artefacts; (iii) annotation-free reporting of operational alarm rates at multiple analysis units (per-comparison, per-signature pool, per-document) with Wilson 95% confidence intervals; (iv) per-firm stratification of the reported rates to surface heterogeneity that aggregate metrics conceal; and (v) explicit disclosure of the unsupervised setting's limits — in particular, the inability to estimate true error rates without signature-level ground-truth labels.
-Many prior similarity-based classification studies rely on ad-hoc thresholds---declaring two images equivalent above a hand-picked cosine cutoff, for example---without principled statistical justification.
-Such thresholds are fragile and invite reviewer skepticism, particularly in an archival-data setting where the cost of misclassification propagates into downstream inference.
-A defensible approach requires (i) a transparent threshold anchored to an empirical reference population drawn from the target corpus; (ii) statistical diagnostics that characterise the *shape* of the underlying similarity distribution and so motivate the choice of anchor; and (iii) external validation against naturally-occurring anchor populations---byte-level identical pairs as a conservative gold positive subset and large random inter-CPA pairs as a gold negative population---reported with Wilson 95% confidence intervals on per-rule capture / FAR rates, since precision and $F_1$ are not meaningful when the positive and negative anchor populations are sampled from different units.
 
-Despite the significance of the problem for audit quality and regulatory oversight, no prior work has specifically addressed non-hand-signing detection in financial audit documents at scale with these methodological safeguards.
+Despite the significance of the problem for audit quality and regulatory oversight, to our knowledge no prior work has specifically addressed non-hand-signing detection in financial audit documents at scale with these methodological safeguards. Woodruff et al. [9] developed an automated pipeline for signature analysis in corporate filings for anti-money-laundering investigations, but their work focused on author clustering rather than detecting image reuse. Copy-move forgery detection methods [10], [11] address duplicated regions within or across images but are designed for natural images and do not account for the specific characteristics of scanned document signatures. Research on near-duplicate image detection using perceptual hashing combined with deep learning [12], [13] provides relevant methodological foundations but has not been applied to document forensics or signature analysis. From the statistical side, the methods we adopt for distributional characterisation — the Hartigan dip test [37] and finite mixture modelling via the EM algorithm [40], [41], complemented by a Burgstahler-Dichev / McCrary density-smoothness diagnostic [38], [39] — have been developed in statistics and accounting-econometrics but have not been combined as a joint diagnostic toolkit for document-forensics threshold characterisation.
-Woodruff et al. [9] developed an automated pipeline for signature analysis in corporate filings for anti-money-laundering investigations, but their work focused on author clustering (grouping signatures by signer identity) rather than detecting reuse of a stored image.
-Copy-move forgery detection methods [10], [11] address duplicated regions within or across images but are designed for natural images and do not account for the specific characteristics of scanned document signatures, where legitimate visual similarity between a signer's authentic signatures is expected and must be distinguished from image reproduction.
-Research on near-duplicate image detection using perceptual hashing combined with deep learning [12], [13] provides relevant methodological foundations but has not been applied to document forensics or signature analysis.
-From the statistical side, the methods we adopt for distributional characterisation---the Hartigan dip test [37] and finite mixture modelling via the EM algorithm [40], [41], complemented by a Burgstahler-Dichev / McCrary density-smoothness diagnostic [38], [39]---have been developed in statistics and accounting-econometrics but have not, to our knowledge, been combined as a joint diagnostic toolkit for document-forensics threshold selection.
 
-In this paper, we present a fully automated, end-to-end pipeline for detecting non-hand-signed CPA signatures in audit reports at scale.
+In this paper we present a fully automated, end-to-end pipeline for screening non-hand-signed CPA signatures in audit reports at scale, together with an anchor-calibrated screening framework that characterises the pipeline's operational behaviour under explicit unsupervised assumptions. The pipeline processes raw PDF documents through (1) signature page identification with a Vision-Language Model; (2) signature region detection with a trained YOLOv11 object detector; (3) deep feature extraction via a pre-trained ResNet-50; (4) dual-descriptor similarity (cosine + independent-minimum dHash); (5) anchor-based threshold calibration at three units of analysis (per-comparison, pool-normalised per-signature, per-document) against an inter-CPA negative-anchor coincidence-rate proxy (§III-L); (6) firm-stratified per-rule reporting and a within-firm cross-CPA hit-matrix analysis (§III-L.4); (7) a composition decomposition that establishes the absence of a within-population bimodal antimode in the descriptor distributions (§III-I.4); and (8) disclosure of each diagnostic's untested assumption (§III-M).
-Our approach processes raw PDF documents through the following stages:
-(1) signature page identification using a Vision-Language Model (VLM);
-(2) signature region detection using a trained YOLOv11 object detector;
-(3) deep feature extraction via a pre-trained ResNet-50 convolutional neural network;
-(4) dual-descriptor similarity computation combining cosine similarity on deep embeddings with difference hash (dHash) distance;
-(5) signature-level distributional characterisation using two threshold estimators---KDE antimode with a Hartigan unimodality test and finite Beta mixture via EM with a logit-Gaussian robustness check---complemented by a Burgstahler-Dichev / McCrary density-smoothness diagnostic, used to read the structure of the per-signature similarity distribution and to motivate a percentile-based operational anchor rather than a mixture-fit crossing; and
-(6) validation against a pixel-identical anchor, a low-similarity anchor, and a replication-dominated Big-4 calibration firm.
 
-The dual-descriptor verification is central to our contribution.
+A key empirical finding is that the descriptor distributions do not support a within-population natural threshold. The apparent multimodality in the Big-4 accountant-level distribution is explained by between-firm location-shift effects (Firm A's mean dHash of $2.73$ versus Firms B/C/D's $6.46$, $7.39$, $7.21$) and integer mass-point artefacts on the integer-valued dHash axis. After joint firm-mean centring and uniform integer-tie jitter, the pooled dHash dip-test rejection disappears ($p_{\text{median}} = 0.35$ across five seeds). Within-firm diagnostics in every Big-4 firm fail to reveal stable bimodal structure after accounting for integer ties; eligible non-Big-4 firms provide corroborating raw-axis evidence on the cosine dimension (§III-I.4). We therefore treat mixture fits as descriptive summaries of firm-compositional structure rather than threshold-generating mechanisms, and calibrate the deployed operating rules using inter-CPA coincidence-rate anchors.
-Cosine similarity of deep feature embeddings captures high-level visual style similarity---it can identify signatures that share similar stroke patterns and spatial layouts---but cannot distinguish between a CPA who signs consistently and one whose signature is reproduced from a stored image.
-Perceptual hashing (specifically, difference hashing) encodes structural-level image gradients into compact binary fingerprints that are robust to scan noise but sensitive to substantive content differences.
-By requiring convergent evidence from both descriptors, we can differentiate *style consistency* (high cosine but divergent dHash) from *image reproduction* (high cosine with low dHash), resolving an ambiguity that neither descriptor can address alone.
 
-A second distinctive feature is our framing of the calibration reference.
+In place of distributional anchoring, we adopt an anchor-based inter-CPA coincidence-rate (ICCR) calibration. At the per-comparison unit, the cos$>0.95$ operating point yields ICCR $= 0.00060$ on a $5 \times 10^5$-pair Big-4 sample; the dHash$\leq 5$ structural cutoff yields ICCR $= 0.00129$; the joint rule cos$>0.95$ AND dHash$\leq 5$ yields joint ICCR $= 0.00014$ (any-pair semantics, matching the deployed extrema rule). At the pool-normalised per-signature unit, the same rule's effective coincidence rate is materially higher because the deployed classifier takes max-cosine and min-dHash over a same-CPA pool: pooled Big-4 any-pair ICCR is $0.1102$ (Wilson 95% CI $[0.1086, 0.1118]$; CPA-block bootstrap 95% $[0.0908, 0.1330]$). At the per-document unit, the operational HC$+$MC alarm fires on $33.75\%$ of Big-4 documents under the inter-CPA candidate-pool counterfactual.
-One major Big-4 accounting firm in Taiwan (hereafter "Firm A") was selected as a candidate calibration reference based on practitioner-knowledge motivation; its benchmark status is then evaluated using the image evidence reported in this paper, not asserted by the practitioner-knowledge motivation itself.
-We therefore treat Firm A as a *replication-dominated* calibration reference rather than a pure positive class.
-This framing is important because the statistical signature of a replication-dominated population is visible in our data: Firm A's per-signature cosine distribution is unimodal with a long left tail (Hartigan dip $p = 0.17$), 92.5% of Firm A signatures exceed cosine 0.95 with the remaining 7.5% forming the left tail, and 145 Firm A signatures across 50 distinct partners are byte-identical to a same-CPA match in a different audit report (35 spanning different fiscal years).
-Adopting the replication-dominated framing---rather than a near-universal framing that would have to absorb the 7.5% residual as noise---ensures internal coherence between the byte-level pixel-identity evidence and the signature-level distributional shape.
 
-A third distinctive feature is the empirical reading we take from the per-signature distributional analysis.
+The pooled per-signature and per-document rates conceal striking firm heterogeneity. A logistic regression of the per-signature hit indicator on firm dummies (Firm A reference) and centred log pool size yields odds ratios of $0.053$ (Firm B), $0.010$ (Firm C), and $0.027$ (Firm D) — Firms B/C/D are an order of magnitude below Firm A even after controlling for the pool-size confound. Cross-firm hit matrix analysis under the deployed any-pair rule shows within-firm collision concentrations of $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D (Table XXV; the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms). The pattern is consistent with firm-specific template, stamp, or document-production reuse mechanisms — though not by itself diagnostic of deliberate sharing. The deployed five-way box rule defines a reproducible screening classifier; the calibration contribution is to characterise its multi-level inter-CPA coincidence behaviour rather than to derive new thresholds. The high-confidence sub-rule (cos $> 0.95$ AND dHash $\leq 5$) and moderate-confidence sub-rule (cos $> 0.95$ AND $5 < \text{dHash} \leq 15$) are explicit decision rules whose calibrated false-positive and false-negative error rates remain unknown in the absence of signature-level labels.
-Three diagnostics applied to the per-signature similarity distribution---the Hartigan dip test, an EM-fitted Beta mixture (with logit-Gaussian robustness check), and the Burgstahler-Dichev / McCrary density-smoothness procedure---jointly indicate that no two-mechanism mixture cleanly explains per-signature similarity: the dip test fails to reject unimodality for Firm A, BIC strongly prefers a 3-component over a 2-component Beta fit, and the BD/McCrary candidate transition lies *inside* the non-hand-signed mode rather than between modes (and is not bin-width-stable; Appendix A).
-The substantive reading is that *pixel-level output quality* is a continuous spectrum shaped by firm-specific reproduction technologies (administrative stamping in early years, firm-level e-signing later) and scan conditions, rather than a discrete class cleanly separated from hand-signing.
-This reading motivates anchoring the operational classifier on a percentile heuristic over the Firm A reference distribution rather than on a mixture-fit crossing, and it motivates the byte-level pixel-identity anchor (Section IV-F.1) as a threshold-free positive reference that does not depend on resolving signature-level mixture structure.
 
-We apply this pipeline to 90,282 audit reports filed by publicly listed companies in Taiwan between 2013 and 2023, extracting and analyzing 182,328 individual CPA signatures from 758 unique accountants.
+Three feature-derived scores converge on the per-CPA descriptor-position ranking with Spearman $\rho \geq 0.879$: the K=3 mixture posterior (a firm-compositional position score under §III-J's reading, not a mechanism cluster posterior), a reverse-anchor cosine percentile relative to a strictly-out-of-target non-Big-4 reference, and the box-rule less-replication-dominated rate. The three scores are deterministic functions of the same per-CPA descriptor pair, so the convergence is documented as internal consistency among feature-derived ranks rather than external validation. A conservative hard-positive subset for image replication is provided by 262 byte-identical signatures in the Big-4 subset (Firm A 145, Firm B 8, Firm C 107, Firm D 2), against which all three candidate checks achieve $0\%$ positive-anchor miss rate (Wilson 95% upper bound $1.45\%$). For the box rule this result is close to tautological at byte-identity; we discuss the conservative-subset caveat in §V-G.
-To our knowledge, this represents the largest-scale forensic analysis of signature authenticity in financial documents reported in the literature.
 
-The contributions of this paper are summarized as follows:
+We apply this pipeline to 90,282 audit reports filed by publicly listed companies in Taiwan between 2013 and 2023, extracting and analyzing 182,328 individual CPA signatures from 758 unique accountants. The Big-4 sub-corpus comprises 437 CPAs and 150,442 signatures with both descriptors available.
 
-1. **Problem formulation.** We formally define non-hand-signing detection as distinct from signature forgery detection and argue that it requires an analytical framework focused on intra-signer similarity distributions rather than genuine-versus-forged classification.
+The contributions of this paper are:
 
-2. **End-to-end pipeline.** We present a pipeline that processes raw PDF audit reports through VLM-based page identification, YOLO-based signature detection, deep feature extraction, and dual-descriptor similarity computation, with automated inference requiring no manual intervention after initial training and annotation.
+1. **Problem formulation.** We define non-hand-signing detection as distinct from signature forgery detection and frame it as a detection problem on intra-signer similarity distributions.
 
-3. **Dual-descriptor verification.** We demonstrate that combining deep-feature cosine similarity with perceptual hashing resolves the fundamental ambiguity between style consistency and image reproduction, and we validate the backbone choice through an ablation study comparing three feature-extraction architectures.
+2. **End-to-end pipeline.** We present a pipeline that processes raw PDF audit reports through VLM-based page identification, YOLO-based signature detection, ResNet-50 feature extraction, and dual-descriptor similarity computation, with automated inference and no manual intervention after initial training.
 
-4. **Percentile-anchored operational threshold.** We anchor the operational classifier's cosine cut on the whole-sample Firm A P7.5 percentile (cos $> 0.95$), a transparent and reproducible reference drawn from a replication-dominated reference population, and complement it with dHash structural cuts derived from the same reference distribution. Operational thresholds are therefore explained by an empirical reference rather than asserted.
+3. **Dual-descriptor similarity.** We demonstrate that combining deep-feature cosine similarity with independent-minimum dHash resolves the ambiguity between *style consistency* and *image reproduction*, and we support the backbone choice through a feature-backbone ablation.
 
-5. **Distributional characterisation of per-signature similarity.** We apply three statistical diagnostics---a Hartigan dip test, an EM-fitted Beta mixture with logit-Gaussian robustness check, and a Burgstahler-Dichev / McCrary density-smoothness procedure---to characterise the shape of the per-signature similarity distribution. The three diagnostics jointly find that per-signature similarity forms a continuous quality spectrum, which both motivates the percentile-based operational anchor over a mixture-fit crossing and is itself a substantive finding for the document-forensics literature on similarity-threshold selection.
+4. **Composition decomposition disproves the distributional-threshold path.** We show via a 2×2 factorial diagnostic (firm-mean centring × integer-tie jitter) that the apparent multimodality of the Big-4 accountant-level descriptor distribution is fully attributable to between-firm location shifts and integer mass-point artefacts. The descriptor distributions contain no within-population bimodal antimode; a distributional "natural threshold" reading of the operating points is not empirically supported.
 
-6. **Replication-dominated calibration methodology.** We introduce a calibration strategy using a replication-dominated reference group, distinguishing *replication-dominated* from *replication-pure* anchors; and we validate classification using byte-level pixel identity as an annotation-free gold positive, requiring no manual labeling.
+5. **Anchor-based multi-level inter-CPA coincidence-rate calibration.** We characterise the deployed five-way classifier at three units of analysis: per-comparison ICCR (cos$>0.95$: $0.0006$; dHash$\leq 5$: $0.0013$; joint: $0.00014$), pool-normalised per-signature ICCR ($0.11$ for the deployed any-pair high-confidence rule), and per-document ICCR ($0.34$ for the operational HC$+$MC alarm). We adopt "inter-CPA coincidence rate" as the metric name throughout and reserve "False Acceptance Rate" for terminology that requires ground-truth negative labels, which the corpus does not provide.
 
-7. **Large-scale empirical analysis.** We report findings from the analysis of over 90,000 audit reports spanning a decade, providing the first large-scale empirical evidence on non-hand-signing practices in financial reporting under a methodology designed for peer-review defensibility.
+6. **Firm heterogeneity quantification and within-firm cross-CPA collision concentration.** Per-document D2 inter-CPA proxy ICCRs differ by an order of magnitude across firms (Firm A: $0.62$ versus Firms B/C/D: $0.09$–$0.16$); a per-signature logistic regression of the any-pair HC hit indicator on firm dummies and centred log pool size confirms the firm gap persists after pool-size control. Cross-firm hit matrix analysis shows within-firm collision concentrations of $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D under the deployed any-pair rule (the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms); the pattern is consistent with firm-specific template, stamp, or document-production reuse mechanisms — a descriptive finding about deployed-rule behaviour, not a claim of deliberate template sharing.
 
-The remainder of this paper is organized as follows.
+7. **K=3 as descriptive firm-compositional partition; three-score convergent internal consistency.** We fit a K=3 Gaussian mixture as a descriptive partition of the Big-4 accountant-level distribution (interpreted as firm-compositional structure, not as three mechanism clusters). Three feature-derived scores agree on the per-CPA descriptor-position ranking at Spearman $\rho \geq 0.879$; we report this as internal consistency rather than external validation, given that the scores share the underlying descriptor pair.
-Section II reviews related work on signature verification, document forensics, perceptual hashing, and the statistical methods we adopt for distributional characterisation.
+
-Section III describes the proposed methodology.
+8. **Annotation-free positive-anchor capture check and unsupervised-setting disclosure.** We achieve $0\%$ positive-anchor miss rate (Wilson 95% upper bound $1.45\%$) on 262 byte-identical Big-4 signatures, with the conservative-subset caveat that byte-identical pairs are by construction near cos$=1$ and dHash$=0$. Each supporting diagnostic in §III-M addresses one specific failure mode of an unsupervised screening classifier — composition artefacts, inter-CPA coincidence, pool-size confounding, firm heterogeneity, threshold sensitivity, or positive-anchor capture — with an explicitly disclosed untested assumption. We do not claim a validated forensic detector; we position the system as a specificity-proxy-anchored screening framework with human-in-the-loop review.
-Section IV presents experimental results including the signature-level distributional characterisation, pixel-identity validation, and backbone ablation study.
+
-Section V discusses the implications and limitations of our findings.
+The remainder of the paper is organized as follows. Section II reviews related work on signature verification, document forensics, perceptual hashing, and the statistical methods used. Section III describes the proposed methodology. Section IV presents the experimental results — distributional characterisation, mixture fits, convergent internal-consistency checks, leave-one-firm-out reproducibility, pixel-identity positive-anchor check, and full-dataset robustness. Section V discusses the implications and limitations. Section VI concludes with directions for future work.
-Section VI concludes with directions for future work.

paper/paper_a_methodology_v3.md

@@ -4,19 +4,19 @@
 
 We propose a six-stage pipeline for large-scale non-hand-signed auditor signature detection in scanned financial documents.
 Fig. 1 illustrates the overall architecture.
-The pipeline takes as input a corpus of PDF audit reports and produces, for each document, a classification of its CPA signatures along a confidence continuum anchored on whole-sample Firm A percentile heuristics and validated against a byte-level pixel-identity positive anchor and a large random inter-CPA negative anchor.
+The pipeline takes as input a corpus of PDF audit reports and produces five-way operational screening labels (§III-H.1) whose behaviour is characterised by pixel-identity positive-anchor capture checks and inter-CPA coincidence-rate calibration (§III-L).
 
 Throughout this paper we use the term *non-hand-signed* rather than "digitally replicated" to denote any signature produced by reproducing a previously stored image of the partner's signature---whether by administrative stamping workflows (dominant in the early years of the sample) or firm-level electronic signing systems (dominant in the later years).
 From the perspective of the output image the two workflows are equivalent: both can reproduce one or more stored signature images, producing same-CPA signatures that are identical or near-identical up to reproduction, scanning, compression, and template-variant noise.
 
 <!--
 [Figure 1: Pipeline Architecture - clean vector diagram]
-90,282 PDFs → VLM Pre-screening → 86,072 PDFs
+90,282 PDFs → VLM Pre-screening → 86,084 candidates → processing checks → 86,071 PDFs
 → YOLOv11 Detection → 182,328 signatures
 → ResNet-50 Features → 2048-dim embeddings
 → Dual-Descriptor Verification (Cosine + dHash)
-→ Firm A P7.5-anchored Classifier → Five-way classification
+→ Anchor-Calibrated Five-Way Classifier → Five-way classification
-→ Pixel-identity + Inter-CPA + Held-Out Firm A validation
+→ Pixel-identity Positive Anchor + Inter-CPA Coincidence-Rate Negative Anchor
 -->
 
 ## B. Data Collection
@@ -29,15 +29,16 @@ Each report is a multi-page PDF document containing, among other content, the au
 CPA names, affiliated accounting firms, and audit engagement tenure were obtained from a publicly available audit-firm tenure registry encompassing 758 unique CPAs across 15 document types, with the majority (86.4%) being standard audit reports.
 Table I summarizes the dataset composition.
 
-<!-- TABLE I: Dataset Summary
+**Table I.** Dataset Summary.
+
 | Attribute | Value |
 |-----------|-------|
 | Total PDF documents | 90,282 |
 | Date range | 2013–2023 |
-| Documents with signatures | 86,072 (95.4%) |
+| Signature-page candidates (VLM-positive) | 86,084 (95.3%) |
+| Processed for signature extraction | 86,071 (95.3%) |
 | Unique CPAs identified | 758 |
 | Accounting firms | >50 |
--->
 
 ## C. Signature Page Identification
 
@@ -47,8 +48,8 @@ The model was configured with temperature 0 for deterministic output.
 
 The scanning range was restricted to the first quartile of each document's page count, reflecting the regulatory structure of Taiwanese audit reports in which the auditor's report page is consistently located in the first quarter of the document.
 Scanning terminated upon the first positive detection.
-This process identified 86,072 documents with signature pages; the remaining 4,198 documents (4.6%) were classified as having no signatures and excluded.
+This process identified 86,084 documents with signature pages; the remaining 4,198 documents (4.6%) were classified as having no signatures and excluded.
-An additional 12 corrupted PDFs were excluded, yielding a final set of 86,071 documents.
+An additional 13 PDFs that could not be rendered (corruption or read errors) were excluded, yielding a final set of 86,071 documents.
 
 Cross-validation between the VLM and subsequent YOLO detection confirmed high agreement: YOLO successfully detected signature regions in 98.8% of VLM-positive documents.
 The 1.2% disagreement reflects the combined rate of (i) VLM false positives (pages incorrectly flagged as containing signatures) and (ii) YOLO false negatives (signature regions missed by the detector), and we do not attempt to attribute the residual to either source without further labeling.
@@ -61,20 +62,19 @@ A region was labeled as "signature" if it contained any Chinese handwritten cont
 
 The model was trained for 100 epochs on a 425/75 training/validation split with COCO pre-trained initialization, achieving strong detection performance (Table II).
 
-<!-- TABLE II: YOLO Detection Performance
+**Table II.** YOLO Detection Performance.
+
 | Metric | Value |
 |--------|-------|
 | Precision | 0.97–0.98 |
 | Recall | 0.95–0.98 |
 | mAP@0.50 | 0.98–0.99 |
 | mAP@0.50:0.95 | 0.85–0.90 |
--->
 
 Batch inference on all 86,071 documents extracted 182,328 signature images at a rate of 43.1 documents per second (8 workers).
 A red stamp removal step was applied to each cropped signature using HSV color-space filtering, replacing detected red regions with white pixels to isolate the handwritten content.
 
-Each signature was matched to its corresponding CPA using positional order (first or second signature on the page) against the official CPA registry, achieving a 92.6% match rate (168,755 of 182,328 signatures).
+Each signature was matched to its corresponding CPA using positional order (first or second signature on the page) against the official CPA registry, achieving a 92.6% match rate (168,755 of 182,328 signatures). The matched records assume standard two-signature ordering; residual order-mismatch risk remains for nonstandard layouts. The remaining 7.4% (13,573 signatures) could not be matched to a registered CPA name---typically because the auditor's report page format deviates from the standard two-signature layout, or because OCR of the printed CPA name on the page returns a name not present in the registry---and these signatures are excluded from all subsequent same-CPA pairwise analyses (a same-CPA best-match statistic is undefined when a signature has no assigned CPA). The 92.6% matched subset forms the candidate pool for same-CPA analyses, before the Big-4 and descriptor-completeness restrictions described in §III-G.
-The remaining 7.4% (13,573 signatures) could not be matched to a registered CPA name---typically because the auditor's report page format deviates from the standard two-signature layout, or because OCR of the printed CPA name on the page returns a name not present in the registry---and these signatures are excluded from all subsequent same-CPA pairwise analyses (a same-CPA best-match statistic is undefined when a signature has no assigned CPA). The 92.6% matched subset is the sample that flows into Sections IV-D through IV-H; the unmatched 7.4% are excluded for definitional reasons rather than discarded as noise.
 
 ## E. Feature Extraction
 
@@ -84,8 +84,8 @@ The final classification layer was removed, yielding the 2048-dimensional output
 Preprocessing consisted of resizing to 224×224 pixels with aspect-ratio preservation and white padding, followed by ImageNet channel normalization.
 All feature vectors were L2-normalized, ensuring that cosine similarity equals the dot product.
 
-The choice of ResNet-50 without fine-tuning was motivated by three considerations: (1) the task is similarity comparison rather than classification, making general-purpose discriminative features sufficient; (2) ImageNet features have been shown to transfer effectively to document analysis tasks [20], [21]; and (3) avoiding domain-specific fine-tuning reduces the risk of overfitting to dataset-specific artifacts, though we note that a fine-tuned model could potentially improve discriminative performance (see Section V-G).
+The choice of ResNet-50 without fine-tuning was motivated by three considerations: (1) the task is similarity comparison rather than classification, making general-purpose discriminative features sufficient; (2) ImageNet features have been shown to transfer effectively to document analysis tasks [20], [21]; and (3) avoiding domain-specific fine-tuning reduces the risk of overfitting to dataset-specific artifacts, though we note that a fine-tuned model could potentially improve discriminative performance (see Section V-H, Engineering-level caveats).
-This design choice is validated by an ablation study (Section IV-I) comparing ResNet-50 against VGG-16 and EfficientNet-B0.
+This design choice is supported by an ablation study (Section IV-L) comparing ResNet-50 against VGG-16 and EfficientNet-B0.
 
 ## F. Dual-Method Similarity Descriptors
 
@@ -105,190 +105,349 @@ Unlike DCT-based perceptual hashes, dHash is computationally lightweight and par
 
 These descriptors provide partially independent evidence.
 Cosine similarity is sensitive to the full feature distribution and reflects fine-grained execution variation; dHash captures only coarse perceptual structure and is robust to scanner-induced noise.
-Non-hand-signing yields extreme similarity under *both* descriptors, since the underlying image is identical up to reproduction noise.
+Non-hand-signing is expected to yield extreme similarity under *both* descriptors, since the underlying image is identical up to reproduction noise; scan-stage noise can in principle push a replicated pair off either extremum but rarely both.
-Hand-signing, by contrast, yields high dHash similarity (the overall layout of a signature is preserved across writing occasions) but measurably lower cosine similarity (fine execution varies).
+Hand-signing, by contrast, often yields high dHash similarity (the overall layout of a signature is typically preserved across writing occasions) but measurably lower cosine similarity (fine execution varies).
 Convergence of the two descriptors is therefore a natural robustness check; when they disagree, the case is flagged as borderline.
 
-We specifically excluded SSIM (Structural Similarity Index) [30] after empirical testing showed it to be unreliable for scanned documents: the calibration firm (Section III-H) exhibited a mean SSIM of only 0.70 due to scan-induced pixel-level variations, despite near-identical visual content.
+We do not use SSIM (Structural Similarity Index) [30] or pixel-level comparison as primary descriptors. SSIM was developed as a perceptual quality index for natural images and is by construction sensitive to the local-luminance and local-contrast perturbations routine in a print-scan cycle (JPEG block artefacts, scan-noise speckle, scanner-rule ghosts) — properties that penalise identically-reproduced signature crops at the very margins SSIM is designed to weight most heavily. Pixel-level distances ($L_1$, $L_2$, pixel-identity counting) are defined on geometrically aligned images at a common resolution and inflate under the sub-pixel offsets that scanner DPI, paper-handling alignment, and PDF-page rasterisation routinely introduce, so two scans of the same physical document cannot score near-identically. The supplementary materials contain the full design-level argument; pixel-identity counting is retained only as a threshold-free positive anchor (§III-K), because byte-identical pairs are necessarily produced by literal file reuse and so do not interact with the alignment-fragility argument.
-Cosine similarity and dHash are both robust to the noise introduced by the print-scan cycle.
 
-## G. Unit of Analysis and Summary Statistics
+Cosine similarity on L2-normalised deep embeddings and dHash both remain stable across the print-scan-rasterise cycle by design [14], [19], [21], [27]; together they constitute the dual descriptor used throughout the rest of this paper.
 
-Two unit-of-analysis choices are relevant for this study, ordered from finest to coarsest: (i) the *signature*---one signature image extracted from one report; and (ii) the *auditor-year*---all signatures by one CPA within one fiscal year.
+## G. Unit of Analysis and Scope
-The signature is the operational unit of classification (Section III-K) and of all primary statistical analyses (Section IV-D, IV-F, IV-G).
-The auditor-year is used in the partner-level similarity ranking of Section IV-G.2 as a within-year aggregation unit: each auditor-year's mean is computed over its own fiscal-year signatures, although the per-signature best-match cosine that feeds the mean is computed against the full same-CPA cross-year pool (Section III-G's max-cosine / min-dHash definition).
-We do not use a coarser CPA-level cross-year unit, because pooling a CPA's signatures across the full 2013--2023 sample period would conflate distinct signing-mechanism regimes whenever a CPA's practice changes during the sample, and we make no claim about the within-CPA stability of signing mechanisms over time.
 
-For per-signature classification we compute, for each signature, the maximum pairwise cosine similarity and the minimum dHash Hamming distance against every other signature attributed to the same CPA (over the full same-CPA set, not restricted to the same fiscal year).
+We analyse signatures at two units of resolution. The **signature** — one signature image extracted from one report — is the operational unit of classification (§III-H.1) and of the signature-level analyses in §IV (notably §IV-J for the five-way per-signature category counts and the inter-CPA negative-anchor coincidence-rate analysis referenced in §IV-I). The **accountant** — one CPA aggregated over all of their signatures in the corpus — is the unit of mixture-model characterisation (§III-J), of per-CPA internal-consistency analysis (§III-K), and of the leave-one-firm-out reproducibility check (§III-K). At the accountant level we compute, for each CPA with $n_{\text{sig}} \geq 10$ signatures, the per-CPA mean of the per-signature best-match cosine ($\overline{\text{cos}}_a$) and the per-CPA mean of the independent-minimum dHash ($\overline{\text{dHash}}_a$). The minimum threshold of 10 signatures per CPA is required for the per-CPA mean to be a stable summary; CPAs below this threshold are excluded from the accountant-level analyses but remain in the per-signature analyses.
-The max/min (rather than mean) formulation reflects the identification logic for non-hand-signing: if even one other signature of the same CPA is a pixel-level reproduction, that pair will dominate the extremes and reveal the non-hand-signed mechanism.
-Mean statistics would dilute this signal.
 
-For the dHash dimension we use the *independent minimum dHash*: the minimum Hamming distance from a signature to *any* other signature of the same CPA (over the full same-CPA set).
+We make no within-year or across-year uniformity assumption about CPA signing mechanisms. Per-signature labels are signature-level quantities throughout this paper; we do not translate them to per-report or per-partner mechanism assignments, and we abstain from partner-level frequency inferences (such as "X% of CPAs hand-sign") that would require such a translation. A CPA's per-CPA mean is a *summary statistic* of their observed signatures, not a claim that all of their signatures share a single mechanism.
-The independent minimum is unconditional on the cosine-nearest pair and is therefore the conservative structural-similarity statistic; it is the dHash statistic used throughout the operational classifier (Section III-K) and all reported capture-rate analyses.
 
-We make one stipulation about same-CPA pair detectability.
+We adopt one stipulation about same-CPA pair detectability:
 
-**(A1) Pair-detectability.** *If a CPA uses image replication anywhere in the corpus, then at least one same-CPA signature pair is near-identical (after reproduction noise) within the cross-year same-CPA pool used by the max-cosine / min-dHash computation above.*
+> **(A1) Pair-detectability.** *If a CPA uses image replication anywhere in the corpus, then at least one same-CPA signature pair is near-identical (after reproduction noise) within the cross-year same-CPA pool used by the max-cosine / min-dHash computation.*
-This is plausible for high-volume stamping or firm-level electronic-signing workflows---where a stored image is typically reused many times under similar scan and compression conditions---but it is *not* guaranteed when (i) the corpus contains only one observed replicated report for a CPA, (ii) multiple template variants are in use simultaneously, or (iii) scan-stage noise pushes a replicated pair outside the detection regime.
-A1 is a *cross-year pair-existence* property, not a within-year uniformity claim, and is the only assumption the per-signature detector requires to be sensitive to replication.
 
-We make *no* within-year or across-year uniformity assumption about CPA signing mechanisms.
+A1 is plausible for high-volume stamping or firm-level electronic signing workflows but is not guaranteed when (i) the corpus contains only one observed replicated report for a CPA, (ii) multiple template variants are used in parallel, or (iii) scan-stage noise pushes a replicated pair outside the detection regime. A1 is the only assumption the per-signature detector requires to be sensitive to replication.
-Per-signature labels are signature-level quantities throughout this paper; we do not translate them to per-report or per-partner mechanism assignments, and we abstain from partner-level frequency inferences (such as "X% of CPAs hand-sign") that would require such a translation.
-A CPA's signing output within a single fiscal year may reflect a single replication template, multiple templates used in parallel (e.g., different stored images for different engagement positions or reporting pipelines), within-year mechanism mixing, or a combination; our signature-level analyses remain valid under all of these regimes, since they do not attempt mechanism attribution at the partner or report level.
 
-The intra-report consistency analysis in Section IV-G.3 is a firm-level homogeneity check---whether the *two co-signing CPAs on the same report* receive the same signature-level label under the operational classifier---rather than a test of within-partner or within-year uniformity.
+**Scope: the Big-4 sub-corpus.** The primary analyses (§III-I, §III-J, §III-K, §III-L, and the corresponding §IV-D through §IV-J and §IV-M tables) are restricted to the four largest accounting firms in Taiwan, pseudonymously labelled Firm A through Firm D throughout the manuscript. §IV-A through §IV-C and §IV-L report the corpus-wide pipeline performance and feature-backbone ablation that support the descriptor choice of §III-F; §IV-K reports a deliberately narrow full-dataset cross-check at $n = 686$ CPAs. The Big-4 sub-corpus comprises 437 CPAs (171 / 112 / 102 / 52 across Firms A through D) with $n_{\text{sig}} \geq 10$ — the threshold for accountant-level analyses — totalling 150,442 Big-4 signatures with both pre-computed descriptors available. Restricting the primary analyses to Big-4 is a methodological choice driven by four considerations:
 
-## H. Calibration Reference: Firm A as a Replication-Dominated Population
+1. **Restricted generalisability claim and Big-4 institutional comparability.** The primary claims are scoped to the Big-4 audit-report context, where the four firms share comparable institutional scale, document-production infrastructure, and CPA-volume regime; we do not assert that the same descriptive mixture structure or operational alert behaviour extends to mid/small firms. The 249 non-Big-4 CPAs enter only (a) as an external reference population in §III-H.2's reverse-anchor internal-consistency check, (b) as a robustness comparison in §IV-K, and (c) as a corroborating-population check on the dHash discrete-mass-point artefact in §III-I.4. Generalisation beyond Big-4 is left as future work.
 
-A distinctive aspect of our methodology is the use of Firm A---a major Big-4 accounting firm in Taiwan---as an empirical calibration reference.
+2. **Within-firm cross-CPA collision structure analysis.** §III-L.4 reports a Big-4 cross-firm hit-matrix analysis that quantifies the within-firm cross-CPA template-like collision pattern. The four-firm setting affords the cleanest signal for this analysis; replicating the same matrix structure on the heterogeneous mid/small-firm tail is left as future work.
-Rather than treating Firm A as a synthetic or laboratory positive control, we treat it as a naturally occurring *replication-dominated population*: a CPA population whose aggregate signing behavior is dominated by non-hand-signing but is not a pure positive class.
 
-Practitioner knowledge motivated treating Firm A as a candidate calibration reference: the firm is understood within the audit profession to reproduce a stored signature image for the majority of certifying partners---originally via administrative stamping workflows and later via firm-level electronic signing systems---while not ruling out that a minority of partners may continue to hand-sign some or all of their reports.
+3. **Firm A as templated-end case study.** Firm A is empirically the firm whose CPAs are most concentrated in the high-cosine, low-dHash corner of the descriptor plane (§III-J K=3 component cross-tab; byte-level pair analysis referenced in §III-H.2). We retain Firm A within the Big-4 scope as a descriptive case study of the templated end rather than as the calibration anchor for thresholds.
-This practitioner background is *non-load-bearing* in our analysis: the evidentiary basis used in this paper is the observable image evidence reported below---byte-identical same-CPA pairs, the Firm A per-signature similarity distribution, partner-ranking concentration, and intra-report consistency---which does not depend on any claim about signing practice beyond what the audit-report images themselves show.
 
-We establish Firm A's replication-dominated status through two primary independent quantitative analyses plus a third strand comprising three complementary checks, each of which can be reproduced from the public audit-report corpus alone:
+4. **Leave-one-firm-out fold feasibility.** §III-K reports leave-one-firm-out (LOOO) cross-validation of the Big-4 K=3 fit. The Big-4 sub-corpus permits a four-fold LOOO at the firm level (one fold per Big-4 firm). No analogous firm-level fold is available outside Big-4 because mid/small firms have CPA counts of $O(1)$–$O(30)$ per firm.
 
-First, *automated byte-level pair analysis* (Section IV-F.1; reproduced by `signature_analysis/28_byte_identity_decomposition.py` with output in `reports/byte_identity_decomp/byte_identity_decomposition.json`) identifies 145 Firm A signatures that are byte-identical to at least one other same-CPA signature from a different audit report, distributed across 50 distinct Firm A partners (of 180 registered); 35 of these byte-identical matches span different fiscal years.
+**Sample-size reconciliation.** Two Big-4 signature counts appear in this section and §IV: $n = 150{,}442$ for analyses using the pre-computed per-signature descriptors $\text{cos}_s$ (`max_similarity_to_same_accountant`) and $\text{dHash}_s$ (`min_dhash_independent`), and $n = 150{,}453$ for analyses recomputing pair-level metrics directly from the stored feature and dHash byte vectors (Scripts 40b, 43, 44). The $11$-signature difference reflects descriptor-completion status: $11$ signatures have feature vectors and dHash byte vectors stored but lack the pre-computed extrema. The $11$ signatures are negligible at population scale and do not affect any reported coincidence rate within $0.01$ percentage point. The CPA counts $468$ (all Big-4 CPAs with both vectors stored) and $437$ (Big-4 CPAs with $n_{\text{sig}} \geq 10$ for accountant-level stability) likewise reflect a single uniform exclusion rule rather than analysis-specific subsetting.
-Byte-identity implies pixel-identity by construction, and independent hand-signing cannot produce pixel-identical images across distinct reports---these pairs therefore establish image reuse as a concrete, threshold-free phenomenon within Firm A and confirm that replication is widespread (50 of 180 registered partners) rather than confined to a handful of CPAs.
 
-Second, *signature-level distributional evidence*: Firm A's per-signature best-match cosine distribution fails to reject unimodality (Hartigan dip test $p = 0.17$, $N = 60{,}448$ Firm A signatures; Section IV-D) and exhibits a long left tail, consistent with a dominant high-similarity regime plus residual within-firm heterogeneity rather than two cleanly separated mechanisms.
+## H. Operational Classifier and Reference Populations
-92.5% of Firm A's per-signature best-match cosine similarities exceed 0.95 and the remaining 7.5% form the long left tail (we do not disaggregate partner-level mechanism here; see Section III-G for the scope of claims).
-The unimodal-long-tail shape, not the precise 92.5/7.5 split, is the structural evidence: it predicts that Firm A is replication-dominated rather than a clean two-class population, and a noise-only explanation of the left tail would predict a shrinking share as scan/PDF technology matured over 2013--2023, which is not what we observe (Section IV-G.1).
 
-Third, we additionally validate the Firm A benchmark through three complementary analyses reported in Section IV-G. Only the partner-level ranking is fully threshold-free; the longitudinal-stability and intra-report analyses use the operational classifier and are interpreted as consistency checks on its firm-level output:
+### H.1. Deployed Operational Rule
-  (a) *Longitudinal stability (Section IV-G.1).* The share of Firm A per-signature best-match cosine values below 0.95 is stable at 6-13% across 2013-2023, with the lowest share in 2023. The 0.95 cutoff is the whole-sample Firm A P7.5 heuristic (Section III-K; 92.5% of whole-sample Firm A signatures exceed this cutoff); the substantive finding here is the *temporal stability* of the rate, not the absolute rate at any single year.
-  (b) *Partner-level similarity ranking (Section IV-G.2).* When every auditor-year is ranked globally by its per-auditor-year mean best-match cosine (across all firms: Big-4 and Non-Big-4), Firm A auditor-years account for 95.9% of the top decile against a baseline share of 27.8% (a 3.5$\times$ concentration ratio), and this over-representation is stable across 2013-2023. This analysis uses only the ordinal ranking and is independent of any absolute cutoff.
-  (c) *Intra-report consistency (Section IV-G.3).* Because each Taiwanese statutory audit report is co-signed by two engagement partners, firm-wide stamping practice predicts that both signers on a given Firm A report should receive the same signature-level label under the classifier. Firm A exhibits 89.9% intra-report agreement against 62-67% at the other Big-4 firms. This test uses the operational classifier and is therefore a *consistency* check on the classifier's firm-level output rather than a threshold-free test; the cross-firm gap (not the absolute rate) is the substantive finding.
 
-We emphasize that the 92.5% figure is a within-sample consistency check rather than an independent validation of Firm A's status; the validation role is played by the byte-level pixel-identity evidence, the unimodal-long-tail dip-test result, the three complementary analyses above, and the held-out Firm A fold (described in Section III-J; fold-level rate differences are disclosed in Section IV-F.2).
+Each Big-4 signature is assigned to one of five categories using the per-signature descriptor pair $(\text{cos}_s, \text{dHash}_s)$ where $\text{cos}_s$ is the maximum cosine similarity to another signature by the same CPA and $\text{dHash}_s$ is the minimum independent dHash to another signature by the same CPA:
 
-We emphasize that Firm A's replication-dominated status was *not* derived from the thresholds we calibrate against it.
+1. **High-confidence non-hand-signed (HC):** Cosine $> 0.95$ AND $\text{dHash}_{\text{indep}} \leq 5$. Both descriptors converge on strong replication evidence.
-Its identification rests on the byte-level pair evidence and the dip-test-confirmed unimodal-long-tail shape, both of which are independent of any threshold choice.
+2. **Moderate-confidence non-hand-signed (MC):** Cosine $> 0.95$ AND $5 < \text{dHash}_{\text{indep}} \leq 15$. Feature-level evidence is strong; structural similarity is present but below the high-confidence cutoff.
-The "replication-dominated, not pure" framing is important both for internal consistency---it predicts and explains the long left tail observed in Firm A's cosine distribution (Section IV-D)---and for avoiding overclaim in downstream inference.
+3. **High style consistency (HSC):** Cosine $> 0.95$ AND $\text{dHash}_{\text{indep}} > 15$. High feature-level similarity without structural corroboration — consistent with a CPA who signs very consistently but not via image reproduction.
+4. **Uncertain (UN):** Cosine between the all-pairs intra/inter KDE crossover ($0.837$) and $0.95$.
+5. **Likely hand-signed (LH):** Cosine $\leq 0.837$.
 
-## I. Signature-Level Threshold Characterisation
+Document-level labels are aggregated via the worst-case rule: each audit report inherits the most-replication-consistent category among its certifying-CPA signatures (rank order HC > MC > HSC > UN > LH). The thresholds ($\text{cos} = 0.95$ as the cosine operating point, $\text{cos} = 0.837$ as the all-pairs KDE crossover, $\text{dHash} = 5$ and $15$ as structural-similarity sub-band cutoffs) retain their prior calibration provenance (see supplementary materials). These thresholds define the deployed screening rule; the present analysis does not re-derive them as optimal cutoffs but characterises their behaviour under inter-CPA coincidence anchors (developed in §III-L).
 
-This section describes how we set the operational classifier's similarity threshold and how we characterise the per-signature similarity distribution that supports it.
+The remainder of this section (§III-H.2) describes the reference populations used to calibrate and cross-check this rule. §III-I demonstrates that the descriptor distributions do not provide a within-population natural threshold; §III-J–§III-K develop the descriptive partition and internal-consistency cross-checks; §III-L develops the anchor-based threshold calibration; §III-M discloses the unsupervised-setting limits.
-The two roles are kept separate by design.
 
-> **Operational threshold (used by the classifier).** The cosine cut is anchored on the whole-sample Firm A P7.5 percentile (cos $> 0.95$; Section III-K).
+### H.2. Reference Populations
->
-> **Statistical characterisation (used to motivate the choice of anchor and to describe the distributional structure).** A Hartigan dip test, an EM-fitted Beta mixture (with logit-Gaussian robustness check), and a Burgstahler-Dichev / McCrary density-smoothness procedure---all applied at the per-signature level (Section IV-D).
 
-The reason for the split is empirical.
+The calibration distinguishes two reference populations: Firm A as a within-Big-4 templated-end case study, and the 249 non-Big-4 CPAs as an out-of-target reference for internal-consistency checking.
-The three statistical diagnostics jointly find that per-signature similarity forms a continuous quality spectrum (Section IV-D, summarised below): the dip test fails to reject unimodality for Firm A; BIC strongly prefers a 3-component over a 2-component Beta fit, so the 2-component crossing is a forced fit; and the BD/McCrary candidate transition lies inside the non-hand-signed mode rather than between modes (and is not bin-width-stable; Appendix A).
-Under these conditions the natural anchor for an operational cosine cut is a transparent percentile of a replication-dominated reference population (Firm A) rather than a mixture-fit crossing whose location depends on parametric assumptions the data do not support.
 
-We describe the three diagnostics and the assumptions underlying each in the subsections below.
+**Internal reference: Firm A as the templated-end case study.** Firm A is empirically the firm whose CPAs are most concentrated in the high-cosine, low-dHash corner of the Big-4 descriptor plane. In the Big-4 K=3 descriptive partition (§III-J; Scripts 35, 38), Firm A accounts for 0% of the C1 component (low-cos / high-dHash corner; cos $\approx 0.946$, dHash $\approx 9.17$, weight $\approx 0.143$), 17.5% of the C2 component (central region), and 82.5% of the C3 component (high-cos / low-dHash corner); the opposite pattern holds at Firm C (Script 35: 23.5% C1, 75.5% C2, 1.0% C3, hereafter referred to as "the Firm whose CPAs are most concentrated in C1"). Byte-level decomposition of these signatures (see supplementary materials) identifies 145 Firm A pixel-identical signatures, spanning 50 distinct Firm A partners of the 180 registered, with 35 byte-identical matches occurring across different fiscal years; the 145 are the Firm A portion of the 262 byte-identical Big-4 signatures.
-The two threshold estimators rest on decreasing-in-strength assumptions: the KDE antimode/crossover requires only smoothness; the Beta mixture additionally requires a parametric specification, and the logit-Gaussian cross-check reports sensitivity to that form.
-The Burgstahler-Dichev / McCrary procedure is applied to the same distribution as a *density-smoothness diagnostic*: it would identify a sharp local density discontinuity if one existed at the boundary between two cleanly separated mechanisms.
-Because all three diagnostics are applied to the same sample rather than to independent experiments, agreement or disagreement among them is read as evidence about distributional structure rather than as a formal statistical guarantee.
 
-### 1) Method 1: KDE Antimode / Crossover with Unimodality Test
+Firm A is *not* the calibration anchor for the operational threshold. Firm A enters the Big-4 mixture on equal footing with Firms B through D; the K=3 components are derived from the joint Big-4 distribution (§III-J), not from Firm A alone. Firm A's role in the methodology is descriptive: it is the Big-4 firm whose CPAs are most concentrated in the high-cosine, low-dHash corner of the descriptor plane, and the byte-level pair evidence above provides the firm-level signature-reuse evidence that anchors §III-K's pixel-identity positive-anchor miss rate.
 
-We use two closely related KDE-based threshold estimators and apply each where it is appropriate.
+**External reference: non-Big-4 as the reverse-anchor reference for internal-consistency checking.** The 249 non-Big-4 CPAs ($n_{\text{sig}} \geq 10$, drawn from $\sim$30 mid- and small-firms) constitute a population strictly outside the Big-4 target. Their per-CPA $(\overline{\text{cos}}_a, \overline{\text{dHash}}_a)$ distribution defines a 2D Gaussian reference (fit by Minimum Covariance Determinant with support fraction 0.85 for robustness; Script 38). This reference is used in §III-K's reverse-anchor internal-consistency check: each Big-4 CPA's location relative to the reference centre, measured as the marginal cosine cumulative-distribution-function value under the reference, is one of three feature-derived scores used as a cross-check on the per-signature classifier. The reverse-anchor reference is *not* a positive or negative anchor for threshold derivation — its role is to provide a strictly out-of-target benchmark against which the within-Big-4 mixture-derived ranking can be internally cross-checked.
-When two labeled populations are available (e.g., the all-pairs intra-class and inter-class similarity distributions of Section IV-C), the *KDE crossover* is the intersection point of the two kernel density estimates under Scott's rule for bandwidth selection [28]; under equal priors and symmetric misclassification costs it approximates the Bayes-optimal decision boundary between the two classes.
-When a single distribution is analysed (e.g., the per-signature best-match cosine distribution of Section IV-D) the *KDE antimode* is the local density minimum between two modes of the fitted density; it serves the same decision-theoretic role when the distribution is multimodal but is undefined when the distribution is unimodal.
-In either case we use the Hartigan & Hartigan dip test [37] as a formal test of unimodality.
-The dip test asks one question: *is the distribution single-peaked?*
-A non-significant $p$-value means we cannot reject the single-peak null (the data are consistent with one peak); a significant $p$-value means the distribution has *more than one peak* (it could be two, three, or more---the test does not specify how many).
-We use the test to decide whether a KDE antimode is well-defined (it is, only when there is more than one peak), not to assert any particular number of components.
-We additionally perform a sensitivity analysis varying the bandwidth over $\pm 50\%$ of the Scott's-rule value to verify threshold stability.
 
-### 2) Method 2: Finite Mixture Model via EM
+The reverse-anchor reference centre is at $\overline{\text{cos}} = 0.935$, $\overline{\text{dHash}} = 9.77$ (Script 38). The reference sits at a lower cosine and higher dHash than the Big-4 K=3 low-cos / high-dHash component (cos $= 0.946$, dHash $= 9.17$; §III-J); compared to the Big-4 high-cos / low-dHash component (cos $= 0.983$, dHash $= 2.41$; §III-J) the reference is markedly less replication-dominated. The reverse-anchor metric for a given Big-4 CPA is the percentile of $\overline{\text{cos}}_a$ within the reference marginal cosine distribution, sign-flipped so that lower percentile (further into the left tail of the reference) corresponds to a Big-4 CPA whose mean cosine sits further from the templated end of the descriptor plane. This is a "deviation in the less-replication-dominated descriptor-position direction" measure, not a "deviation toward the templated descriptor-position" measure; the reference is the less-replication-dominated population.
 
-We fit a two-component Beta mixture to the cosine distribution via the EM algorithm [40] using method-of-moments M-step estimates (which are numerically stable for bounded proportion data).
+## I. Distributional Diagnostics: Why the Composition Path Does Not Yield a Natural Threshold
-The first component represents non-hand-signed signatures (high mean, narrow spread) and the second represents hand-signed signatures (lower mean, wider spread).
-Under the fitted model the threshold is the crossing point of the two weighted component densities,
 
-$$\pi_1 \cdot \text{Beta}(x; \alpha_1, \beta_1) = (1 - \pi_1) \cdot \text{Beta}(x; \alpha_2, \beta_2),$$
+This section characterises the joint distribution of accountant-level descriptor means $(\overline{\text{cos}}_a, \overline{\text{dHash}}_a)$ across the 437 Big-4 CPAs of §III-G and tests whether the distribution provides distributional support — in the form of within-population bimodality — for the deployed operational thresholds. We apply four diagnostic procedures in turn: a univariate unimodality test on each accountant-level marginal; a 2D Gaussian mixture fit (developed in §III-J); a density-smoothness diagnostic; and a composition decomposition that distinguishes within-population multimodality from between-firm location-shift artefacts. The four diagnostics jointly imply that the operational thresholds are *not* anchored by distributional bimodality: §III-L develops an anchor-based calibration framework that does not require this assumption.
 
-solved numerically via bracketed root-finding.
+**1. Hartigan dip test on each accountant-level marginal.** We apply the Hartigan & Hartigan dip test [37] to each of the two marginal distributions $\{\overline{\text{cos}}_a\}_{a=1}^{437}$ and $\{\overline{\text{dHash}}_a\}_{a=1}^{437}$, with bootstrap-based $p$-value estimation ($n_{\text{boot}} = 2000$). In both cases no bootstrap replicate exceeded the observed dip statistic, so the empirical $p$-value is bounded above by $5 \times 10^{-4}$; we report this in tables as $p < 5 \times 10^{-4}$ rather than $p = 0$ to reflect the bootstrap resolution (Script 34). For comparison, no rejection of unimodality holds in the comparison scopes tested in Script 32: Firm A pooled alone ($p_{\text{cos}} = 0.992$, $p_{\text{dHash}} = 0.924$, $n = 171$); Firms B + C + D pooled ($p_{\text{cos}} = 0.998$, $p_{\text{dHash}} = 0.906$, $n = 266$); all non-Firm-A CPAs pooled ($p_{\text{cos}} = 0.998$, $p_{\text{dHash}} = 0.907$, $n = 515$). Single-firm dip tests for Firms B, C, and D were not separately computed; the comparison scopes above sufficed to establish that no narrower-than-Big-4 *tested* scope at the accountant level rejected unimodality. The accountant-level Big-4 rejection is a descriptive observation; §III-I.4 below shows that the rejection is fully explained by between-firm location-shift effects rather than within-population bimodality.
-As a robustness check against the Beta parametric form we fit a parallel two-component Gaussian mixture to the *logit-transformed* similarity, following standard practice for bounded proportion data.
-White's [41] quasi-MLE consistency result justifies interpreting the logit-Gaussian estimates as asymptotic approximations to the best Gaussian-family fit under misspecification; we use the cross-check between Beta and logit-Gaussian crossings as a diagnostic of parametric-form sensitivity rather than as a guarantee of distributional recovery.
 
-We fit 2- and 3-component variants of each mixture and report BIC for model selection.
+**2. K=2 / K=3 Gaussian mixture fits (descriptive partition).** A 2-component 2D Gaussian Mixture Model (full covariance, $n_{\text{init}} = 15$, fixed seed 42; Script 34) recovers components at $(\overline{\text{cos}}, \overline{\text{dHash}}) = (0.954, 7.14)$, weight $0.689$, and $(0.983, 2.41)$, weight $0.311$. The marginal crossings of the K=2 fit are $\overline{\text{cos}}^* = 0.9755$ and $\overline{\text{dHash}}^* = 3.755$, with bootstrap 95% confidence intervals $[0.9742, 0.9772]$ and $[3.48, 3.97]$ over $n_{\text{boot}} = 500$ resamples. The 3-component fit (§III-J) is BIC-preferred — using the convention that lower BIC is preferred, $\text{BIC}(K{=}3) - \text{BIC}(K{=}2) = -3.48$ (Script 36). The $\Delta$BIC magnitude is small in absolute terms; we do not treat $\Delta\text{BIC} = 3.5$ alone as decisive evidence for K=3 as a population mixture. Following §III-I.4 we treat both K=2 and K=3 fits as *descriptive partitions* of the joint Big-4 distribution that reflect firm-composition structure (Firm A vs others; §III-J) rather than as inferential evidence for two or three latent population modes.
-When BIC prefers the 3-component fit, the 2-component assumption itself is a forced fit, and the Bayes-optimal threshold derived from the 2-component crossing should be treated as an upper bound rather than a definitive cut.
 
-### 3) Density-Smoothness Diagnostic: Burgstahler-Dichev / McCrary
+**3. Burgstahler-Dichev / McCrary density-smoothness diagnostic.** We apply the discontinuity test of [38, 39] as a *density-smoothness diagnostic* (rather than as a threshold estimator) on each accountant-level marginal axis (cosine in bins of $0.002$, dHash in integer bins). At the Big-4 scope, the diagnostic identifies no significant transition on either marginal at $\alpha = 0.05$ (Script 34). Outside Big-4, the diagnostic does flag dHash transitions in some subsets (Script 32: `big4_non_A` dHash transition at $10.8$; `all_non_A` dHash transition at $6.6$; pre-2018 and post-2020 time-stratified variants also exhibit one or more dHash transitions), but no cosine transition is identified in any subset. The Big-4-scope null on both axes is consistent with §III-I.4 below: under the composition decomposition the Big-4 marginals are unimodal once between-firm and integer-tie confounds are removed, so a local-discontinuity test correctly fails to flag a within-population transition.
 
-Complementing the two threshold estimators above, we apply the discontinuity test of Burgstahler and Dichev [38], made asymptotically rigorous by McCrary [39], as a *density-smoothness diagnostic* rather than as a third threshold estimator.
+**4. Composition decomposition (Scripts 39b–39e).** §III-I.1 establishes that the accountant-level marginals reject unimodality at the Big-4 sub-corpus. The remaining question is whether the rejection reflects (a) genuine within-population bimodality at the signature or accountant level, (b) between-firm location-shift artefacts (firms with different mean descriptor positions pool to a multi-peaked distribution), or (c) integer mass-point artefacts on the integer-valued dHash axis (the dHash dip statistic is sensitive to spikes at integer values). We apply four diagnostics that decompose the rejection into these candidate sources:
-We discretize each distribution (cosine into bins of width 0.005; $\text{dHash}_\text{indep}$ into integer bins) and compute, for each bin $i$ with count $n_i$, the standardized deviation from the smooth-null expectation of the average of its neighbours,
 
-$$Z_i = \frac{n_i - \tfrac{1}{2}(n_{i-1} + n_{i+1})}{\sqrt{N p_i (1-p_i) + \tfrac{1}{4} N (p_{i-1}+p_{i+1})(1 - p_{i-1} - p_{i+1})}},$$
+*Within-firm signature-level dip (Scripts 39b, 39c).* Repeating the dip test at the signature level inside each individual Big-4 firm (Script 39b) and inside each individual non-Big-4 firm with $\geq 500$ signatures (Script 39c) yields a consistent picture. The cosine marginal *fails* to reject unimodality in every single firm tested — all four Big-4 firms ($p_{\text{cos}} \in \{0.176, 0.991, 0.551, 0.976\}$ for Firms A through D; Script 39b) and ten non-Big-4 firms with $\geq 500$ signatures ($p_{\text{cos}} \in [0.59, 0.99]$; Script 39c). The raw dHash marginal *does* reject unimodality in every firm tested ($p < 5 \times 10^{-4}$ in all $14$ firms), but the raw dHash values are integer-valued in $\{0, 1, \ldots, 64\}$, leaving open the possibility of an integer-tie artefact.
 
-which is approximately $N(0,1)$ under the null of distributional smoothness.
+*Integer-jitter robustness (Scripts 39d, 39e).* Adding independent uniform jitter $\sim \mathrm{U}[-0.5, +0.5]$ to break exact dHash ties and re-running the dip test on the perturbed signature cloud (5 seeds, $n_{\text{boot}} = 2000$; Script 39d) eliminates the dHash within-firm rejection in every Big-4 firm tested (Firm A jittered $p_{\text{median}} = 0.999$; B $0.996$; C $0.999$; D $0.9995$; $0$/$5$ seeds reject at $\alpha = 0.05$ in any firm). The pooled-Big-4 dHash dip *does* survive jitter alone ($p_{\text{median}} = 0$, $5$/$5$ seeds reject), but Firm A's mean dHash ($2.73$) is substantially below Firms B/C/D's ($6.46$, $7.39$, $7.21$) — a between-firm location shift. Script 39e applies a 2 \times 2 factorial correction (firm-mean centring $\times$ integer jitter) on the Big-4 pooled dHash:
-A candidate transition is identified at an adjacent bin pair where $Z_{i-1}$ is significantly negative and $Z_i$ is significantly positive (cosine) or the reverse (dHash).
-Appendix A reports a bin-width sensitivity sweep covering $\text{bin} \in \{0.003, 0.005, 0.010, 0.015\}$ for cosine and $\text{bin} \in \{1, 2, 3\}$ for dHash; the sweep shows that signature-level BD transitions are not bin-width-stable, consistent with histogram-resolution artifacts rather than a genuine cross-mode density discontinuity.
-We therefore do not treat the BD/McCrary procedure as a threshold estimator in our application but as diagnostic evidence about distributional smoothness.
 
-### 4) Reading the Three Diagnostics Together
+| Condition | Firm-mean centred | Integer jitter | Median dip $p$ |  Reject at $\alpha = 0.05$ |
+|---|---|---|---|---|
+| 1 raw | — | — | $< 5 \times 10^{-4}$ | $5/5$ |
+| 2 centred only | $\checkmark$ | — | $< 5 \times 10^{-4}$ | $5/5$ |
+| 3 jittered only | — | $\checkmark$ | $< 5 \times 10^{-4}$ | $5/5$ |
+| 4 centred and jittered | $\checkmark$ | $\checkmark$ | $\mathbf{0.35}$ | $\mathbf{0/5}$ |
 
-The two threshold estimators rest on decreasing-in-strength assumptions: the KDE antimode/crossover requires only smoothness; the Beta mixture additionally requires a parametric specification (with logit-Gaussian as a robustness cross-check against that form).
+Removing *both* the between-firm location shift *and* the integer mass points eliminates the Big-4 dHash rejection. The Big-4 pooled dHash multimodality is therefore fully attributable to firm-composition contrast (primarily Firm A's mean $\text{dHash} = 2.73$ versus Firms B/C/D $\approx 6.5$–$7.4$) and integer-density artefacts, with no residual continuous within-firm bimodality.
-If the two estimated thresholds were to differ by less than a practically meaningful margin and the BD/McCrary procedure were to identify a sharp transition at the same level, that pattern would constitute convergent evidence for a clean two-mechanism boundary at that location.
 
-This is *not* the pattern we observe at the per-signature level.
+*Cosine analogue.* The cosine axis follows the same pattern by construction: the within-firm signature-level cosine dip tests above (Scripts 39b, 39c) fail to reject in every Big-4 firm and in every eligible non-Big-4 firm, so any pooled cosine multimodality must arise from between-firm composition rather than from within-population bimodality.
-The two threshold estimators yield crossings spread across a wide range (Section IV-D); the BIC clearly prefers a 3-component over a 2-component Beta fit, indicating that the 2-component crossing is a forced fit and should be read as an upper bound rather than a definitive cut; and the BD/McCrary procedure locates its candidate transition *inside* the non-hand-signed mode rather than between modes (Appendix A).
-We interpret this jointly as evidence that per-signature similarity is a continuous quality spectrum rather than a clean two-mechanism mixture, and we accordingly anchor the operational classifier's cosine cut on whole-sample Firm A percentile heuristics (Section III-K) rather than on a mixture-fit crossing.
 
-## J. Pixel-Identity, Inter-CPA, and Held-Out Firm A Validation (No Manual Annotation)
+*Integer-histogram valleys (Script 39d).* A genuine within-firm dHash antimode would appear as a strict local minimum in the count histogram with deep relative depth. Within each of the four Big-4 firms, the dHash histogram on bins $0$–$20$ exhibits no strict local minimum; the Big-4 pooled histogram exhibits one shallow valley at $\text{dHash} = 4$ with relative depth $0.021$ (a $2.1\%$ count drop). No valley near the deployed $\text{dHash} = 5$ operational boundary appears within any individual firm. The hypothesised dHash antimode near $\text{dHash} \approx 5$ is not empirically supported by the histogram analysis.
 
-Rather than construct a stratified manual-annotation validation set, we validate the classifier using four naturally occurring reference populations that require no human labeling:
+**5. Conclusion: no natural threshold from the descriptor distribution.** §III-I.4 jointly establishes that (a) the Big-4 accountant-level dip rejection is fully attributable to between-firm composition and integer mass-point artefacts; (b) within the Big-4 firms, the descriptor marginals at the signature level are unimodal once integer ties are broken (Scripts 39b, 39d); (c) eligible non-Big-4 checks provide corroborating raw-axis evidence on the cosine dimension (Script 39c) and corroborate the integer-mass-point reading of raw dHash, but are not used as calibration evidence for the deployed thresholds; and (d) no integer-histogram valley near the deployed $\text{dHash} = 5$ operational boundary exists within any Big-4 firm. The descriptor distributions therefore do not contain a within-population bimodal antimode that could anchor an operational threshold. The K=2 / K=3 mixture fits of §III-I.2 and §III-J are retained as *descriptive partitions* that reflect firm-composition contrast, not as inferential evidence for two or three population modes. §III-L develops the anchor-based threshold calibration framework, which derives operational rates from inter-CPA pair-level negative-anchor coincidences rather than from a distributional antimode.
 
-1. **Pixel-identical anchor (gold positive, conservative subset):** signatures whose nearest same-CPA match is byte-identical after crop and normalization.
+## J. K=3 as a Descriptive Partition of Firm-Composition Contrast
-Handwriting physics makes byte-identity impossible under independent signing events, so a byte-identical same-CPA pair is pair-level proof of image reuse and---for the byte-identical subset---conservative ground truth for non-hand-signed signatures; the narrow exception, in which a genuinely hand-signed exemplar was subsequently reused as the stamping or e-signature template, is discussed as a Limitation in Section V-G.
-We further emphasize that this anchor is a *subset* of the true positive class---only those non-hand-signed signatures whose nearest match happens to be byte-identical---and perfect recall against this anchor therefore does not establish recall against the full non-hand-signed population (Section V-G discusses this further).
 
-2. **Inter-CPA negative anchor (large gold negative):** $\sim$50,000 pairs of signatures randomly sampled from *different* CPAs.
+This section develops the K=2 and K=3 Gaussian mixture fits to the Big-4 accountant-level distribution and clarifies their role. **Both fits are descriptive partitions of the joint Big-4 distribution; they reflect firm-composition contrast — primarily Firm A versus Firms B, C, D — rather than within-population mechanism modes.** §III-I.4 demonstrates that the apparent multimodality of the accountant-level marginals is fully explained by between-firm location shifts and integer mass-point artefacts, leaving no residual evidence for two or three latent within-population mechanism classes. Neither mixture is used to assign signature-level or document-level labels in the primary analysis. The operational classifier of §III-H.1 is calibrated in §III-L via inter-CPA negative-anchor coincidence rates, not via mixture-derived antimodes.
-Inter-CPA pairs cannot arise from reuse of a single signer's stored signature image, so this population is a reliable negative class for threshold sweeps.
-This anchor is substantially larger than a simple low-similarity-same-CPA negative and yields tight Wilson 95% confidence intervals on FAR at each candidate threshold.
 
-3. **Firm A anchor (replication-dominated prior positive):** Firm A signatures, treated as a majority-positive reference with within-firm heterogeneity in the left tail, as evidenced by the 7.5% of Firm A signatures whose per-signature best-match cosine falls at or below 0.95 (Section III-H, Section IV-D).
+**K=2 fit.** Two components at $(\overline{\text{cos}}, \overline{\text{dHash}}) = (0.954, 7.14)$ (weight $0.689$) and $(0.983, 2.41)$ (weight $0.311$) (Script 34). $\text{BIC}(K{=}2) = -1108.45$. Marginal crossings: $\overline{\text{cos}}^* = 0.9755$, $\overline{\text{dHash}}^* = 3.755$. We refer to the components by index rather than by mechanism labels, since §III-I.4 establishes that the K=2 separation is firm-compositional rather than mechanistic.
-Because Firm A is both used for empirical percentile calibration in Section III-H and as a validation anchor, we make the within-Firm-A sampling variance visible by splitting Firm A CPAs randomly (at the CPA level, not the signature level) into a 70% *calibration* fold and a 30% *heldout* fold.
-The calibration-fold percentiles used in thresholding---cosine median, P1, and P5 (lower-tail, since higher cosine indicates greater similarity), and dHash_indep median and P95 (upper-tail, since lower dHash indicates greater similarity)---are derived from the 70% calibration fold only.
-The heldout fold is used exclusively to report post-hoc capture rates with Wilson 95% confidence intervals.
 
-4. **Low-similarity same-CPA anchor (supplementary negative):** signatures whose maximum same-CPA cosine similarity is below 0.70.
+**K=3 fit.** Three components, sorted by ascending cosine mean (Script 35; Script 38 reproduces):
-This anchor is retained for continuity with prior work but is small in our dataset ($n = 35$) and is reported only as a supplementary reference; its confidence intervals are too wide for quantitative inference.
 
-From these anchors we report FAR with Wilson 95% confidence intervals against the inter-CPA negative anchor.
+| Component | $\overline{\text{cos}}$ | $\overline{\text{dHash}}$ | weight | descriptive position |
-We do not report an Equal Error Rate or FRR column against the byte-identical positive anchor, because byte-identical pairs have cosine $\approx 1$ by construction and any FRR computed against that subset is trivially $0$ at every threshold below $1$; the conservative-subset role of the byte-identical anchor is instead discussed qualitatively in Section V-F.
+|---|---|---|---|---|
-Precision and $F_1$ are not meaningful in this anchor-based evaluation because the positive and negative anchors are constructed from different sampling units (intra-CPA byte-identical pairs vs random inter-CPA pairs), so their relative prevalence in the combined set is an arbitrary construction rather than a population parameter; we therefore omit precision and $F_1$ from Table X.
+| C1 | 0.9457 | 9.17 | 0.143 | low-cos / high-dHash corner |
-The 70/30 held-out Firm A fold of Section IV-F.2 additionally reports capture rates with Wilson 95% confidence intervals computed within the held-out fold, which is a valid population for rate inference.
+| C2 | 0.9558 | 6.66 | 0.536 | central region |
+| C3 | 0.9826 | 2.41 | 0.321 | high-cos / low-dHash corner |
 
-## K. Per-Document Classification
+$\text{BIC}(K{=}3) = -1111.93$, lower than $K{=}2$ by $3.48$ (mild numerical preference for K=3 under standard BIC interpretation, but not by itself decisive). The "descriptive position" column refrains from any mechanism interpretation: §III-I.4 establishes that the cosine and dHash axes both lack within-population bimodality, so component centres are best interpreted as locations in a continuous descriptor space rather than as latent mechanism modes.
 
-The per-signature classifier operates at the signature level with operational thresholds anchored on whole-sample Firm A percentile heuristics: cos $> 0.95$ (Firm A P7.5) for the cosine dimension and dHash$_\text{indep} \leq 5$ / $> 15$ (Firm A median+P75 / style-consistency ceiling) for the structural dimension.
+**Per-firm component composition (Script 35 firm × cluster cross-tab).** The K=3 partition is dominated by firm membership:
-This percentile-based anchor is the natural choice given the continuous-spectrum shape of the per-signature similarity distribution documented in Section IV-D; sensitivity to nearby alternatives is reported in Section IV-F.3.
-All dHash references in this section refer to the *independent-minimum* dHash defined in Section III-G---the smallest Hamming distance from a signature to any other same-CPA signature.
-We use a single dHash statistic throughout the operational classifier and the supporting capture-rate analyses (Tables IX, XI, XII, XVI), which keeps the classifier definition and its empirical evaluation arithmetically consistent.
 
-We assign each signature to one of five signature-level categories using convergent evidence from both descriptors:
+- Firm A: $0\%$ C1, $17.5\%$ C2, $82.5\%$ C3
+- Firm B: $8.9\%$ C1, $\sim 78\%$ C2, $\sim 13\%$ C3
+- Firm C: $23.5\%$ C1, $75.5\%$ C2, $1.0\%$ C3
+- Firm D: $11.5\%$ C1, $\sim 84\%$ C2, $\sim 4.5\%$ C3
 
-1. **High-confidence non-hand-signed:** Cosine $> 0.95$ AND $\text{dHash}_\text{indep} \leq 5$.
+Firm A accounts for $141$ of the $143$ C3-assigned CPAs; Firm C accounts for $24$ of the $40$ C1-assigned CPAs. The K=3 partition is therefore well-described as a firm-compositional decomposition: C3 is essentially "Firm A and any non-Firm-A CPA whose mean descriptors happen to land in the high-cos / low-dHash corner"; C1 is essentially "non-Firm-A CPAs whose mean descriptors land in the low-cos / high-dHash corner." The composition contrast that K=3 captures at the accountant level reappears at the deployment level in the cross-firm hit matrix of §III-L.4 (Script 44): under the deployed any-pair rule, within-firm collision concentration is $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms). The K=3 partition and the cross-firm hit matrix therefore describe the same underlying firm-compositional structure at two different units of analysis.
-Both descriptors converge on strong replication evidence.
 
-2. **Moderate-confidence non-hand-signed:** Cosine $> 0.95$ AND $5 < \text{dHash}_\text{indep} \leq 15$.
+**Leave-one-firm-out stability (Scripts 36, 37).** Leave-one-firm-out cross-validation shows that K=2 is unstable across folds: holding Firm A out gives a fold rule cos $> 0.938$ AND dHash $\leq 8.79$, while holding any single non-Firm-A Big-4 firm out gives a fold rule near cos $> 0.975$ AND dHash $\leq 3.76$ (Script 36). The maximum absolute deviation of the four fold cosine crossings from their across-fold mean is $0.028$ (the corresponding pairwise across-fold range is $0.0376$, from $0.9380$ for the held-out-Firm-A fold to $0.9756$ for the held-out-Firm-D fold; Script 36 stability summary). The $0.028$ value is $5.6\times$ the report's $0.005$ across-fold stability tolerance. K=3 in contrast has a *reproducible component shape*: across the four folds the C1 cosine mean varies by at most $0.005$, the C1 dHash mean by at most $0.96$, and the C1 weight by at most $0.023$ (Script 37). K=3 hard-posterior membership for the held-out firm is composition-sensitive — for Firm C the held-out C1 rate is $36.3\%$ vs the full-Big-4 baseline of $23.5\%$, an absolute difference of $12.8$ pp; for Firm A the held-out C1 rate is $4.7\%$ vs baseline $0.0\%$; the report's own legend classifies this pattern as `P2_PARTIAL` ("the C1 cluster exists but membership is not well-predicted by the held-out fit"). We accordingly do not use K=3 hard-posterior membership as an operational label.
-Feature-level evidence is strong; structural similarity is present but below the high-confidence cutoff, potentially due to scan variations.
 
-3. **High style consistency:** Cosine $> 0.95$ AND $\text{dHash}_\text{indep} > 15$.
+We take the joint K=2 / K=3 LOOO evidence as supporting the following descriptive claims, all of which are used in §III-K and §V but none of which underwrites the operational classifier:
-High feature-level similarity without structural corroboration---consistent with a CPA who signs very consistently but not via image reproduction.
 
-4. **Uncertain:** Cosine between the all-pairs intra/inter KDE crossover (0.837) and 0.95 without sufficient convergent evidence for classification in either direction.
+- The Big-4 K=2 marginal crossing $(0.975, 3.76)$ is essentially a firm-mass separator between Firm A and Firms B + C + D, not a within-Big-4 mechanism boundary.
+- The Big-4 K=3 mixture exhibits a reproducible three-component component shape across LOOO folds at the descriptor-position level, with C1 reproducibly located at $\overline{\text{cos}} \approx 0.946$, $\overline{\text{dHash}} \approx 9.17$.
+- Hard-posterior K=3 membership is composition-sensitive across folds (max absolute deviation $12.8$ pp); K=3 is therefore not used to assign operational labels to CPAs.
 
-5. **Likely hand-signed:** Cosine below the all-pairs KDE crossover threshold.
+The operational signature-level classifier of §III-L is calibrated against inter-CPA pair-level negative-anchor coincidence rates, not against mixture-derived antimodes. Cross-checks between the deployed five-way box rule and the K=3 partition appear in §III-K.
 
-We note three conventions about the thresholds.
+## K. Convergent Internal-Consistency Checks
-First, the cosine cutoff $0.95$ corresponds to approximately the whole-sample Firm A P7.5 of the per-signature best-match cosine distribution---that is, 92.5% of whole-sample Firm A signatures exceed this cutoff and 7.5% fall at or below it (Section III-H)---chosen as a round-number lower-tail boundary whose complement (92.5% above) has a transparent interpretation in the whole-sample reference distribution; the cosine crossover $0.837$ is the all-pairs intra/inter KDE crossover; both are derived from whole-sample distributions rather than from the 70% calibration fold, so the classifier inherits its operational cosine cuts from the whole-sample Firm A and all-pairs distributions.
-Section IV-F.3 reports a sensitivity check confirming that replacing $0.95$ with the nearby rounded sensitivity cut $0.945$ (motivated by the calibration-fold P5 = 0.9407, see Section IV-F.2) shifts whole-Firm-A dual-rule capture by 1.19 percentage points, so the round-number heuristic is robust to nearby percentile-based alternatives.
-Section IV-F.2 reports both calibration-fold and held-out-fold capture rates for this classifier so that fold-level sampling variance is visible.
-Second, the dHash cutoffs $\leq 5$ and $> 15$ are chosen from the whole-sample Firm A $\text{dHash}_\text{indep}$ distribution: $\leq 5$ captures the upper tail of the high-similarity mode (whole-sample Firm A median $\text{dHash}_\text{indep} = 2$, P75 $\approx 4$, so $\leq 5$ is the band immediately above median), while $> 15$ marks the regime in which independent-minimum structural similarity is no longer indicative of image reproduction.
-Third, the signature-level threshold-estimator outputs of Section IV-D (KDE antimode, Beta-mixture and logit-Gaussian crossings, BD/McCrary diagnostic) are *not* the operational thresholds of this classifier: they are descriptive characterisation of the per-signature similarity distribution, and Section IV-D shows they do not converge to a clean two-mechanism boundary at the per-signature level---which is why the operational cosine cut is anchored on the whole-sample Firm A percentile rather than on any mixture-fit crossing.
 
-Because each audit report typically carries two certifying-CPA signatures (Section III-D), we aggregate signature-level outcomes to document-level labels using a worst-case rule: the document inherits the *most-replication-consistent* signature label (i.e., among the two signatures, the label rank ordered High-confidence $>$ Moderate-confidence $>$ Style-consistency $>$ Uncertain $>$ Likely-hand-signed determines the document's classification).
+The descriptive partition of §III-J is supported by three feature-derived per-CPA scores and a conservative hard-positive subset analysis. We caution at the outset that the three scores are **not statistically independent measurements** — all three are deterministic functions of the same per-CPA descriptor means $(\overline{\text{cos}}_a, \overline{\text{dHash}}_a)$ — so their high pairwise rank correlations are partly a mechanical consequence of shared inputs. Per §III-I.4, none of the three scores has a within-population bimodality interpretation; they are firm-compositional position scores at the accountant level. The checks below therefore document **internal consistency among feature-derived ranks**, not external validation against an independent hand-signed ground truth (which the corpus does not provide).
-This rule is consistent with the detection goal of flagging any potentially non-hand-signed report rather than requiring all signatures on the report to converge.
 
-## L. Data Source and Firm Anonymization
+**1. Three feature-derived per-CPA scores (Script 38).** For each Big-4 CPA we compute:
+
+- **Score 1 (K=3 posterior on the low-cos / high-dHash component):** $P(\text{C1})$ from the K=3 fit of §III-J. Per §III-J this is a firm-compositional position score on the (cos, dHash) plane (not a probability of any latent "hand-signing mechanism") — a function of both descriptor means.
+- **Score 2 (reverse-anchor cosine percentile):** the marginal cosine CDF value of $\overline{\text{cos}}_a$ under the non-Big-4 reference Gaussian of §III-H.2, sign-flipped so that lower percentile (further into the reference's left tail) corresponds to a Big-4 CPA whose mean cosine sits further from the templated end. This is a function of $\overline{\text{cos}}_a$ alone.
+- **Score 3 (deployed binary high-confidence box rule rate):** the per-CPA fraction of signatures that do **not** satisfy the deployed binary high-confidence box rule (cos $> 0.95$ AND dHash $\leq 5$). This is a per-signature-aggregated function of the same descriptors.
+
+Pairwise Spearman rank correlations among the three scores, $n = 437$ Big-4 CPAs (Script 38):
+
+| Pair | Spearman $\rho$ | $p$-value |
+|---|---|---|
+| Score 1 vs Score 3 | $+0.9627$ | $< 10^{-248}$ |
+| Score 2 vs Score 3 | $+0.8890$ | $< 10^{-149}$ |
+| Score 1 vs Score 2 | $+0.8794$ | $< 10^{-142}$ |
+
+We read this as the strongest internal-consistency signal in the analysis: three different summarisations of the same descriptor pair agree on the per-CPA descriptor-position ranking with $\rho > 0.87$. The three scores agree on placing Firm A as the most replication-dominated descriptor position and the three non-Firm-A Big-4 firms further from the templated end, but they do not all rank the non-Firm-A firms identically: the K=3 posterior P(C1) and the box-rule less-replication-dominated rate (Scores 1 and 3) place Firm C at the less-replication-dominated end of Big-4 (mean P(C1) $= 0.311$; mean box-rule less-replication-dominated rate $= 0.790$), while the reverse-anchor cosine percentile (Score 2) places Firm D fractionally higher than Firm C (mean reverse-anchor score $-0.7125$ vs Firm C $-0.7672$, with higher value indicating deeper into the reference left tail). The mean values for Firms B and D sit between Firms A and C on Scores 1 and 3 (Script 38 per-firm summary). We do not claim this constitutes external validation of any operational classifier; the deployed box rule is calibrated separately (§III-L), and the convergence above shows that a mixture-derived score and a reverse-anchor score concur with the box rule's per-CPA-aggregated outputs on the directional ordering, with a modest disagreement at the less-replication-dominated end between the three non-A Big-4 firms.
+
+**2. Per-signature consistency (Script 39).** Per-CPA aggregation could in principle reflect averaging across within-CPA heterogeneity rather than coherent within-CPA behaviour. We test this by repeating the K=3 fit at the signature level — fitting a fresh K=3 GMM to the 150,442 Big-4 signature-level $(\text{cos}, \text{dHash}_{\text{indep}})$ points (Script 39) — and comparing labels. The per-CPA and per-signature K=3 fits recover a broadly similar three-component ordering; per-CPA C1 is at $\overline{\text{cos}} = 0.946$, $\overline{\text{dHash}} = 9.17$ vs per-signature C1 at $\overline{\text{cos}} = 0.928$, $\overline{\text{dHash}} = 9.75$ (an absolute cosine drift of $0.018$). Cohen $\kappa$ on the binary collapse (replication-dominated vs less-replication-dominated):
+
+| Pair | Cohen $\kappa$ |
+|---|---|
+| Deployed binary high-confidence box rule vs per-CPA K=3 hard label | $0.662$ |
+| Deployed binary high-confidence box rule vs per-signature K=3 hard label | $0.559$ |
+| Per-CPA K=3 vs per-signature K=3 | $0.870$ |
+
+The $\kappa = 0.870$ between per-CPA-fit and per-signature-fit K=3 binary labels indicates that per-CPA aggregation does not collapse the broad three-component ordering. The lower $\kappa = 0.56\text{–}0.66$ between the binary box rule and either K=3 fit is consistent with two factors: different decision geometries (rectangular box vs Gaussian-mixture posterior boundary), and the fact that the binary box rule is a strict subset of the five-way rule. This comparison checks only the binary high-confidence rule (cos $> 0.95$ AND dHash $\leq 5$); §III-K does not directly check the five-way rule's `5 < \text{dHash} \leq 15` moderate-confidence band, whose calibration and capture-rate evidence is reported in the supplementary materials and not regenerated on the Big-4 subset.
+
+**3. Leave-one-firm-out reproducibility (Scripts 36, 37).** Discussed in §III-J above. We summarise the joint result for cross-reference:
+
+- *K=2 LOOO is unstable.* The maximum absolute deviation of the four fold cosine crossings from their across-fold mean is $0.028$, against the report's $0.005$ across-fold stability tolerance (Script 36; pairwise fold range $0.0376$, from $0.9380$ to $0.9756$). When Firm A is held out, the fold rule classifies $171/171$ of held-out Firm A CPAs as templated; when any non-Firm-A Big-4 firm is held out, the fold rule classifies $0$ of the held-out firm's CPAs as templated. This pattern indicates the K=2 boundary is essentially a Firm-A-vs-others separator rather than a within-Big-4 mechanism boundary.
+
+- *K=3 LOOO is partially stable.* The C1 (low-cos / high-dHash) component shape is reproducible across folds: max deviation from the full-Big-4 baseline is $0.005$ in cosine, $0.96$ in dHash, and $0.023$ in mixture weight (Script 37). Hard-posterior membership remains composition-sensitive — observed absolute differences are $1.8$–$12.8$ pp across the four folds, with the Firm C fold exceeding the report's $5$ pp viability bar; the report's own screening label is `P2_PARTIAL` ("K=3 is not predictively useful as an operational classifier"). We accordingly do not use K=3 hard-posterior membership as an operational label.
+
+**4. Positive-anchor miss rate on byte-identical signatures (Script 40).** The corpus provides one conservative hard-positive subset: signatures whose nearest same-CPA match is byte-identical after crop and normalisation. Independent hand-signing cannot produce pixel-identical images, so byte-identical signatures are a conservative hard-positive subset for image replication. The Big-4 byte-identical subset comprises $n = 262$ signatures ($145 / 8 / 107 / 2$ across Firms A through D; Script 40).
+
+We report each candidate check's *positive-anchor miss rate* — the fraction of byte-identical signatures classified as belonging to the less-replication-dominated descriptor positions. This is a one-sided check against a conservative positive subset, **not a paired specificity metric in the usual two-class sense**; we do not report a paired negative-anchor metric here because no signature-level hand-signed ground truth exists. The corresponding signature-level inter-CPA negative-anchor ICCR evidence is developed in §III-L.1 (Big-4 sample) and the corpus-wide version cited at §IV-I:
+
+| Candidate check | Pixel-identity miss rate (Wilson 95% CI) |
+|---|---|
+| Deployed binary high-confidence box rule (cos $> 0.95$ AND dHash $\leq 5$) | $0\%$ $[0\%, 1.45\%]$ |
+| K=3 per-CPA hard label (C3 high-cos / low-dHash corner; descriptive only) | $0\%$ $[0\%, 1.45\%]$ |
+| Reverse-anchor with prevalence-calibrated cut | $0\%$ $[0\%, 1.45\%]$ |
+
+All three candidate scores correctly assign every byte-identical signature to the replicated class. We caution that for the box rule this result is close to tautological: byte-identical nearest-neighbour signatures have cosine $\approx 1$ and dHash $\approx 0$ by construction, so any threshold strictly below cos $= 1$ and strictly above dHash $= 0$ will capture them. The positive-anchor miss rate is therefore a necessary check (a classifier that *failed* this check would be disqualified), not a sufficient validation of the classifier's behaviour on the non-byte-identical replicated population. The reverse-anchor cut here is chosen by prevalence calibration against the box rule's overall replicated rate ($49.58\%$ of Big-4 signatures); this is a documented limitation since no signature-level hand-signed ground truth exists to permit direct ROC optimisation.
+
+## L. Anchor-Based Threshold Calibration
+
+The operational classifier defined in §III-H.1 is calibrated by characterising the deployed thresholds' inter-CPA pair-level negative-anchor coincidence behaviour and their pool-normalised per-signature and per-document alert behaviour, at multiple units of analysis. §III-I.4 establishes that the descriptor distributions do not contain a within-population bimodal antimode that could anchor an operational threshold; the K=3 mixture of §III-J is a descriptive firm-compositional partition, not a mechanism-cluster model. Throughout this section we report **inter-CPA coincidence rates** rather than "False Acceptance Rates"; we explain the terminological choice in §III-L.0.
+
+### L.0. Calibration methodology
+
+**Calibration role of the present analysis.** The deployed thresholds of §III-H.1 preserve continuity with the existing literature and the supplementary calibration evidence. §III-I.4 establishes that a recalibration cannot be anchored on distributional antimodes (no within-population bimodality exists); §III-L.1 below characterises the cosine threshold's specificity-proxy behaviour at the inter-CPA pair level and the structural-dimension threshold $\text{dHash} \leq 5$'s pair-level coincidence behaviour. The sub-band thresholds ($\text{dHash} = 15$, $\text{cos} = 0.837$) retain their supplementary calibration evidence; the present calibration does not provide independent rates for those sub-bands.
+
+**Three units of analysis.** We report inter-CPA negative-anchor coincidence behaviour at three units, each addressing a different operational question:
+
+- *Per comparison.* For a randomly drawn pair of signatures from different CPAs, what fraction satisfies the rule (cos $>$ cos\_threshold and / or dHash $\leq$ dHash\_threshold)? This is the conventional pairwise calibration unit in biometric verification. We report it for both the cosine and dHash dimensions, marginally and jointly (§III-L.1).
+- *Per signature pool.* For a Big-4 source signature $s$ with same-CPA pool of size $n_{\text{pool}}(s)$, what is the probability that the deployed rule fires *under the counterfactual* of replacing the source's same-CPA pool with $n_{\text{pool}}(s)$ random non-same-CPA candidates? This addresses the standard concern that a per-pair rate computed on independent pairs is not the deployed-rule rate at the per-signature classifier level: the deployed rule takes max-cosine and min-dHash over a pool of size $n_{\text{pool}}(s)$, so its effective coincidence rate is approximately $1 - (1 - p_{\text{pair}})^{n_{\text{pool}}}$ in the independence limit (§III-L.2).
+- *Per document.* For an audit report aggregated via the worst-case rule, what fraction of documents have at least one signature whose deployed pool-normalised rule fires under the same inter-CPA candidate-replacement counterfactual? This is the operational alarm-rate unit (§III-L.3).
+
+**Any-pair vs same-pair semantics.** The deployed rule uses independent extrema: a signature satisfies the HC rule if $\max_{\text{pool}} \text{cos} > 0.95$ AND $\min_{\text{pool}} \text{dHash} \leq 5$, *not* if a single candidate in the pool satisfies both. We refer to this as the **any-pair** rule. A stricter alternative — the **same-pair** rule — requires a single candidate to satisfy both inequalities; the deployed rule is any-pair, but we report same-pair as a stricter alternative classifier where useful (§III-L.2, §III-L.4).
+
+**Terminological note on "FAR".** The biometric-verification literature speaks of "False Acceptance Rate" (FAR) for a per-pair rate computed on independent inter-CPA pairs. We adopt **inter-CPA coincidence rate (ICCR)** as the metric name and *do not* use "FAR" in the manuscript prose, for two reasons: (a) FAR has a specific biometric-verification meaning that requires ground-truth negative labels (which the corpus does not provide at the signature level); (b) §III-L.4 shows that the inter-CPA negative-anchor assumption — that inter-CPA pairs are negative — is partially violated by within-firm cross-CPA template-like collision structures. Reading "inter-CPA coincidence rate" as a *specificity proxy* under an explicitly disclosed assumption is faithful to the evidence; reading it as a true biometric FAR would overstate the evidence.
+
+### L.1. Per-comparison inter-CPA coincidence rate (Script 40b)
+
+We sample $5 \times 10^5$ inter-CPA pairs uniformly at random from Big-4 signatures, computing for each pair the cosine similarity (feature dot product) and Hamming distance between the dHash byte vectors. Marginal and joint rates at threshold $k$ are reported with Wilson 95% confidence intervals (Script 40b).
+
+| Threshold | Per-comparison inter-CPA coincidence rate | 95% Wilson CI |
+|---|---|---|
+| Cosine $> 0.95$ | $0.00060$ | $[0.00053, 0.00067]$ |
+| Cosine $> 0.945$ (alternative operating point from supplementary calibration evidence) | $0.00081$ | $[0.00073, 0.00089]$ |
+| Cosine $> 0.97$ | $0.00024$ | $[0.00020, 0.00029]$ |
+| Cosine $> 0.98$ | $0.00009$ | $[0.00007, 0.00012]$ |
+| dHash $\leq 5$ | $0.00129$ | $[0.00120, 0.00140]$ |
+| dHash $\leq 4$ | $0.00050$ | $[0.00044, 0.00057]$ |
+| dHash $\leq 3$ | $0.00019$ | $[0.00015, 0.00023]$ |
+| dHash $\leq 2$ | $0.00006$ | $[0.00004, 0.00008]$ |
+| Joint: cos $> 0.95$ AND dHash $\leq 5$ (any-pair semantics) | $0.00014$ | $[0.00011, 0.00018]$ |
+| Joint: cos $> 0.95$ AND dHash $\leq 4$ (any-pair) | $0.00011$ | $[0.00008, 0.00014]$ |
+
+The cosine row at $\text{cos} > 0.95$ is consistent with the corpus-wide per-comparison rate of $0.0005$ reported in §IV-I on a similarly-sized inter-CPA sample; the present $5 \times 10^5$-pair sample yields $0.00060$, within that precision. The dHash row and joint row are reported here for the first time; the corpus-wide spike did not provide an inter-CPA pair-level coincidence rate for the structural dimension or the joint rule.
+
+The all-firms-scope sample yields slightly lower per-comparison coincidence rates (cos $> 0.95$: $0.00031$; dHash $\leq 5$: $0.00073$; joint: $0.00007$); the all-firms sample weights small CPAs more heavily under CPA-uniform pair sampling, so we treat the Big-4 sample as the primary calibration scope and report all-firms as a corroborating-scope robustness check.
+
+**Conditional inter-CPA coincidence rate.** A natural follow-up question is whether the dHash dimension provides marginal specificity beyond the cosine gate. For pairs with cos $> 0.95$, the conditional rate of dHash $\leq 5$ is $0.234$ (Wilson 95% CI $[0.190, 0.285]$; $70$ of $299$ pairs in the Big-4 sample). At cos $> 0.95$, dHash provides $\sim 4.3\times$ further per-comparison specificity (joint $0.00014$ vs cos-only $0.00060$).
+
+The per-comparison rate is a useful *specificity-proxy calibration* for the deployed rule's pair-level behaviour. It does *not* directly translate to the deployed-rule specificity at the per-signature classifier level, because the deployed classifier takes extrema over a same-CPA pool of size $n_{\text{pool}}$. The pool-normalised inter-CPA alert rate is reported in §III-L.2.
+
+### L.2. Pool-normalised inter-CPA alert rate (Script 43)
+
+The deployed rule uses $\max_{\text{pool}} \text{cos}$ and $\min_{\text{pool}} \text{dHash}$ over the same-CPA pool of size $n_{\text{pool}}(s)$ for each signature $s$. A per-comparison rate is therefore not the rate at which the deployed classifier fires per signature. To compute the per-signature inter-CPA-equivalent rate, for each Big-4 source signature $s$ we simulate one realisation of an inter-CPA candidate pool of the same size $n_{\text{pool}}(s)$, drawn uniformly from non-same-CPA signatures across all firms, compute the deployed extrema and rule indicator, and aggregate (Script 43; $n_{\text{sig}} = 150{,}453$ vector-complete in this analysis; CPA-block bootstrap 95% CIs reported below).
+
+**Headline rates (deployed any-pair rule, HC = cos $> 0.95$ AND dHash $\leq 5$).** Wilson 95% CIs on the point estimate, CPA-block bootstrap 95% CI on $n_{\text{boot}} = 1000$ replicates:
+
+| Rule semantics | Per-signature ICCR | Wilson 95% CI | CPA-bootstrap 95% CI |
+|---|---|---|---|
+| Any-pair (deployed) | $0.1102$ | $[0.1086, 0.1118]$ | $[0.0908, 0.1330]$ |
+| Same-pair (stricter alternative) | $0.0827$ | $[0.0813, 0.0841]$ | $[0.0668, 0.1021]$ |
+
+Per-firm any-pair rates (no bootstrap; descriptive):
+
+| Firm | $n_{\text{sig}}$ | Any-pair ICCR | Same-pair ICCR |
+|---|---|---|---|
+| Firm A | $60{,}450$ | $0.2594$ | $0.2018$ |
+| Firm B | $34{,}254$ | $0.0147$ | $0.0023$ |
+| Firm C | $38{,}616$ | $0.0053$ | $0.0019$ |
+| Firm D | $17{,}133$ | $0.0110$ | $0.0051$ |
+
+**Pool-size decile dependence.** The deployed rule's pool-normalised rate is monotonically (broadly) increasing in $n_{\text{pool}}$, consistent with the $1 - (1 - p_{\text{pair}})^{n_{\text{pool}}}$ form expected under inter-CPA independence (Script 43 decile table). Decile 1 (smallest pools, $n_{\text{pool}} \in [0, 201]$): any-pair ICCR $= 0.0249$. Decile 10 (largest, $n_{\text{pool}} \in [846, 1115]$): any-pair ICCR $= 0.1905$. The trend is broadly monotonic with two minor non-monotone reversals (decile 5 and decile 9 dip below their predecessors).
+
+**Threshold sensitivity at per-signature unit.** Tightening the HC rule from $\text{dHash} \leq 5$ to $\text{dHash} \leq 3$ (same-pair) reduces the per-signature ICCR from $0.0827$ to $0.0449$ (Big-4 pooled); tightening to $\text{dHash} \leq 4$ gives $0.0639$ (same-pair). A stricter operating point of dHash $\leq 3$ same-pair would correspond to a per-signature ICCR of $\approx 0.05$; the deployed HC any-pair rule with $\text{dHash} \leq 5$ corresponds to $\approx 0.11$. Stakeholders requiring a tighter specificity proxy could consider the dHash $\leq 3$ same-pair variant, with the unsupervised-setting caveats of §III-M.
+
+### L.3. Document-level inter-CPA proxy alert rate (Script 45)
+
+The deployed worst-case aggregation classifies each document by the most-replication-consistent category among its constituent signatures (§III-H.1). Three operationally meaningful document-level alarm definitions are reported, each as the fraction of documents whose worst-case signature category falls in the alarm set under the same inter-CPA candidate-pool counterfactual as §III-L.2 (Script 45; $n_{\text{docs}} = 75{,}233$ Big-4 documents):
+
+| Alarm definition | Alarm set | Document-level ICCR | Wilson 95% CI |
+|---|---|---|---|
+| D1 | HC only | $0.1797$ | $[0.1770, 0.1825]$ |
+| D2 | HC + MC ("any non-hand-signed screening label") | $0.3375$ | $[0.3342, 0.3409]$ |
+| D3 | HC + MC + HSC | $0.3384$ | $[0.3351, 0.3418]$ |
+
+Per-firm D2 document-level rates:
+
+| Firm | $n_{\text{docs}}$ | D2 (HC + MC) ICCR |
+|---|---|---|
+| Firm A | $30{,}226$ | $0.6201$ |
+| Firm B | $17{,}127$ | $0.1600$ |
+| Firm C | $19{,}501$ | $0.1635$ |
+| Firm D | $8{,}379$ | $0.0863$ |
+
+The document-level D2 rate of $33.75\%$ pooled over Big-4 is the most operationally relevant alarm-rate metric: it is the fraction of audit documents that would carry at least one signature flagged HC or MC under the counterfactual of inter-CPA candidate-pool replacement. The non-trivial per-document inter-CPA alarm rate (and its concentration in Firm A at $62\%$) motivates the positioning of the operational system as a **screening framework with human-in-the-loop review**, not as an autonomous forensic classifier (§III-M).
+
+### L.4. Firm heterogeneity (Script 44)
+
+§III-L.2 and §III-L.3 report large per-firm variation in the deployed rule's pool-normalised behaviour: Firm A's any-pair per-signature ICCR is $0.2594$, an order of magnitude larger than Firm B's $0.0147$, Firm C's $0.0053$, Firm D's $0.0110$. A natural alternative explanation is the pool-size confound: Firm A's median pool size ($\sim 285$) is larger than other firms', and pool size monotonically (broadly) increases the per-signature rate (§III-L.2 decile trend). We test the firm-vs-pool confound with a logistic regression of the per-signature hit indicator (any-pair HC) on firm dummies (Firm A = reference) and centred log pool size (Script 44):
+
+| Term | Odds ratio (vs Firm A) | Direction | Magnitude |
+|---|---|---|---|
+| Firm B | $0.053$ | $< 1$ | $\sim 19\times$ lower odds than Firm A |
+| Firm C | $0.010$ | $< 1$ | $\sim 100\times$ lower odds than Firm A |
+| Firm D | $0.027$ | $< 1$ | $\sim 37\times$ lower odds than Firm A |
+| log(pool size, centred) | $4.01$ | $> 1$ | $\sim 4\times$ higher odds per unit log pool size |
+
+The Firm B/C/D odds ratios are very small after controlling for pool size, indicating that firm membership accounts for a large multiplicative effect on the per-signature rate that is *not* explained by pool size alone. (We report odds ratios rather than $z$-scores because per-signature observations are clustered by CPA and firm, and naive standard errors would be unreliable under within-cluster correlation; a cluster-robust standard error analysis is left as a robustness check.)
+
+The per-decile per-firm breakdown (Script 44) confirms the pattern: within every pool-size decile, Firms B/C/D have rates of $0.0006$–$0.0358$, while Firm A's rate ranges $0.0541$–$0.5958$ across deciles. The firm gap is large within matched pool sizes, not driven by pool composition.
+
+**Cross-firm hit matrix.** Among Big-4 source signatures whose any-pair rule fires under the inter-CPA candidate-pool counterfactual, the candidate firm of the max-cosine partner is distributed as follows (Script 44):
+
+| Source firm | Firm A candidate | Firm B | Firm C | Firm D | non-Big-4 | hits |
+|---|---|---|---|---|---|---|
+| Firm A | $14{,}447$ | $95$ | $44$ | $19$ | $17$ | $14{,}622$ |
+| Firm B | $92$ | $371$ | $8$ | $4$ | $9$ | $484$ |
+| Firm C | $16$ | $7$ | $149$ | $5$ | $1$ | $178$ |
+| Firm D | $22$ | $2$ | $6$ | $106$ | $1$ | $137$ |
+
+For the same-pair joint event (a single candidate satisfying both $\text{cos} > 0.95$ and $\text{dHash} \leq 5$), the candidate firm is even more strongly concentrated within the source firm: Firm A source $\to$ Firm A candidate in $11{,}314$ of $11{,}319$ same-pair hits ($99.96\%$); Firm B source $\to$ Firm B candidate in $85$ of $87$ ($97.7\%$); Firm C source $\to$ Firm C candidate in $54$ of $55$ ($98.2\%$); Firm D source $\to$ Firm D candidate in $64$ of $66$ ($97.0\%$).
+
+**Interpretation.** Under the deployed any-pair rule, the within-firm collision concentration is $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D — Firm A's pattern is markedly more within-firm-concentrated than the other three firms', though every Big-4 firm still has more than three quarters of its any-pair collisions falling on candidates within the same firm. The stricter same-pair joint event — a single candidate satisfying both cos $> 0.95$ and dHash $\leq 5$ — saturates at $97.0$–$99.96\%$ within-firm across all four firms. This pattern is consistent with — but not by itself diagnostic of — firm-specific template, stamp, or document-production reuse: within-firm scanning workflows, common form templates, and shared report-generation infrastructure could produce visually similar signature crops across different CPAs within the same firm. Byte-level decomposition of Firm A's $145$ pixel-identical signatures across $\sim 50$ distinct certifying partners (supplementary materials; §III-H.2) provides direct evidence of image-level reuse among Firm A signatures; the distribution across many partners is consistent with a firm-level template or production workflow, and the broader inter-CPA collision pattern in §III-L.4 is consistent with similar, milder production-related reuse patterns at Firms B/C/D. We report this as "inter-CPA collision concentration is within-firm" — a descriptive observation about deployed-rule behaviour — and refrain from inferring that the within-firm hits constitute deliberate or systematic template sharing.
+
+This connects back to §III-J: the K=3 firm-composition contrast at the accountant level (Firm A dominating C3; Firm C dominating C1) reappears at the deployment level in the cross-firm hit matrix, where the within-firm collision concentration is the dominant pattern at all four Big-4 firms — most strongly at Firm A ($98.8\%$ any-pair, $99.96\%$ same-pair) and at materially lower but still majority levels at Firms B/C/D ($76.7$–$83.7\%$ any-pair; $97.0$–$98.2\%$ same-pair).
+
+### L.5. Alert-rate sensitivity around deployed thresholds (Script 46)
+
+To test whether the deployed cosine threshold $0.95$ and dHash threshold $5$ coincide with a low-gradient (plateau-stable) region of the deployed-rule alert-rate surface — which would be weak distributional evidence that the deployed thresholds are stable operating points — we sweep each threshold across a range and report the per-signature alert rate on actual observed Big-4 same-CPA pools (not inter-CPA-replaced pools), comparing the local gradient at the deployed threshold to the median gradient across the sweep (Script 46).
+
+At the deployed HC operating point cos $> 0.95$ AND dHash $\leq 5$, the local gradient of the per-signature alert rate is substantially larger than the median gradient across the sweep (cosine: ratio $\approx 25\times$ at the $0.95$ point relative to median; dHash: ratio $\approx 3.8\times$ at the $5$ point relative to median; both Script 46). Reading these ratios descriptively, the deployed HC threshold is *locally sensitive* rather than plateau-stable: small threshold perturbations materially change the deployed alert rate (cosine sweep at dHash $\leq 5$ yields rates of $0.5091$ at cos $> 0.945$ vs $0.4789$ at cos $> 0.955$, a $3.0$ pp swing across a $0.01$ cosine perturbation; dHash sweep at cos $> 0.95$ yields rates of $0.4207$ at dHash $\leq 4$ vs $0.5639$ at dHash $\leq 6$, a $14.3$ pp swing across a single integer step). The local-gradient-to-median-gradient ratios are descriptive diagnostics, not formal plateau tests; the primary evidence for "no within-population bimodal antimode at these thresholds" comes from §III-I.4's composition decomposition, not from §III-L.5.
+
+The MC/HSC boundary at dHash $= 15$, by contrast, *is* in a low-gradient region (ratio $\approx 0.08$ to the median); the plateau-like behaviour around dHash $= 15$ is corroborating evidence that the high-end structural threshold lies in a regime where the rule's alert rate is approximately saturated, consistent with the high-dHash tail behaviour expected once near-identical pairs have been exhausted. The §III-L.5 non-plateau / local-sensitivity finding therefore applies specifically to the HC cutoff (cos $= 0.95$, dHash $= 5$); the MC/HSC sub-band boundary at dHash $= 15$ exhibits the opposite behaviour and is plateau-like.
+
+We interpret the deployed HC thresholds as **specificity-anchored operating points** chosen for the specificity-vs-alert-yield tradeoff (§III-L.1), *not* as distributional antimodes. Alternative operating points on the tradeoff curve can be characterised by inverting the per-comparison or pool-normalised ICCR curves (§III-L.1, §III-L.2) at the preferred specificity target.
+
+### L.6. Observed deployed alert rate on actual same-CPA pools
+
+The pool-normalised inter-CPA rates of §III-L.2 and §III-L.3 use the counterfactual of replacing the source signature's same-CPA pool with random non-same-CPA candidates. The **observed deployed alert rate** uses the source's actual same-CPA pool, i.e., the rate at which the deployed rule fires on the real corpus. For Big-4, the deployed HC any-pair rule fires on $49.58\%$ of signatures and $62.28\%$ of documents (Script 46; Script 42 reproduces the per-signature rate at $49.58\%$).
+
+The per-signature observed-deployed rate is $\sim 4.5\times$ the pool-normalised inter-CPA rate ($0.4958$ vs $0.1102$); the per-document observed-deployed rate is $\sim 3.5\times$ the pool-normalised inter-CPA D1 (HC) rate ($0.6228$ vs $0.1797$). We refer to this multiplicative gap as the **deployed-rate excess over the inter-CPA proxy**:
+
+- Per-signature: $0.4958 - 0.1102 = 0.3856$ ($38.6$ pp excess)
+- Per-document HC: $0.6228 - 0.1797 = 0.4431$ ($44.3$ pp excess)
+
+We *do not* interpret the deployed-rate excess as a presumed true-positive rate; the inferential limits of this interpretation are developed in §III-M. The deployed-rate excess is best read as a *same-CPA repeatability signal* — a quantity that exceeds what random inter-CPA candidate replacement would produce — rather than as an estimate of true replication prevalence.
+
+## M. Unsupervised Diagnostic Strategy and Limits
+
+The corpus lacks signature-level ground-truth replication labels: no signature is annotated as definitively hand-signed or definitively templated. The conservative positive anchor (pixel-identical same-CPA signatures; §III-K.4) is by construction near $\text{cos} = 1$ and $\text{dHash} = 0$, providing a tautological capture-check rather than a sensitivity estimate for the non-byte-identical replicated class. The corpus therefore does not admit standard supervised classifier validation: we cannot report False Rejection Rate, sensitivity, recall, Equal Error Rate, ROC-AUC, or precision against ground truth.
+
+Each diagnostic reported in this paper therefore addresses one specific failure mode of an unsupervised screening classifier (Table XXVII), with an explicitly disclosed untested assumption:
+
+**Table XXVII.** Diagnostics, failure mode addressed, and disclosed untested assumption.
+
+| Diagnostic | Failure mode addressed | Disclosed untested assumption |
+|---|---|---|
+| Composition decomposition (§III-I.4; Scripts 39b–39e) | Whether descriptor multimodality is within-population (mechanism) or between-group (composition + integer artefact); $p_{\text{median}} = 0.35$ under joint firm-mean centring + integer-tie jitter | Integer-tie jitter and firm-mean centring are unbiased over the descriptor support; corroborated by Big-4 per-firm jitter (Script 39d; per-firm dHash rejection disappears under jitter at every Big-4 firm) and Big-4 pooled centred + jittered ($n_{\text{seeds}} = 5$; Script 39e) |
+| Per-comparison inter-CPA coincidence rate (§III-L.1; Script 40b) | Pair-level specificity proxy under a random-pair negative anchor | Inter-CPA pairs are negative (i.e., not template-related); partially violated by within-firm sharing (§III-L.4) |
+| Pool-normalised per-signature ICCR (§III-L.2; Script 43) | Deployed-rule specificity proxy at per-signature unit, accounting for pool size | Same as above + that pool replacement preserves the negative-anchor property |
+| Document-level ICCR (§III-L.3; Script 45) | Operational alarm rate proxy at per-document unit under three alarm definitions | Same as above |
+| Firm-heterogeneity logistic regression (§III-L.4; Script 44) | Multiplicative effect of firm membership on per-signature rate, controlling for pool size | Per-signature observations are clustered by CPA/firm; naïve standard errors unreliable; cluster-robust analysis is a future check |
+| Cross-firm hit matrix (§III-L.4; Script 44) | Concentration of inter-CPA collisions within source firm | Concentration depends on deployed-rule semantics (the stricter same-pair joint event yields $97.0$–$99.96\%$ within-firm at all four firms versus $76.7$–$98.8\%$ under any-pair; §III-L.4); per-document per-firm assignment uses Script 45's mode-of-firms tie-break (§IV-M.4 footnote) |
+| Alert-rate sensitivity sweep (§III-L.5; Script 46) | Local sensitivity of deployed rule to threshold perturbation | Gradient comparison is descriptive, not a formal plateau test |
+| Convergent score Spearman ranking (§III-K.1; Script 38) | Internal-consistency of three feature-derived per-CPA scores | Scores share underlying inputs and are not statistically independent |
+| Pixel-identical conservative positive capture (§III-K.4; Script 40) | Trivial sanity check on the conservative positive anchor | Anchor is tautologically captured by any reasonable threshold |
+| LOOO firm-level reproducibility (§III-K.3; Scripts 36, 37) | Algorithmic stability of K=2 / K=3 partition across firm folds | Stability is necessary but not sufficient for classification validity |
+
+No single diagnostic provides ground-truth validation; together they define the limits of what can be supported in this corpus without signature-level ground truth.
+
+**Limits of the present analysis.** We do not claim a validated forensic detector or an autonomous classification system. We do not report False Rejection Rate, sensitivity, recall, EER, ROC-AUC, precision, or positive predictive value against ground truth, because no ground truth exists at the signature level. We do not interpret the deployed-rate excess of §III-L.6 as a presumed true-positive rate: that interpretation would require assuming that the within-firm same-CPA pool's collision rate equals the inter-CPA proxy rate in the absence of replication (i.e., that genuine same-CPA hand-signing would produce a collision rate no higher than random inter-CPA pairs). Two factors make the assumption unsafe: (a) a CPA who signs consistently can produce stylistically similar signatures across years that exceed inter-CPA similarity at the cosine axis; (b) within-firm template sharing (§III-L.4 cross-firm hit matrix; byte-level evidence of Firm A's pixel-identical signatures across partners, supplementary materials) places a substantial inter-CPA collision floor that itself reflects template-like reuse rather than independent inter-CPA random matching. We do not infer that the within-firm collision concentration of §III-L.4 constitutes deliberate template sharing; we describe it as "inter-CPA collision concentration is within-firm" and treat the mechanism as an open empirical question.
+
+**Scope of the present analysis.** The deployed signature-replication screening rule is characterised at three units of analysis (per-comparison, per-signature pool, per-document) against an inter-CPA negative-anchor coincidence-rate calibration. The per-comparison rates ($\leq 0.0006$ at cos $> 0.95$; $\leq 0.0013$ at dHash $\leq 5$; $\leq 0.00014$ jointly) are specificity-proxy-anchored operating points consistent with biometric-verification convention, with the proxy nature recorded in §III-L.0 and §III-M. The per-signature and per-document rates ($0.11$ and $0.34$ respectively under the deployed any-pair HC + MC alarm) are operationally meaningful **alarm-yield** indicators rather than true error rates. Per-firm rates show substantial heterogeneity (Firm A's per-document HC + MC alarm at $0.62$ vs Firm B/C/D at $0.09$–$0.16$), driven by firm-level rather than pool-size effects, and concentrated in within-firm cross-CPA candidate matching. The framework is positioned as a **specificity-proxy-anchored screening tool with human-in-the-loop review**, not as a validated forensic classifier.
+
+**Specificity-alert-yield tradeoff.** Because sensitivity is unobservable, stakeholders cannot derive an operating point by optimising a ROC criterion. Instead, the specificity-proxy-anchored framework offers a *specificity-alert-yield tradeoff*: tighter operating points (e.g., cos $> 0.98$ AND dHash $\leq 3$) reduce both per-comparison ICCR (to $\approx 5 \times 10^{-5}$; §III-L.1 inversion) and per-signature alert yield (to $\approx 0.05$; §III-L.2), with an unknown effect on actual replication-detection recall. Tighter operating points are not necessarily preferable: any tightening reduces the alert rate but may also miss true replicated signatures whose noise has pushed them outside the tighter envelope. The deployment decision depends on the relative cost of manual review (per alarm) and missed-replication risk (per false negative) — neither directly observable from corpus data.
+
+## N. Data Source and Firm Anonymization
 
 **Audit-report corpus.** The 90,282 audit-report PDFs analyzed in this study were obtained from the Market Observation Post System (MOPS) operated by the Taiwan Stock Exchange Corporation.
 MOPS is the statutory public-disclosure platform for Taiwan-listed companies; every audit report filed on MOPS is already a publicly accessible regulatory document.

paper/paper_a_references_v3.md

@@ -1,6 +1,6 @@
 # References
 
-<!-- IEEE numbered style, sequential by first appearance in text. v3 adds statistical-method refs (37–41). -->
+<!-- IEEE numbered style, sequential by first appearance in text. -->
 
 [1] Taiwan Certified Public Accountant Act (會計師法), Art. 4; FSC Attestation Regulations (查核簽證核准準則), Art. 6. Available: https://law.moj.gov.tw/ENG/LawClass/LawAll.aspx?pcode=G0400067
 
@@ -84,4 +84,10 @@
 
 [41] H. White, "Maximum likelihood estimation of misspecified models," *Econometrica*, vol. 50, no. 1, pp. 1–25, 1982.
 
-<!-- Total: 41 references (v2: 36 + 5 new statistical methods refs) -->
+[42] M. Stone, "Cross-validatory choice and assessment of statistical predictions," *J. R. Statist. Soc. B*, vol. 36, no. 2, pp. 111–147, 1974.
+
+[43] S. Geisser, "The predictive sample reuse method with applications," *J. Amer. Statist. Assoc.*, vol. 70, no. 350, pp. 320–328, 1975.
+
+[44] A. Vehtari, A. Gelman, and J. Gabry, "Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC," *Stat. Comput.*, vol. 27, no. 5, pp. 1413–1432, 2017.
+
+<!-- Total: 44 references -->

paper/paper_a_related_work_v3.md

@@ -15,8 +15,8 @@ Hafemann et al. [16] further addressed the practical challenge of adapting to ne
 A common thread in this literature is the assumption that the primary threat is *identity fraud*: a forger attempting to produce a convincing imitation of another person's signature.
 Our work addresses a fundamentally different problem---detecting whether the *legitimate signer's* stored signature image has been reproduced across many documents---which requires analyzing the upper tail of the intra-signer similarity distribution rather than modeling inter-signer discriminability.
 
-Brimoh and Olisah [8] proposed a consensus-threshold approach that derives classification boundaries from known genuine reference pairs, the methodology most closely related to our calibration strategy.
+Brimoh and Olisah [8] are closest in spirit in using reference evidence to discipline threshold choice.
-However, their method operates on standard verification benchmarks with laboratory-collected signatures, whereas our approach applies threshold calibration using a replication-dominated subpopulation identified through domain expertise in real-world regulatory documents.
+Their setting, however, uses standard verification benchmarks with known genuine references, whereas our archival setting lacks signature-level labels and therefore characterises a fixed deployed screening rule through inter-CPA coincidence-rate anchors.
 
 ## B. Document Forensics and Copy Detection
 
@@ -51,13 +51,13 @@ Chamakh and Bounouh [22] confirmed that a simple ResNet backbone with cosine sim
 Babenko et al. [23] established that CNN-extracted neural codes with cosine similarity provide an effective framework for image retrieval and matching, a finding that underpins our feature-comparison approach.
 These findings collectively suggest that pre-trained CNN features, when L2-normalized and compared via cosine similarity, provide a robust and computationally efficient representation for signature comparison---particularly suitable for large-scale applications where the computational overhead of Siamese training or metric learning is impractical.
 
-## E. Statistical Methods for Threshold Determination
+## E. Statistical Methods for Threshold Characterisation and Calibration
 
-Our threshold-determination framework combines three families of methods developed in statistics and accounting-econometrics.
+Our threshold-characterisation and calibration framework combines three families of methods developed in statistics and accounting-econometrics.
 
 *Non-parametric density estimation.*
 Kernel density estimation [28] provides a smooth estimate of a similarity distribution without parametric assumptions.
-Where the distribution is bimodal, the local density minimum (antimode) between the two modes is the Bayes-optimal decision boundary under equal priors.
+In idealized two-class mixture settings with equal priors and equal misclassification costs, the local density minimum (antimode) between the two modes coincides with the Bayes-optimal decision boundary.
 The statistical validity of the unimodality-vs-multimodality dichotomy can be tested via the Hartigan & Hartigan dip test [37], which tests the null of unimodality; we use rejection of this null as evidence consistent with (though not a direct test for) bimodality.
 
 *Discontinuity tests on empirical distributions.*
@@ -71,9 +71,12 @@ When the empirical distribution is viewed as a weighted sum of two (or more) lat
 For observations bounded on $[0,1]$---such as cosine similarity and normalized Hamming-based dHash similarity---the Beta distribution is the natural parametric choice, with applications spanning bioinformatics and Bayesian estimation.
 Under mild regularity conditions, White's quasi-MLE result [41] supports interpreting maximum-likelihood estimates under a mis-specified parametric family as consistent estimators of the pseudo-true parameter that minimizes the Kullback-Leibler divergence to the data-generating distribution within that family; we use this result to justify the Beta-mixture fit as a principled approximation rather than as a guarantee that the true distribution is Beta.
 
-The present study combines all three families, using each to produce an independent threshold estimate and treating cross-method convergence---or principled divergence---as evidence of where in the analysis hierarchy the mixture structure is statistically supported.
+The present study uses these tools diagnostically: first to test whether the descriptor distribution supports a natural operating boundary, and then, when that support fails under composition decomposition, to motivate anchor-based ICCR calibration of a fixed deployed rule.
+
+*Cross-validation in a small-cluster scope.*
+Cross-validation methodology in the leave-one-out tradition has been developed extensively in statistics since Stone [42] and Geisser [43], and modern surveys including Vehtari et al. [44] discuss its application to mixture models. In document-forensics calibration the technique has been used selectively, typically with the individual document or signature as the hold-out unit. Our application in §III-K differs in two respects from the standard usage: (i) the hold-out unit is the *firm* (not the individual CPA or signature), so the analysis directly probes cross-firm reproducibility of the fitted mixture rather than within-firm sampling variance; and (ii) the held-out predictions are interpreted as a *composition-sensitivity band* on the candidate mixture boundary, not as a sufficiency claim for the deployed five-way operational classifier (§III-H.1; calibrated separately in §III-L). We treat LOOO drift as descriptive information about how the mixture characterisation moves when training composition changes, not as a pass/fail test for the operational classifier.
 <!--
-REFERENCES for Related Work (see paper_a_references_v3.md for full list):
+REFERENCES for Related Work (full list in the References section):
 [3]  Bromley et al. 1993 — Siamese TDNN (NeurIPS)
 [4]  Dey et al. 2017 — SigNet
 [5]  Kao & Wen 2020 — Single-sample SV with forgery detection
@@ -101,4 +104,7 @@ REFERENCES for Related Work (see paper_a_references_v3.md for full list):
 [39] McCrary 2008 — density discontinuity test
 [40] Dempster, Laird & Rubin 1977 — EM algorithm
 [41] White 1982 — quasi-MLE consistency
+[42] Stone 1974 — cross-validatory choice
+[43] Geisser 1975 — predictive sample reuse
+[44] Vehtari, Gelman & Gabry 2017 — practical Bayesian LOO/WAIC
 -->

paper/paper_a_results_v3.md

@@ -1,5 +1,7 @@
 # IV. Experiments and Results
 
+Section IV reports the empirical results that calibrate and characterise the operational classifier of §III-H.1 (calibration developed in §III-L). The primary analyses (§IV-D through §IV-J, and the anchor-based ICCR calibration consolidated in §IV-M) are scoped to the Big-4 sub-corpus (Firms A–D, $n = 437$ CPAs with $n_{\text{sig}} \geq 10$, totalling 150,442 signatures with both descriptors available) per the methodology choice articulated in §III-G. §IV-K reports a full-dataset (686 CPAs) robustness check on the K=3 mixture and per-CPA score-rank convergence; §IV-A through §IV-C and §IV-L report the corpus-wide pipeline performance and feature-backbone ablation that support the descriptor choice of §III-F.
+
 ## A. Experimental Setup
 
 Experiments used mixed hardware: YOLOv11n training and inference for signature detection, and ResNet-50 forward inference for feature extraction over all 182,328 detected signatures, were performed on an Nvidia RTX 4090 (CUDA); the downstream statistical analyses (KDE antimode, Hartigan dip test, Beta-mixture EM with logit-Gaussian robustness check, Burgstahler-Dichev/McCrary density-smoothness diagnostic, and pairwise cosine/dHash computations) were performed on an Apple Silicon workstation with Metal Performance Shaders (MPS) acceleration.
@@ -11,10 +13,11 @@ Because all steps rely on deterministic forward inference over fixed pre-trained
 
 The YOLOv11n model achieved high detection performance on the validation set (Table II), with all loss components converging by epoch 60 and no significant overfitting despite the relatively small training set (425 images).
 We note that Table II reports validation-set metrics, as no separate hold-out test set was reserved given the small annotation budget (500 images total).
-However, the subsequent production deployment provides practical validation: batch inference on 86,071 documents yielded 182,328 extracted signatures (Table III), with an average of 2.14 signatures per document, consistent with the standard practice of two certifying CPAs per audit report.
+However, the subsequent production deployment provides a practical consistency check: batch inference on 86,071 documents yielded 182,328 extracted signatures (Table III), with an average of 2.14 signatures per document, consistent with the standard practice of two certifying CPAs per audit report.
 The high VLM--YOLO agreement rate (98.8%) further corroborates detection reliability at scale.
 
-<!-- TABLE III: Extraction Results
+**Table III.** Extraction Results.
+
 | Metric | Value |
 |--------|-------|
 | Documents processed | 86,071 |
@@ -23,15 +26,17 @@ The high VLM--YOLO agreement rate (98.8%) further corroborates detection reliabi
 | Avg. signatures per document | 2.14 |
 | CPA-matched signatures | 168,755 (92.6%) |
 | Processing rate | 43.1 docs/sec |
--->
+
+The Big-4 subset of the detection output yields 150,442 signatures with both descriptors (cosine and independent dHash) successfully computed; this is the per-signature population used in the primary analyses of §IV-D through §IV-J.
 
 ## C. All-Pairs Intra-vs-Inter Class Distribution Analysis
 
 Fig. 2 presents the cosine similarity distributions computed over the full set of *pairwise comparisons* under two groupings: intra-class (all signature pairs belonging to the same CPA) and inter-class (signature pairs from different CPAs).
-This all-pairs analysis is a different unit from the per-signature best-match statistics used in Sections IV-D onward; we report it first because it supplies the reference point for the KDE crossover used in per-document classification (Section III-K).
+This all-pairs analysis is a different unit from the per-signature best-match statistics used in Sections IV-D onward; we report it first because it supplies the reference point for the KDE crossover used in per-document classification (Section III-L).
 Table IV summarizes the distributional statistics.
 
-<!-- TABLE IV: Cosine Similarity Distribution Statistics
+**Table IV.** Cosine Similarity Distribution Statistics.
+
 | Statistic | Intra-class | Inter-class |
 |-----------|-------------|-------------|
 | N (pairs) | 41,352,824 | 500,000 |
@@ -40,349 +45,261 @@ Table IV summarizes the distributional statistics.
 | Median | 0.836 | 0.774 |
 | Skewness | −0.711 | −0.851 |
 | Kurtosis | 0.550 | 1.027 |
--->
 
 Both distributions are left-skewed and leptokurtic.
 Shapiro-Wilk and Kolmogorov-Smirnov tests rejected normality for both ($p < 0.001$), confirming that parametric thresholds based on normality assumptions would be inappropriate.
-Distribution fitting identified the lognormal distribution as the best parametric fit (lowest AIC) for both classes, though we use this result only descriptively; all subsequent threshold-estimator outputs reported in Section IV-D are derived via the methods of Section III-I to avoid single-family distributional assumptions.
+Distribution fitting identified the lognormal distribution as the best parametric fit (lowest AIC) for both classes, though we use this result only descriptively; the subsequent distributional diagnostics in Section IV-D are produced via the methods of Section III-I to avoid single-family distributional assumptions.
 
-The KDE crossover---where the two density functions intersect---was located at 0.837 (Table V).
+The KDE crossover---where the two density functions intersect---was located at 0.837.
-Under equal prior probabilities and equal misclassification costs, this crossover approximates the Bayes-optimal boundary between the two classes.
+Under equal prior probabilities and equal misclassification costs, this crossover is a candidate decision boundary between the two classes; we adopt it only as the operational LH/UN boundary in §III-H.1, not as a natural distributional threshold.
 Statistical tests confirmed significant separation between the two distributions (Cohen's $d = 0.669$, Mann-Whitney [36] $p < 0.001$, K-S 2-sample $p < 0.001$).
 
 We emphasize that pairwise observations are not independent---the same signature participates in multiple pairs---which inflates the effective sample size and renders $p$-values unreliable as measures of evidence strength.
 We therefore rely primarily on Cohen's $d$ as an effect-size measure that is less sensitive to sample size.
 A Cohen's $d$ of 0.669 indicates a medium effect size [29], confirming that the distributional difference is practically meaningful, not merely an artifact of the large sample count.
 
-## D. Signature-Level Distributional Characterisation
+## D. Big-4 Accountant-Level Distributional Characterisation
-
+
-This section applies the threshold-estimator and density-smoothness diagnostic of Section III-I to the per-signature similarity distribution.
+This section reports the empirical evidence for §III-I's distributional diagnostics at the Big-4 accountant level. The accountant-level dip-test rejection reported in Table V is, per §III-I.4, fully attributable to between-firm location shifts and integer mass-point artefacts rather than to within-population bimodality; the composition-decomposition diagnostics that establish this finding are tabulated in §IV-M below alongside the anchor-based ICCR calibration.
-The joint reading is that per-signature similarity is a continuous quality spectrum rather than a clean two-mechanism mixture, which is why the operational classifier (Section III-K) anchors its cosine cut on the whole-sample Firm A P7.5 percentile rather than on any mixture-fit crossing.
+
-
+**Table V.** Hartigan dip-test results, accountant-level marginals (Big-4 primary; comparison scopes from Script 32).
-### 1) Hartigan Dip Test: Unimodality at the Signature Level
+
-
+| Population | $n$ CPAs | $p_{\text{cos}}$ | $p_{\text{dHash}}$ | Interpretation |
-Applying the Hartigan & Hartigan dip test [37] to the per-signature best-match distributions reveals a critical structural finding (Table V).
+|---|---|---|---|---|
-The $N = 168{,}740$ count used in Table V and in the downstream same-CPA per-signature best-match analyses (Tables V and XII, and the Firm-A per-signature rows of Tables XIII and XVIII) is $15$ signatures smaller than the $168{,}755$ CPA-matched count reported in Table III: these $15$ signatures belong to CPAs with exactly one signature in the entire corpus, for whom no same-CPA pairwise best-match statistic can be computed, and are therefore excluded from all same-CPA similarity analyses.
+| **Big-4 pooled (primary)** | 437 | $< 5 \times 10^{-4}$ | $< 5 \times 10^{-4}$ | reject unimodality on both axes |
-
+| Firm A pooled alone | 171 | 0.992 | 0.924 | unimodal |
-<!-- TABLE V: Hartigan Dip Test Results
+| Firms B + C + D pooled | 266 | 0.998 | 0.906 | unimodal |
-| Distribution | N | dip | p-value | Verdict (α=0.05) |
+| All non-Firm-A pooled | 515 | 0.998 | 0.907 | unimodal |
-|--------------|---|-----|---------|------------------|
+
-| Firm A cosine (max-sim) | 60,448 | 0.0019 | 0.169 | Unimodal |
+Bootstrap implementation: $n_{\text{boot}} = 2000$; for the Big-4 cells, no bootstrap replicate exceeded the observed dip statistic, so the empirical $p$-value is bounded above by the bootstrap resolution $1 / 2000 = 5 \times 10^{-4}$ (Script 34 reports this as $p = 0.0000$; we report $p < 5 \times 10^{-4}$ to reflect the resolution). Single-firm dip statistics for Firms B, C, and D were not separately computed.
-| Firm A min dHash (independent) | 60,448 | 0.1051 | <0.001 | Multimodal |
+
-| All-CPA cosine (max-sim) | 168,740 | 0.0035 | <0.001 | Multimodal |
+**Table VI.** Burgstahler-Dichev / McCrary density-smoothness diagnostic on accountant-level marginals (cosine in 0.002 bins; dHash in integer bins; $\alpha = 0.05$, two-sided).
-| All-CPA min dHash (independent) | 168,740 | 0.0468 | <0.001 | Multimodal |
+
--->
+| Population | Cosine: significant transition? | dHash: significant transition? |
-
+|---|---|---|
-Firm A's per-signature cosine distribution *fails to reject unimodality* ($p = 0.17$), a pattern consistent with a dominant high-similarity regime plus a long left tail attributable to within-firm heterogeneity in signing outputs (Section III-G discusses the scope of partner-level claims).
+| **Big-4 pooled (primary)** | none ($p > 0.05$) | none ($p > 0.05$) |
-The all-CPA cosine distribution, which mixes many firms with heterogeneous signing practices, is *multimodal* ($p < 0.001$).
+| Firm A pooled alone | none | none |
-The Firm A unimodal-long-tail finding is, in conjunction with the byte-identity, partner-ranking, and intra-report evidence reported below, consistent with the replication-dominated framing (Section III-H): a dominant high-similarity regime plus residual within-firm heterogeneity, rather than two cleanly separated mechanisms.
+| Firms B + C + D pooled | none | one transition at $\overline{\text{dHash}} = 10.8$ |
-
+| All non-Firm-A pooled | none | one transition at $\overline{\text{dHash}} = 6.6$ |
-### 2) Burgstahler-Dichev / McCrary Density-Smoothness Diagnostic
+
-
+The Big-4-scope null on both axes is consistent with the §IV-E mixture evidence: the K=3 components overlap in their tails rather than separating sharply, so a local-discontinuity test does not flag a transition. Outside Big-4, dHash transitions appear in some subsets but no cosine transition is identified in any tested subset (Script 32 sweeps; pre-2018 and post-2020 stratified variants exhibit dHash transitions at varying locations). These off-Big-4 dHash transitions are scope-dependent and are not used as operational thresholds; we do not claim a specific structural interpretation for them without an explicit bin-width sensitivity sweep at those scopes.
-Applying the BD/McCrary procedure (Section III-I.3) to the per-signature cosine distribution yields a nominally significant $Z^- \rightarrow Z^+$ transition at cosine 0.985 for Firm A and 0.985 for the full sample; the min-dHash distributions exhibit a transition at Hamming distance 2 for both Firm A and the full sample under the bin width ($0.005$ / $1$) used here.
+
-Two cautions, however, prevent us from treating these signature-level transitions as thresholds.
+## E. Big-4 K=2 / K=3 Mixture Fits
-First, the cosine transition at 0.985 lies *inside* the non-hand-signed mode rather than at the separation between two mechanisms, consistent with the dip-test finding that per-signature cosine is not cleanly bimodal.
+
-Second, Appendix A documents that the signature-level transition locations are not bin-width-stable (Firm A cosine drifts across 0.987, 0.985, 0.980, 0.975 as the bin width is widened from 0.003 to 0.015, and full-sample dHash transitions drift across 2, 10, 9 as bin width grows from 1 to 3), which is characteristic of a histogram-resolution artifact rather than of a genuine density discontinuity between two mechanisms.
+This section reports the K=2 and K=3 2D Gaussian mixture fits to the Big-4 accountant-level distribution and the bootstrap stability of their marginal crossings.
-We therefore use BD/McCrary as a density-smoothness diagnostic rather than as an independent threshold estimator.
+
-
+**Table VII.** Big-4 K=2 mixture components (descriptive partition; not mechanism clusters per §III-J) and marginal-crossing bootstrap 95% confidence intervals.
-### 3) Beta Mixture at Signature Level: A Forced Fit
+
-
+| K=2 component | $\overline{\text{cos}}$ | $\overline{\text{dHash}}$ | weight |
-Fitting 2- and 3-component Beta mixtures to Firm A's per-signature cosine via EM yields a clear BIC preference for the 3-component fit ($\Delta\text{BIC} = 381$), with a parallel preference under the logit-GMM robustness check.
+|---|---|---|---|
-For the full-sample cosine the 3-component fit is likewise strongly preferred ($\Delta\text{BIC} = 10{,}175$).
+| K=2-a (low-cos / high-dHash position) | 0.954 | 7.14 | 0.689 |
-Under the forced 2-component fit the Firm A Beta crossing lies at 0.977 and the logit-GMM crossing at 0.999---values sharply inconsistent with each other, indicating that the 2-component parametric structure is not supported by the data.
+| K=2-b (high-cos / low-dHash position) | 0.983 | 2.41 | 0.311 |
-Under the full-sample 2-component forced fit no Beta crossing is identified; the logit-GMM crossing is at 0.980.
+
-
+Marginal crossings (point + bootstrap 95% CI, $n_{\text{boot}} = 500$):
-### 4) Joint Reading of the Three Diagnostics
+
-
+| Axis | Point | Bootstrap median | 95% CI | CI half-width |
-The three diagnostics agree that per-signature similarity does not form a clean two-mechanism mixture:
+|---|---|---|---|---|
-(i) the Hartigan dip test fails to reject unimodality for Firm A and rejects it for the heterogeneous-firm pooled sample;
+| cos | 0.9755 | 0.9754 | $[0.9742, 0.9772]$ | 0.0015 |
-(ii) BIC strongly prefers a 3-component over a 2-component Beta fit, so the 2-component crossing is a *forced fit* and the Beta-vs-logit-Gaussian disagreement (0.977 vs 0.999 for Firm A) reflects parametric-form sensitivity rather than a stable two-mechanism boundary;
+| dHash | 3.755 | 3.763 | $[3.476, 3.969]$ | 0.246 |
-(iii) the BD/McCrary procedure locates its candidate transition *inside* the non-hand-signed mode rather than between modes, and the transition is not bin-width-stable.
+
-
+$\text{BIC}(K{=}2) = -1108.45$ (Script 34).
-Table VI summarises the signature-level threshold-estimator outputs for cross-method comparison.
+
-
+**Table VIII.** Big-4 K=3 mixture components (descriptive firm-compositional partition per §III-J; not mechanism clusters).
-<!-- TABLE VI: Signature-Level Threshold-Estimator Summary
+
-| Method | Cosine | dHash |
+| K=3 component | $\overline{\text{cos}}$ | $\overline{\text{dHash}}$ | weight | descriptive position |
-|--------|--------|-------|
+|---|---|---|---|---|
-| All-pairs KDE crossover (Section IV-C)                                          | 0.837              | —      |
+| C1 | 0.9457 | 9.17 | 0.143 | low-cos / high-dHash corner |
-| Beta-2 EM crossing (Firm A; forced fit, BIC prefers $K{=}3$)                    | 0.977              | —      |
+| C2 | 0.9558 | 6.66 | 0.536 | central region |
-| logit-GMM-2 crossing (Full sample; forced fit)                                  | 0.980              | —      |
+| C3 | 0.9826 | 2.41 | 0.321 | high-cos / low-dHash corner |
-| BD/McCrary transition (diagnostic; bin-unstable, Appendix A)                    | 0.985              | 2.0    |
+
-| Firm A whole-sample P7.5 (operational anchor; Section III-K)                    | 0.95               | —      |
+$\text{BIC}(K{=}3) = -1111.93$, lower than $K{=}2$ by $3.48$ (mild support; not by itself decisive). The full-fit K=3 baseline above is reproduced in Scripts 35, 37, and 38 with identical hyperparameters; Script 37 additionally fits K=3 on each leave-one-firm-out training set (those fold-specific components differ from the full-fit baseline by design and are reported separately in §IV-G Table XIII). Operational use of the K=2 / K=3 fits is governed by §III-J and §III-L; §IV-G reports the LOOO reproducibility evidence that motivates reporting both fits descriptively.
-| Firm A whole-sample dHash_indep P75 (operational $\leq 5$ band lower edge)      | —                  | 4      |
+
-| Firm A whole-sample dHash_indep ceiling (style-consistency boundary)            | —                  | 15     |
+## F. Convergent Internal-Consistency Checks
-| Firm A calibration-fold cosine P5 (held-out validation; Section IV-F.2)         | 0.9407             | —      |
+
-| Firm A calibration-fold dHash_indep P95 (held-out validation)                   | —                  | 9      |
+This section reports the empirical evidence for §III-K's three-score internal-consistency analysis. We re-emphasise the §III-K caveat: the three scores are deterministic functions of the same per-CPA descriptor pair $(\overline{\text{cos}}_a, \overline{\text{dHash}}_a)$ and are *not statistically independent measurements*. The pairwise correlations document internal consistency among feature-derived ranks rather than external validation against an independent ground truth.
--->
+
-
+**Table IX.** Per-CPA Spearman rank correlations among three feature-derived scores, Big-4, $n = 437$.
-Non-hand-signed replication quality is therefore best read as a continuous spectrum produced by firm-specific reproduction technologies (administrative stamping in early years, firm-level e-signing later) acting on a common stored exemplar.
+
-This finding has a direct methodological pay-off: it is *why* the operational cosine cut is anchored on the whole-sample Firm A P7.5 percentile (Section III-K), and it is *why* the byte-level pixel-identity anchor (Section IV-F.1) is the natural threshold-free positive reference for downstream validation.
+| Score pair | Spearman $\rho$ | $p$-value |
-
+|---|---|---|
-## E. Calibration Validation with Firm A
+| K=3 P(C1) vs deployed box-rule less-replication-dominated rate | $+0.9627$ | $< 10^{-248}$ |
-
+| Reverse-anchor cosine percentile vs deployed box-rule less-replication-dominated rate | $+0.8890$ | $< 10^{-149}$ |
-Fig. 3 presents the per-signature cosine and dHash distributions of Firm A compared to the overall population.
+| K=3 P(C1) vs Reverse-anchor cosine percentile | $+0.8794$ | $< 10^{-142}$ |
-Table IX reports the proportion of Firm A signatures crossing each candidate threshold; these rates play the role of calibration-validation metrics (what fraction of a known replication-dominated population does each threshold capture?).
+
-
+(Source: Script 38.) Reverse-anchor reference: 2D Gaussian fit by MCD (support fraction 0.85) on $n = 249$ non-Big-4 CPAs; reference centre $\overline{\text{cos}} = 0.935$, $\overline{\text{dHash}} = 9.77$.
-<!-- TABLE IX: Firm A Whole-Sample Capture Rates (consistency check, NOT external validation)
+
-| Rule | Firm A rate | k / N |
+**Table X.** Per-firm summary across the three feature-derived scores, Big-4.
-|------|-------------|-------|
+
-| cosine > 0.837 (all-pairs KDE crossover)              | 99.93% | 60,408 / 60,448 |
+| Firm | $n$ CPAs | mean $P(\text{C1})$ | mean reverse-anchor score | mean deployed less-replication-dominated rate |
-| cosine > 0.9407 (calibration-fold P5)                 | 95.15% | 57,518 / 60,448 |
+|---|---|---|---|---|
-| cosine > 0.945 (calibration-fold P5 rounded)          | 94.02% | 56,836 / 60,448 |
+| Firm A | 171 | 0.0072 | $-0.9726$ | 0.1935 |
-| cosine > 0.95                                         | 92.51% | 55,922 / 60,448 |
+| Firm B | 112 | 0.1410 | $-0.8201$ | 0.6962 |
-| dHash_indep ≤ 5 (whole-sample upper-tail of mode)     | 84.20% | 50,897 / 60,448 |
+| Firm C | 102 | 0.3110 | $-0.7672$ | 0.7896 |
-| dHash_indep ≤ 8                                       | 95.17% | 57,527 / 60,448 |
+| Firm D | 52 | 0.2406 | $-0.7125$ | 0.7608 |
-| dHash_indep ≤ 15 (style-consistency boundary)         | 99.83% | 60,348 / 60,448 |
+
-| cosine > 0.95 AND dHash_indep ≤ 8 (operational dual)  | 89.95% | 54,370 / 60,448 |
+(Source: Script 38 per-firm summary; reverse-anchor score is sign-flipped so that *higher* values indicate deeper into the reference left tail = less replication-dominated relative to the non-Big-4 reference.)
-
+
-All rates computed exactly from the full Firm A sample (N = 60,448 signatures); per-rule counts and codes are available in the supplementary materials.
+The three scores agree on placing Firm A as the most replication-dominated and the three non-Firm-A firms as less replication-dominated. The K=3 posterior P(C1) and the box-rule less-replication-dominated rate (Score 1 and Score 3) place Firm C at the least-replication-dominated end of Big-4; the reverse-anchor cosine percentile (Score 2) ranks Firm D fractionally above Firm C. This residual within-Big-4-non-A disagreement is a design feature of the reverse-anchor metric: Score 2 measures only the marginal cosine percentile under the non-Big-4 reference, so a firm with a slightly higher cosine but a markedly different dHash distribution (Firm D vs Firm C) can score higher on Score 2 while scoring lower on Scores 1 and 3, both of which use both descriptors.
--->
+
-
+**Table XI.** Per-signature Cohen $\kappa$ (binary collapse, replication-dominated vs less-replication-dominated), $n = 150{,}442$ Big-4 signatures.
-Table IX is a whole-sample consistency check rather than an external validation: the thresholds 0.95, dHash median, and dHash 95th percentile are themselves anchored to the whole-sample Firm A distribution described in Section III-K (the 70/30 calibration-fold thresholds of Table XI are separate and slightly different, e.g., calibration-fold cosine P5 = 0.9407 rather than the whole-sample heuristic 0.95).
+
-The dual rule cosine $> 0.95$ AND dHash $\leq 8$ captures 89.95% of Firm A, a value that is consistent with the dip-test-confirmed unimodal-long-tail shape of Firm A's per-signature cosine distribution (Section IV-D.1) and the 92.5% / 7.5% signature-level split (Section III-H).
+| Pair | Cohen $\kappa$ |
-Section IV-F.2 reports the corresponding rates on the 30% Firm A hold-out fold, which provides the external check these whole-sample rates cannot.
+|---|---|
-
+| deployed binary high-confidence box rule (cos $> 0.95$ AND dHash $\leq 5$) vs per-CPA K=3 hard label | 0.662 |
-## F. Pixel-Identity, Inter-CPA, and Held-Out Firm A Validation
+| deployed binary high-confidence box rule vs per-signature K=3 hard label | 0.559 |
-
+| Per-CPA K=3 hard label vs per-signature K=3 hard label | 0.870 |
-We report three validation analyses corresponding to the anchors of Section III-J.
+
-
+(Source: Script 39.) Per-signature K=3 components ($n = 150{,}442$) sorted by ascending cosine: $(0.928, 9.75, 0.146)$ / $(0.963, 6.04, 0.582)$ / $(0.989, 1.27, 0.272)$, an absolute cosine drift of $0.018$ in C1 and $0.006$ in C3 relative to the per-CPA fit. These convergence checks cover only the binary high-confidence rule (cos $> 0.95$ AND dHash $\leq 5$); the five-way classifier's moderate-confidence band ($5 < \text{dHash} \leq 15$) retains its prior calibration and capture-rate evidence (supplementary materials; cross-referenced in §IV-J).
-### 1) Pixel-Identity Positive Anchor with Inter-CPA Negative Anchor
+
-
+## G. Leave-One-Firm-Out Reproducibility
-Of the 182,328 extracted signatures, 310 have a same-CPA nearest match that is byte-identical after crop and normalization (pixel-identical-to-closest = 1); these form the byte-identity positive anchor---a pair-level proof of image reuse that serves as conservative ground truth for non-hand-signed signatures, subject to the source-template edge case discussed in Section V-G.
+
-Within Firm A specifically, 145 of these byte-identical signatures are distributed across 50 distinct partners (of 180 registered Firm A partners), with 35 of the byte-identical pairs spanning different fiscal years; this Firm A decomposition is reproduced by `signature_analysis/28_byte_identity_decomposition.py` and reported in `reports/byte_identity_decomp/byte_identity_decomposition.json` (Appendix B).
+This section reports the firm-level cross-validation evidence motivating §III-J's "K=3 descriptive, not operational" framing.
-As the gold-negative anchor we sample 50,000 random cross-CPA signature pairs (inter-CPA cosine: mean $= 0.762$, $P_{95} = 0.884$, $P_{99} = 0.913$, max $= 0.988$).
+
-Because the positive and negative anchor populations are constructed from different sampling units (byte-identical same-CPA pairs vs random inter-CPA pairs), their relative prevalence in the combined anchor set is arbitrary, and precision / $F_1$ / recall therefore have no meaningful population interpretation.
+**Table XII.** K=2 leave-one-firm-out across the four Big-4 folds.
-We accordingly report FAR with Wilson 95% confidence intervals against the large inter-CPA negative anchor in Table X.
+
-The primary quantity reported by Table X is FAR: the probability that a random pair of signatures from *different* CPAs exceeds the candidate threshold.
+| Held-out firm | $n_{\text{train}}$ | $n_{\text{held}}$ | Fold rule (cos cut, dHash cut) | Held-out classified as templated by fold rule |
-We do not report an Equal Error Rate: EER is meaningful only when the positive and negative error-rate curves cross in a nontrivial interior region, but byte-identical positives all sit at cosine $\approx 1$ by construction, so FRR against that subset is trivially $0$ at every threshold below $1$. An EER calculation against this anchor would be arithmetic tautology rather than biometric performance, and we therefore omit it.
+|---|---|---|---|---|
-
+| Firm A | 266 | 171 | cos $> 0.9380$ AND dHash $\leq 8.79$ | $171 / 171 = 100.00\%$ ($95\%$ Wilson $[97.80\%, 100.00\%]$) |
-<!-- TABLE X: Cosine Threshold Sweep — FAR Against 50,000 Inter-CPA Negative Pairs
+| Firm B | 325 | 112 | cos $> 0.9744$ AND dHash $\leq 3.98$ | $0 / 112 = 0\%$ ($95\%$ Wilson $[0\%, 3.32\%]$) |
-| Threshold | FAR | FAR 95% Wilson CI |
+| Firm C | 335 | 102 | cos $> 0.9752$ AND dHash $\leq 3.75$ | $0 / 102 = 0\%$ ($95\%$ Wilson $[0\%, 3.63\%]$) |
-|-----------|-----|-------------------|
+| Firm D | 385 | 52 | cos $> 0.9756$ AND dHash $\leq 3.74$ | $0 / 52 = 0\%$ ($95\%$ Wilson $[0\%, 6.88\%]$) |
-| 0.837 (all-pairs KDE crossover) | 0.2062 | [0.2027, 0.2098] |
+
-| 0.900                            | 0.0233 | [0.0221, 0.0247] |
+(Source: Script 36.) Across-fold cosine crossing: pairwise range $[0.9380, 0.9756]$, range = $0.0376$; max absolute deviation from the across-fold mean is $0.028$. This exceeds the report's $0.005$ across-fold stability tolerance by $5.6\times$ and is much larger than the full-Big-4 bootstrap CI half-width of $0.0015$. Together with the all-or-nothing held-out classification pattern (Firm A held out $\Rightarrow$ all held-out CPAs templated; any non-Firm-A firm held out $\Rightarrow$ none templated), this indicates the K=2 boundary is essentially a Firm-A-vs-others separator rather than a within-Big-4 mechanism boundary.
-| 0.945 (calibration-fold P5 rounded) | 0.0008 | [0.0006, 0.0011] |
+
-| 0.950 (whole-sample Firm A P7.5; operational cut)  | 0.0007 | [0.0005, 0.0009] |
+**Table XIII.** K=3 leave-one-firm-out: C1 component shape and held-out membership.
-| 0.973 (signature-level Beta/KDE upper bound)        | 0.0003 | [0.0002, 0.0004] |
+
-| 0.979 (signature-level Beta-2 forced-fit crossing)  | 0.0002 | [0.0001, 0.0004] |
+| Held-out firm | C1 cos (fit) | C1 dHash (fit) | C1 weight (fit) | Held-out C1 hard-label rate | Full-Big-4 baseline C1% | Absolute difference |
-
+|---|---|---|---|---|---|---|
-Table note: We do not include FRR against the byte-identical positive anchor as a column here: the byte-identical subset has cosine $\approx 1$ by construction, so FRR against that subset is trivially $0$ at every threshold below $1$ and carries no biometric information beyond verifying that the threshold does not exceed $1$. The conservative-subset FRR role of the byte-identical anchor is instead discussed qualitatively in Section V-F.
+| Full-Big-4 baseline | 0.9457 | 9.17 | 0.143 | — | — | — |
--->
+| Firm A held out | 0.9425 | 10.13 | 0.145 | $4.68\%$ | $0.00\%$ | $4.68$ pp |
-
+| Firm B held out | 0.9441 | 9.16 | 0.127 | $7.14\%$ | $8.93\%$ | $1.76$ pp |
-Two caveats apply.
+| Firm C held out | 0.9504 | 8.41 | 0.126 | $36.27\%$ | $23.53\%$ | $12.77$ pp |
-First, the byte-identical positive anchor referenced above is a *conservative subset* of the true non-hand-signed population: it captures only those non-hand-signed signatures whose nearest match happens to be byte-identical, not those that are near-identical but not bytewise identical.
+| Firm D held out | 0.9439 | 9.29 | 0.120 | $17.31\%$ | $11.54\%$ | $5.81$ pp |
-A would-be FRR computed against this subset is definitionally $0$ at every threshold below $1$ (since byte-identical pairs have cosine $\approx 1$), so such an FRR is a mathematical boundary check rather than an empirical miss-rate estimate; we discuss the generalization limits of this conservative-subset framing in Section V-F.
+
-Second, the 0.945 / 0.95 thresholds are derived from the Firm A whole-sample and calibration-fold percentiles rather than from this anchor set, so the FAR values in Table X are post-hoc-fit-free evaluations of thresholds that were not chosen to optimize Table X.
+(Source: Script 37; screening label `P2_PARTIAL`.) Component shape is reproducible across folds: max deviation of C1 cosine = $0.005$, C1 dHash = $0.96$, C1 weight = $0.023$. Hard-posterior membership for the held-out firm varies: max absolute difference from the full-Big-4 baseline is $12.77$ pp at the Firm C held-out fold, exceeding the report's $5$ pp viability bar. We accordingly do not use K=3 hard-posterior membership as an operational classifier label (§III-J, §III-L).
-The very low FAR at the operational cut is therefore informative about specificity against a realistic inter-CPA negative population.
+
-
+## H. Pixel-Identity Positive-Anchor Miss Rate
-### 2) Held-Out Firm A Validation (within-Firm-A sampling variance disclosure)
+
-
+This section reports the only conservative hard-positive subset analysis available in the corpus: the positive-anchor miss rate against $n = 262$ Big-4 signatures whose nearest same-CPA match is byte-identical after crop and normalisation. Independent hand-signing cannot produce pixel-identical images, so byte-identical signatures are a conservative hard-positive subset for image replication. The analysis is one-sided (positive-anchor only); a paired false-alarm rate against a hand-signed negative anchor is not available because no signature-level hand-signed ground truth exists in the corpus (§III-K item 4).
-We split Firm A CPAs randomly 70 / 30 at the CPA level into a calibration fold (124 CPAs, 45,116 signatures) and a held-out fold (54 CPAs, 15,332 signatures).
+
-The total of 178 Firm A CPAs differs from the 180 in the Firm A registry by two CPAs whose signatures could not be matched to a single assigned-accountant record because of disambiguation ties in the CPA registry and which we therefore exclude from both folds; this handling is made explicit here.
+**Table XIV.** Positive-anchor miss rate, $n = 262$ Big-4 byte-identical signatures.
-Thresholds are re-derived from calibration-fold percentiles only.
+
-Table XI reports both calibration-fold and held-out-fold capture rates with Wilson 95% CIs and a two-proportion $z$-test.
+| Classifier | Misclassified as less-replication-dominated | Miss rate | Wilson 95% CI |
-
+|---|---|---|---|
-<!-- TABLE XI: Held-Out vs Calibration Firm A Capture Rates and Generalization Test
+| deployed binary high-confidence box rule (cos $> 0.95$ AND dHash $\leq 5$) | $0 / 262$ | $0\%$ | $[0\%, 1.45\%]$ |
-| Rule | Calibration 70% fold (CI) | Held-out 30% fold (CI) | 2-prop z | p | k/n calib | k/n held |
+| K=3 per-CPA hard label (C3 = high-cos / low-dHash; descriptive) | $0 / 262$ | $0\%$ | $[0\%, 1.45\%]$ |
-|------|---------------------------|-------------------------|----------|---|-----------|----------|
+| Reverse-anchor (prevalence-calibrated cut) | $0 / 262$ | $0\%$ | $[0\%, 1.45\%]$ |
-| cosine > 0.837                       | 99.94% [99.91%, 99.96%] | 99.93% [99.87%, 99.96%] | +0.31  | 0.756 n.s. | 45,087/45,116 | 15,321/15,332 |
+
-| cosine > 0.9407 (calib-fold P5)      | 94.99% [94.79%, 95.19%] | 95.63% [95.29%, 95.94%] | -3.19  | 0.001      | 42,856/45,116 | 14,662/15,332 |
+(Source: Script 40.) Per-firm breakdown of the byte-identical subset: Firm A 145; Firm B 8; Firm C 107; Firm D 2. All three candidate scores correctly assign every byte-identical signature to the replicated class.
-| cosine > 0.945 (calib-fold P5 rounded) | 93.77% [93.54%, 93.99%] | 94.78% [94.41%, 95.12%] | -4.54  | <0.001     | 42,305/45,116 | 14,531/15,332 |
+
-| cosine > 0.950 (whole-sample P7.5; operational cut) | 92.14% [91.89%, 92.38%] | 93.61% [93.21%, 93.98%] | -5.97  | <0.001     | 41,570/45,116 | 14,352/15,332 |
+We caution that for the deployed box rule this result is close to tautological (byte-identical nearest-neighbour signatures have cosine $\approx 1$ and dHash $\approx 0$, well inside the rule's high-confidence region). The reverse-anchor cut is chosen by *prevalence calibration* against the box rule's overall replicated rate of $49.58\%$ across Big-4 signatures; this is a documented limitation since no signature-level hand-signed ground truth exists to permit direct ROC optimisation.
-| dHash_indep ≤ 5                      | 82.96% [82.61%, 83.31%] | 87.84% [87.31%, 88.34%] | -14.29 | <0.001     | 37,430/45,116 | 13,467/15,332 |
+
-| dHash_indep ≤ 8                      | 94.84% [94.63%, 95.04%] | 96.13% [95.82%, 96.43%] | -6.45  | <0.001     | 42,788/45,116 | 14,739/15,332 |
+## I. Inter-CPA Pair-Level Coincidence Rate
-| dHash_indep ≤ 9 (calib-fold P95)     | 96.65% [96.48%, 96.81%] | 97.48% [97.22%, 97.71%] | -5.07  | <0.001     | 43,604/45,116 | 14,945/15,332 |
+
-| dHash_indep ≤ 15                     | 99.83% [99.79%, 99.87%] | 99.84% [99.77%, 99.89%] | -0.31  | 0.754 n.s. | 45,040/45,116 | 15,308/15,332 |
+The metric reported here is the inter-CPA pair-level coincidence rate (ICCR). It is the per-pair rate at which two signatures from different CPAs satisfy the deployed rule. We do not label it as a False Acceptance Rate because (a) FAR has a biometric-verification meaning that requires ground-truth negative labels, and (b) the inter-CPA negative-anchor assumption is partially violated by within-firm cross-CPA template-like collision structures (§III-L.4 cross-firm hit matrix).
-| cosine > 0.95 AND dHash_indep ≤ 8    | 89.40% [89.12%, 89.68%] | 91.54% [91.09%, 91.97%] | -7.60  | <0.001     | 40,335/45,116 | 14,035/15,332 |
+
-
+A corpus-wide spike on $\sim 50{,}000$ inter-CPA pairs gives a per-comparison rate of $0.0005$ (Wilson 95% CI $[0.0003, 0.0007]$) at the cosine cut $0.95$. The Big-4-scope spike at higher sample size ($5 \times 10^5$ inter-CPA pairs) replicates this number, adds the structural dimension (dHash), and adds joint-rule rates; the §III-L.1 numbers are referenced rather than duplicated here, and the consolidated ICCR calibration appears in §IV-M Tables XXI–XXVI.
-Calibration-fold thresholds: Firm A cosine median = 0.9862, P1 = 0.9067, P5 = 0.9407; dHash_indep median = 2, P95 = 9. Counts and z/p values are reproducible from the supplementary materials (fixed random seed).
+
--->
+## J. Five-Way Per-Signature + Document-Level Classification Output
-
+
-Table XI reports both calibration-fold and held-out-fold capture rates with Wilson 95% CIs and a two-proportion $z$-test.
+This section reports the five-way per-signature + document-level worst-case classifier output on the Big-4 sub-corpus. See §III-H.1 for the five-way category definitions and the cosine and dHash cuts; calibration is in §III-L.
-We report fold-versus-fold comparisons rather than fold-versus-whole-sample comparisons, because the whole-sample rate is a weighted average of the two folds and therefore cannot, in general, fall inside the Wilson CI of either fold when the folds differ in rate; the correct generalization reference is the calibration fold, which produced the thresholds.
+
-
+**Table XV.** Five-way per-signature category counts, Big-4 sub-corpus, $n = 150{,}442$ classified.
-Under this proper test the two extreme rules agree across folds (cosine $> 0.837$ and $\text{dHash}_\text{indep} \leq 15$; both $p > 0.7$).
+
-The operationally relevant rules in the 85–95% capture band differ between folds by 1–5 percentage points ($p < 0.001$ given the $n \approx 45\text{k}/15\text{k}$ fold sizes).
+| Category | Long name | $n$ signatures | % of classified |
-Both folds nevertheless sit in the same replication-dominated regime: every calibration-fold rate in the 85–99% range has a held-out counterpart in the 87–99% range, and the operational dual rule cosine $> 0.95$ AND $\text{dHash}_\text{indep} \leq 8$ captures 89.40% of the calibration fold and 91.54% of the held-out fold.
+|---|---|---|---|
-The modest fold gap is consistent with within-Firm-A heterogeneity in replication intensity: the random 30% CPA sample evidently contained proportionally more high-replication CPAs.
+| HC | High-confidence non-hand-signed | 74,593 | 49.58% |
-We therefore interpret the held-out fold as confirming the qualitative finding (Firm A is strongly replication-dominated across both folds) while cautioning that exact rates carry fold-level sampling noise that a single 30% split cannot eliminate; the threshold-independent partner-ranking analysis (Section IV-G.2) is the cross-check that is robust to this fold variance.
+| MC | Moderate-confidence non-hand-signed | 39,817 | 26.47% |
-
+| HSC | High style consistency | 314 | 0.21% |
-### 3) Operational-Threshold Sensitivity: cos $> 0.95$ vs cos $> 0.945$
+| UN | Uncertain | 35,480 | 23.58% |
-
+| LH | Likely hand-signed | 238 | 0.16% |
-The per-signature classifier (Section III-K) uses cos $> 0.95$ as its operational cosine cut, anchored on the whole-sample Firm A P7.5 heuristic (i.e., 7.5% of whole-sample Firm A signatures lie at or below 0.95; see Section III-H).
+
-We report a sensitivity check in which this round-number cut is replaced by the slightly stricter calibration-fold P5 rounded value cos $> 0.945$ (calibration-fold P5 = 0.9407, see Table XI).
+(Source: Script 42; 11 of 150,453 loaded Big-4 signatures lacked one or both descriptors and were excluded. The $150{,}442$ vs $150{,}453$ distinction — descriptor-complete vs vector-complete — recurs across §IV: descriptor-complete analyses (§IV-D through §IV-J, all using accountant-level aggregates or per-signature category counts derived from the same 150,442-signature substrate) use $n = 150{,}442$; vector- or pair-recomputed analyses (§IV-M.2 Table XXI, §IV-M.3 Table XXII, §IV-M.5 Tables XXIV–XXV; Scripts 40b, 43, 44) use $n = 150{,}453$ because their pair- or pool-level computations load all vector-complete signatures including those failing the descriptor-complete filter. See §III-G for the sample-size reconciliation.)
-Table XII reports the five-way classifier output under each cut.
+
-
+**Per-firm five-way breakdown (% within firm).**
-<!-- TABLE XII: Classifier Sensitivity to the Operational Cosine Cut (All-Sample Five-Way Output, N = 168,740 signatures)
+
-| Category                                   | cos > 0.95 count (%) | cos > 0.945 count (%) | Δ count |
+| Firm | HC | MC | HSC | UN | LH | total signatures |
-|--------------------------------------------|----------------------|-----------------------|---------|
+|---|---|---|---|---|---|---|
-| High-confidence non-hand-signed            | 76,984 (45.62%)      | 79,278 (46.98%)       | +2,294  |
+| Firm A | 81.70% | 10.76% | 0.05% | 7.42% | 0.07% | 60,448 |
-| Moderate-confidence non-hand-signed        | 43,906 (26.02%)      | 50,001 (29.63%)       | +6,095  |
+| Firm B | 34.56% | 35.88% | 0.29% | 29.09% | 0.18% | 34,248 |
-| High style consistency                     |    546 ( 0.32%)      |    665 ( 0.39%)       |   +119  |
+| Firm C | 23.75% | 41.44% | 0.38% | 34.21% | 0.22% | 38,613 |
-| Uncertain                                  | 46,768 (27.72%)      | 38,260 (22.67%)       | -8,508  |
+| Firm D | 24.51% | 29.33% | 0.22% | 45.65% | 0.29% | 17,133 |
-| Likely hand-signed                         |    536 ( 0.32%)      |    536 ( 0.32%)       |     +0  |
+
--->
+(Source: Script 42 per-firm cross-tab.) The per-firm pattern qualitatively aligns with the K=3 cluster cross-tab of Table XVII: Firm A's signatures concentrate in the HC band (81.70%) while its CPAs concentrate at the accountant level in the K=3 C3 (high-cos / low-dHash) component (82.46%; Table XVII). These two figures address different units (per-signature classification vs per-CPA hard cluster assignment) and are not directly comparable as a like-for-like consistency check; we report the qualitative alignment but do not infer a numerical equivalence. The three non-Firm-A Big-4 firms have markedly lower HC rates than Firm A and substantially higher Uncertain rates, with Firm D having the highest Uncertain rate (45.65%).
-
+
-At the aggregate firm-level, the operational dual rule cos $> 0.95$ AND $\text{dHash}_\text{indep} \leq 8$ captures 89.95% of whole Firm A under the 0.95 cut and 91.14% under the 0.945 cut---a shift of 1.19 percentage points.
+**Document-level worst-case aggregation.** Each audit report typically carries two certifying-CPA signatures. We aggregate signature-level outcomes to document-level labels using the worst-case rule (HC > MC > HSC > UN > LH; §III-H.1), applied to the Big-4 sub-corpus.
-At the per-signature categorization level, replacing 0.95 by 0.945 reclassifies 8,508 signatures (5.04% of the corpus) out of the Uncertain band; 6,095 of them migrate to Moderate-confidence non-hand-signed, 2,294 to High-confidence non-hand-signed, and 119 to High style consistency.
+
-The Likely-hand-signed category is unaffected because it depends only on the fixed all-pairs KDE crossover cosine $= 0.837$.
+**Table XVI.** Document-level worst-case category counts, Big-4 sub-corpus, $n = 75{,}233$ unique PDFs.
-The High-confidence non-hand-signed share grows from 45.62% to 46.98%.
+
-
+| Category | Long name | $n$ documents | % |
-We interpret this sensitivity pattern as indicating that the classifier's aggregate and high-confidence output is robust to the choice of operational cut within a 0.005-cosine neighbourhood of the Firm A P7.5 anchor, and that the movement is concentrated at the Uncertain/Moderate-confidence boundary.
+|---|---|---|---|
-The paper therefore retains cos $> 0.95$ as the primary operational cut for transparency (round-number P7.5 of the whole-sample Firm A reference distribution) and reports the 0.945 results as a sensitivity check rather than as a deployed alternative.
+| HC | High-confidence non-hand-signed | 46,857 | 62.28% |
-
+| MC | Moderate-confidence non-hand-signed | 19,667 | 26.14% |
-## G. Additional Firm A Benchmark Validation
+| HSC | High style consistency | 167 | 0.22% |
-
+| UN | Uncertain | 8,524 | 11.33% |
-The capture rates of Section IV-E are an *internal* consistency check: they ask "how much of Firm A does our threshold capture?", but the threshold was itself derived from Firm A's percentiles, so a high capture rate is not surprising.
+| LH | Likely hand-signed | 18 | 0.02% |
-To go beyond this circular check, we report three further analyses, each chosen so that the *informative quantity* does not depend on the threshold's absolute value:
+
-
+(Source: Script 42 document-level table; 379 of 75,233 PDFs carried signatures from more than one Big-4 firm and are reported in the single-firm-PDF per-firm breakdown of the script CSV but pooled into the overall counts here.)
-- **§IV-G.1 (year-by-year stability).** Holds the cosine cutoff fixed at 0.95 and asks whether the share of Firm A below the cutoff is *stable across years*. The information is in the temporal trend, not in the absolute rate; under a noise-only explanation of the left tail, the share should shrink as scan/PDF technology matured.
+
-- **§IV-G.2 (partner-level similarity ranking).** Uses *no threshold at all*: every auditor-year is ranked by mean similarity, and we measure Firm A's share of the top decile against its baseline share. The information is in the concentration ratio, which is invariant to the choice of cutoff.
+**Per-firm document-level breakdown (single-firm PDFs only).**
-- **§IV-G.3 (intra-report agreement).** Applies the calibrated classifier and measures whether the *two co-signing CPAs on the same Firm A report* receive the same classifier label, then compares Firm A's intra-report agreement rate to the other firms'. The information is in the *cross-firm gap*; the absolute agreement rate at any one firm depends on the cutoff, but the gap is robust to moderate cutoff shifts as long as the same cutoff is applied uniformly across firms.
+
-
+| Firm | HC | MC | HSC | UN | LH | total docs |
-Together these three analyses provide threshold-free or threshold-robust evidence that complements the within-sample capture rates of Section IV-E.
+|---|---|---|---|---|---|---|
-
+| Firm A | 27,600 | 1,857 | 7 | 758 | 4 | 30,226 |
-### 1) Year-by-Year Stability of the Firm A Left Tail
+| Firm B | 8,783 | 6,079 | 57 | 2,202 | 6 | 17,127 |
-
+| Firm C | 7,281 | 8,660 | 77 | 3,099 | 5 | 19,122 |
-Table XIII reports the proportion of Firm A signatures with per-signature best-match cosine below 0.95, disaggregated by fiscal year.
+| Firm D | 3,100 | 2,838 | 22 | 2,416 | 3 | 8,379 |
-Under the replication-dominated interpretation (Section III-H), this signature-level left-tail rate reflects within-firm heterogeneity in signing outputs at Firm A.
+
-Consistent with the scope-of-claims framing in Section III-G, we report the rate as a signature-level quantity without disaggregating the underlying mechanism (which may span a minority of hand-signing partners, multi-template replication workflows within the firm, or a combination); partner-level mechanism attribution is not attempted.
+(Source: Script 42; mixed-firm PDFs $n = 379$ excluded from the per-firm rows but included in the overall counts above.)
-Under the alternative hypothesis that the left tail is an artifact of scan or compression noise, the share should shrink as scanning and PDF-compression technology improved over 2013-2023.
+
-
+The five-way **moderate-confidence non-hand-signed** band (cos $> 0.95$ AND $5 < \text{dHash} \leq 15$) retains its prior calibration (supplementary materials); it is **not separately re-characterised by Scripts 38–40**, which checked only the binary high-confidence rule (cos $> 0.95$ AND dHash $\leq 5$). The moderate-band cuts are not re-derived on the Big-4 subset; we report the Table XV per-firm MC proportions (10.76% / 35.88% / 41.44% / 29.33% across Firms A through D) descriptively. The capture-rate calibration evidence for the moderate band is reported in the supplementary materials and not regenerated on the Big-4 subset. We do not claim that the MC-band per-firm ordering above is a separate validation of the §III-K Spearman convergence, since MC occupancy is not a monotone function of the per-CPA less-replication-dominated ranking (e.g., Firm D's MC fraction is lower than Firm B's while Firm D's reverse-anchor score ranks it as less replication-dominated than Firm B).
-<!-- TABLE XIII: Firm A Per-Year Cosine Distribution
+
-| Year | N sigs | mean best-match cosine | % below 0.95 |
+**Table XVII.** Firm × K=3 cluster cross-tabulation, Big-4 sub-corpus.
-|------|--------|-------------|--------------|
+
-| 2013 | 2,167 | 0.9733 | 12.78% |
+| Firm | $n$ | C1 (low-cos / high-dHash) | C2 (central) | C3 (high-cos / low-dHash) | C1 % | C3 % |
-| 2014 | 5,256 | 0.9781 | 8.69% |
+|---|---|---|---|---|---|---|
-| 2015 | 5,484 | 0.9793 | 7.46% |
+| Firm A | 171 | 0 | 30 | 141 | $0.00\%$ | $82.46\%$ |
-| 2016 | 5,739 | 0.9811 | 6.92% |
+| Firm B | 112 | 10 | 102 | 0 | $8.93\%$ | $0.00\%$ |
-| 2017 | 5,796 | 0.9814 | 6.69% |
+| Firm C | 102 | 24 | 77 | 1 | $23.53\%$ | $0.98\%$ |
-| 2018 | 5,986 | 0.9808 | 6.58% |
+| Firm D | 52 | 6 | 45 | 1 | $11.54\%$ | $1.92\%$ |
-| 2019 | 6,122 | 0.9780 | 8.71% |
+
-| 2020 | 6,122 | 0.9770 | 9.46% |
+(Source: Script 35.) The cross-tab is the accountant-level descriptive output of the K=3 mixture (§III-J / §IV-E). It is reported here as a complement to the five-way per-signature classifier (Table XV), not as an operational classifier output. Reading: Firm A's CPAs are concentrated in the C3 (high-cos / low-dHash) component (no Firm A CPAs in C1); Firm C has the highest C1 (low-cos / high-dHash) concentration of the Big-4 (C1 fraction $23.5\%$); Firms B and D sit between A and C on the K=3 hard-label ordering, broadly consistent with the per-firm Spearman ordering of Table X (with the within-Big-4-non-A reverse-anchor disagreement noted there).
-| 2021 | 5,996 | 0.9792 | 8.37% |
+
-| 2022 | 5,918 | 0.9819 | 6.25% |
+**Document-level worst-case aggregation outputs are reported in Table XVI above.**
-| 2023 | 5,862 | 0.9860 | 3.75% |
+
--->
+## K. Full-Dataset Robustness (light scope)
-
+
-The left tail is stable at 6-13% throughout the sample period and shows no pre/post-2020 level shift: the 2013-2019 mean left-tail share is 8.26% and the 2020-2023 mean is 6.96%.
+This section reports the reproducibility cross-check at the full accountant scope ($n = 686$ CPAs, Big-4 plus mid/small firms). The scope of §IV-K is deliberately narrow: we re-run only the K=3 mixture + deployed operational-rule per-CPA less-replication-dominated rate analysis, sufficient to demonstrate that the K=3 + deployed-rule convergence reproduces at the wider scope. The §III-H.1 five-way classifier and the §IV-G LOOO analyses are not re-run at the full scope. The five-way moderate-confidence band retains its prior calibration (supplementary materials; §IV-J).
-The lowest observed share is in 2023 (3.75%), consistent with firm-level electronic signing systems producing more uniform output than earlier manual scanning-and-stamping, not less.
+
-This stability supports the replication-dominated framing: a persistent within-firm heterogeneity component is consistent with a Beta left tail that is stable across production technologies, whereas a noise-only explanation would predict a shrinking share as technology improved.
+**Table XVIII.** K=3 component comparison, Big-4 sub-corpus vs full dataset.
-
+
-### 2) Partner-Level Similarity Ranking
+| K=3 component | Big-4 (n=437) cos / dHash / weight | Full (n=686) cos / dHash / weight | Drift Big-4 → Full |
-
+|---|---|---|---|
-If Firm A applies firm-wide stamping while the other Big-4 firms use stamping only for a subset of partners, Firm A auditor-years should disproportionately occupy the top of the similarity distribution among all auditor-years (across all firms).
+| C1 (low-cos / high-dHash) | 0.9457 / 9.17 / 0.143 | 0.9278 / 11.17 / 0.284 | $\lvert\Delta\rvert$ cos 0.018, dHash 1.99, wt 0.141 |
-We test this prediction directly.
+| C2 (central) | 0.9558 / 6.66 / 0.536 | 0.9535 / 6.99 / 0.512 | $\lvert\Delta\rvert$ cos 0.002, dHash 0.33, wt 0.024 |
-
+| C3 (high-cos / low-dHash) | 0.9826 / 2.41 / 0.321 | 0.9826 / 2.40 / 0.205 | $\lvert\Delta\rvert$ cos 0.000, dHash 0.01, wt 0.117 |
-For each auditor-year (CPA $\times$ fiscal year) with at least 5 signatures we compute the mean best-match cosine similarity across the year's signatures, yielding 4,629 auditor-years across 2013-2023.
+
-Firm A accounts for 1,287 of these (27.8% baseline share).
+(Source: Script 41; full-dataset $\text{BIC}(K{=}3) = -792.31$ vs Big-4 $\text{BIC}(K{=}3) = -1111.93$; BIC values are not directly comparable across different $n$ and are reported only for completeness.)
-Table XIV reports per-firm occupancy of the top $K\%$ of the ranked distribution.
+
-The per-signature best-match cosine underlying each auditor-year mean is taken over the full same-CPA pool (Section III-G) and may match against signatures from other fiscal years, so the auditor-year mean reflects the year's signatures' position within the CPA's full-sample similarity structure rather than purely within-year similarity; a within-year-restricted sensitivity replication is a natural robustness check and is left to future work.
+**Table XIX.** Spearman rank correlation between K=3 P(C1) and deployed operational less-replication-dominated rate, Big-4 sub-corpus vs full dataset.
-
+
-<!-- TABLE XIV: Top-K Similarity Rank Occupancy by Firm (pooled 2013-2023)
+| Scope | $n$ CPAs | Spearman $\rho$ (P(C1) vs deployed less-replication-dominated rate) | $p$-value |
-| Top-K | k in bucket | Firm A | Firm B | Firm C | Firm D | Non-Big-4 | Firm A share |
+|---|---|---|---|
-|-------|-------------|--------|--------|--------|--------|-----------|--------------|
+| Big-4 (primary) | 437 | $+0.9627$ | $< 10^{-248}$ |
-| 10%   | 462         | 443    | 2      | 3      | 0      | 14        | 95.9% |
+| Full dataset | 686 | $+0.9558$ | $< 10^{-300}$ |
-| 25%   | 1,157       | 1,043  | 32     | 23     | 9      | 50        | 90.1% |
+| $\lvert\rho_{\text{full}} - \rho_{\text{Big-4}}\rvert$ | — | $0.0069$ | — |
-| 50%   | 2,314       | 1,220  | 473    | 273    | 102    | 246       | 52.7% |
+
--->
+(Source: Script 41.)
-
+
-Firm A occupies 95.9% of the top 10% and 90.1% of the top 25% of auditor-years by similarity, against its baseline share of 27.8%---a concentration ratio of 3.5$\times$ at the top decile and 3.2$\times$ at the top quartile.
+**Reading.** The K=3 component ordering and the strong Spearman convergence between K=3 P(C1) and the deployed box-rule less-replication-dominated rate are preserved at the full scope. Component centres shift modestly: C3 (high-cos / low-dHash) is essentially unchanged in centre but loses weight $0.117$ as the full population includes more non-templated CPAs (mid/small firms); C1 (low-cos / high-dHash) gains weight $0.141$ and shifts to lower cosine and higher dHash (centre $(0.928, 11.17)$ vs Big-4 $(0.946, 9.17)$) as the broader population includes mid/small-firm CPAs landing toward the low-cos / high-dHash region that the Big-4-primary scope deliberately excludes. We read this as evidence that the Big-4-primary K=3 + deployed-rule convergence is not a Big-4-specific artefact; we do **not** read it as an endorsement of using full-dataset K=3 component centres or operational thresholds in place of the Big-4-primary analysis. Mid/small-firm composition shifts the component centres meaningfully and the primary methodology is restricted to Big-4 by design (§III-G item 4).
-Year-by-year (Table XV), the top-10% Firm A share ranges from 88.4% (2020) to 100% (2013, 2014, 2017, 2018, 2019), showing that the concentration is stable across the sample period.
+
-
+## L. Ablation Study: Feature Backbone Comparison
-<!-- TABLE XV: Firm A Share of Top-10% Similarity by Year
+
-| Year | N auditor-years | Top-10% k | Firm A in top-10% | Firm A share | Firm A baseline |
+To support the choice of ResNet-50 as the feature extraction backbone, we conducted an ablation study comparing three pre-trained architectures: ResNet-50 (2048-dim), VGG-16 (4096-dim), and EfficientNet-B0 (1280-dim).
-|------|-----------------|-----------|-------------------|--------------|-----------------|
-| 2013 | 324 | 32 | 32 | 100.0% | 32.4% |
-| 2014 | 399 | 39 | 39 | 100.0% | 27.8% |
-| 2015 | 394 | 39 | 38 | 97.4% | 27.7% |
-| 2016 | 413 | 41 | 39 | 95.1% | 26.2% |
-| 2017 | 415 | 41 | 41 | 100.0% | 27.2% |
-| 2018 | 434 | 43 | 43 | 100.0% | 26.5% |
-| 2019 | 429 | 42 | 42 | 100.0% | 27.0% |
-| 2020 | 430 | 43 | 38 | 88.4% | 27.7% |
-| 2021 | 450 | 45 | 44 | 97.8% | 28.7% |
-| 2022 | 467 | 46 | 43 | 93.5% | 28.3% |
-| 2023 | 474 | 47 | 46 | 97.9% | 27.4% |
--->
-
-This over-representation is consistent with firm-wide non-hand-signing practice at Firm A and is not derived from any threshold we subsequently calibrate.
-It therefore constitutes genuine cross-firm evidence for Firm A's benchmark status.
-
-### 3) Intra-Report Consistency
-
-Taiwanese statutory audit reports are co-signed by two engagement partners (a primary and a secondary signer).
-Under firm-wide stamping practice at a given firm, both signers on the same report should receive the same signature-level classification.
-Disagreement between the two signers on a report is informative about whether the stamping practice is firm-wide or partner-specific.
-
-For each report with exactly two signatures and complete per-signature data (84,354 reports total: 83,970 single-firm reports, in which both signers are at the same firm, and 384 mixed-firm reports, in which the two signers are at different firms), we classify each signature using the dual-descriptor rules of Section III-K and record whether the two classifications agree.
-Table XVI reports per-firm intra-report agreement for the 83,970 single-firm reports only (firm-assignment defined by the common firm identity of both signers); the 384 mixed-firm reports (0.46% of the 2-signature corpus) are excluded from the intra-report analysis because firm-level agreement is not well defined when the two signers are at different firms.
-
-<!-- TABLE XVI: Intra-Report Classification Agreement by Firm
-| Firm | Total 2-signer reports | Both non-hand-signed | Both uncertain | Both style | Both hand-signed | Mixed | Agreement rate |
-|------|-----------------------|----------------------|----------------|------------|------------------|-------|----------------|
-| Firm A  | 30,222 | 26,435 | 734 | 0 | 4 | 3,049 | **89.91%** |
-| Firm B  | 17,121 | 9,260  | 2,159| 5 | 6 | 5,691 | 66.76% |
-| Firm C  | 19,112 | 8,983  | 3,035| 3 | 5 | 7,086 | 62.92% |
-| Firm D  | 8,375  | 3,028  | 2,376| 0 | 3 | 2,968 | 64.56% |
-| Non-Big-4 | 9,140  | 1,671  | 3,945| 18| 27| 3,479 | 61.94% |
-
-A report is "in agreement" if both signature labels fall in the same coarse bucket
-(non-hand-signed = high+moderate; uncertain; style consistency; or likely hand-signed).
--->
-
-Firm A achieves 89.9% intra-report agreement, with 87.5% of Firm A reports having *both* signers classified as non-hand-signed and only 4 reports (0.01%) having both classified as likely hand-signed.
-The other Big-4 firms (B, C, D) and non-Big-4 firms cluster at 62-67% agreement, a 23-28 percentage-point gap.
-This 23-28 percentage-point gap in intra-report agreement between Firm A and the other firms is consistent with firm-wide (rather than partner-specific) non-hand-signing practice; we do not claim a sharp discontinuity in the formal sense, since classifier calibration, firm-specific document-production pipelines, and signer-mix differences could each contribute to gap magnitude.
-
-We note that this test uses the calibrated classifier of Section III-K rather than a threshold-free statistic; the substantive evidence lies in the *cross-firm gap* between Firm A and the other firms rather than in the absolute agreement rate at any single firm, and that gap is robust to moderate shifts in the absolute cutoff so long as the cutoff is applied uniformly across firms.
-
-## H. Classification Results
-
-Table XVII presents the final classification results under the dual-descriptor framework with Firm A-calibrated thresholds for 84,386 documents.
-The document count (84,386) differs from the 85,042 documents with any YOLO detection (Table III) because 656 documents carry only a single detected signature, for which no same-CPA pairwise comparison and therefore no best-match cosine / min dHash statistic is available; those documents are excluded from the classification reported here.
-We emphasize that the document-level proportions below reflect the *worst-case aggregation rule* of Section III-K: a report carrying one stamped signature and one hand-signed signature is labeled with the most-replication-consistent of the two signature-level verdicts.
-Document-level rates therefore represent the share of reports in which *at least one* signature is non-hand-signed rather than the share in which *both* are; the intra-report agreement analysis of Section IV-G.3 (Table XVI) reports how frequently the two co-signers share the same signature-level label within each firm, so that readers can judge what fraction of the non-hand-signed document-level share corresponds to fully non-hand-signed reports versus mixed reports.
-
-<!-- TABLE XVII: Document-Level Classification (Dual-Descriptor: Cosine + dHash)
-| Verdict | N (PDFs) | % | Firm A | Firm A % |
-|---------|----------|---|--------|----------|
-| High-confidence non-hand-signed | 29,529 | 35.0% | 22,970 | 76.0% |
-| Moderate-confidence non-hand-signed | 36,994 | 43.8% | 6,311 | 20.9% |
-| High style consistency | 5,133 | 6.1% | 183 | 0.6% |
-| Uncertain | 12,683 | 15.0% | 758 | 2.5% |
-| Likely hand-signed | 47 | 0.1% | 4 | 0.0% |
-
-Per the worst-case aggregation rule of Section III-K, a document with two signatures inherits the most-replication-consistent of the two signature-level labels.
--->
-
-Within the 71,656 documents exceeding cosine $0.95$, the dHash dimension stratifies them into three distinct populations:
-29,529 (41.2%) show converging structural evidence of non-hand-signing (dHash $\leq 5$);
-36,994 (51.7%) show partial structural similarity (dHash in $[6, 15]$) consistent with replication degraded by scan variations;
-and 5,133 (7.2%) show no structural corroboration (dHash $> 15$), suggesting high signing consistency rather than image reproduction.
-A cosine-only classifier would treat all 71,656 identically; the dual-descriptor framework separates them into populations with fundamentally different interpretations.
-
-### 1) Firm A Capture Profile (Consistency Check)
-
-96.9% of Firm A's documents fall into the high- or moderate-confidence non-hand-signed categories, 0.6% into high-style-consistency, and 2.5% into uncertain.
-This pattern is consistent with the replication-dominated framing: the large majority is captured by non-hand-signed rules, while the small residual is consistent with the within-firm heterogeneity implied by the dip-test-confirmed unimodal-long-tail shape of Firm A's per-signature cosine distribution (Section IV-D.1) and the 7.5% signature-level left tail (Section III-H).
-The near-zero "likely hand-signed" rate (4 of 30,226 Firm A documents, 0.013%; the 30,226 count here is documents with at least one Firm A signer under the 84,386-document classification cohort, which differs from the 30,222 single-firm two-signer subset in Table XVI by 4 reports) indicates that the within-firm heterogeneity implied by the 7.5% signature-level left tail (Section IV-D) does not project into the lowest-cosine document-level category under the dual-descriptor rules; it is absorbed instead into the uncertain or high-style-consistency categories at this threshold set.
-We note that because the non-hand-signed thresholds are themselves calibrated to Firm A's empirical percentiles (Section III-H), these rates are an internal consistency check rather than an external validation; the held-out Firm A validation of Section IV-F.2 is the corresponding external check.
-
-### 2) Cross-Firm Comparison of Dual-Descriptor Convergence
-
-Among the 65,514 non-Firm-A signatures with per-signature best-match cosine $> 0.95$, 42.12% have $\text{dHash}_\text{indep} \leq 5$, compared to 88.32% of the 55,922 Firm A signatures meeting the same cosine condition---a $\sim 2.1\times$ difference that the structural-verification layer makes visible.
-The Firm A denominator (55,922) matches Table IX exactly: both Table IX and the cross-firm decomposition define Firm A membership via the CPA registry (`accountants.firm`), and the cross-firm analysis additionally requires a non-null independent-min dHash record, which all 55,922 Firm A cosine-eligible signatures have in the current database.
-This cross-firm gap is consistent with firm-wide non-hand-signing practice at Firm A versus partner-specific or per-engagement replication at other firms; it complements the partner-level ranking (Section IV-G.2) and intra-report consistency (Section IV-G.3) findings.
-Counts and percentages are reproduced by `signature_analysis/28_byte_identity_decomposition.py` and reported in `reports/byte_identity_decomp/byte_identity_decomposition.json` (see Appendix B for the table-to-script provenance map).
-
-## I. Ablation Study: Feature Backbone Comparison
-
-To validate the choice of ResNet-50 as the feature extraction backbone, we conducted an ablation study comparing three pre-trained architectures: ResNet-50 (2048-dim), VGG-16 (4096-dim), and EfficientNet-B0 (1280-dim).
 All models used ImageNet pre-trained weights without fine-tuning, with identical preprocessing and L2 normalization.
-Table XVIII presents the comparison.
+The comparison summary is reported in the supplementary materials (backbone-ablation table; not the same table as Table XIX in this section, which reports Big-4 vs full-dataset Spearman drift in §IV-K).
 
-<!-- TABLE XVIII: Backbone Comparison
+<!-- BACKBONE ABLATION TABLE (rendered in supplementary materials):
 | Metric | ResNet-50 | VGG-16 | EfficientNet-B0 |
 |--------|-----------|--------|-----------------|
 | Feature dim | 2048 | 4096 | 1280 |
@@ -402,11 +319,114 @@ single closest match from the same CPA.
 -->
 
 EfficientNet-B0 achieves the highest Cohen's $d$ (0.707), indicating the greatest statistical separation between intra-class and inter-class distributions.
-However, it also exhibits the widest distributional spread (intra std $= 0.123$ vs. ResNet-50's $0.098$), resulting in lower per-sample classification confidence.
+However, it also exhibits the widest distributional spread (intra std $= 0.123$ vs. ResNet-50's $0.098$), i.e., a wider descriptor dispersion per signature.
 VGG-16 performs worst on all key metrics despite having the highest feature dimensionality (4096), suggesting that additional dimensions do not contribute discriminative information for this task.
 
 ResNet-50 provides the best overall balance:
 (1) Cohen's $d$ of 0.669 is competitive with EfficientNet-B0's 0.707;
-(2) its tighter distributions yield more reliable individual classifications;
+(2) its tighter distributions yield more stable descriptor behaviour at the per-signature level;
-(3) the highest Firm A all-pairs 1st percentile (0.543) indicates that known-replication signatures are least likely to produce low-similarity outlier pairs under this backbone; and
+(3) the highest Firm A all-pairs 1st percentile (0.543) indicates that Firm A replication-dominated signatures are least likely to produce low-similarity outlier pairs under this backbone; and
 (4) its 2048-dimensional features offer a practical compromise between discriminative capacity and computational/storage efficiency for processing 182K+ signatures.
+
+## M. Anchor-Based ICCR Calibration Results
+
+This section consolidates the empirical results that support the §III-L anchor-based threshold calibration framework.
+
+### M.1 Composition decomposition (Scripts 39b–39e)
+
+**Table XX.** Within-firm and between-firm decomposition of the Big-4 accountant-level dip-test rejection.
+
+| Diagnostic | Scope | Statistic | Implication |
+|---|---|---|---|
+| Within-firm signature-level cosine dip | Big-4 (4 firms) | $p_{\text{cos}} \in \{0.176, 0.991, 0.551, 0.976\}$ | 0/4 firms reject; cosine within-firm unimodal |
+| Within-firm signature-level cosine dip | non-Big-4 (10 firms $\geq 500$ sigs) | $p_{\text{cos}} \in [0.59, 0.99]$ | 0/10 firms reject; cosine within-firm unimodal |
+| Within-firm jittered-dHash dip (5 seeds, median) | Big-4 (4 firms) | $p_{\text{med}} \in \{0.999, 0.996, 0.999, 0.9995\}$ | 0/4 firms reject after integer-jitter; raw rejection was integer-tie artefact |
+| Big-4 pooled dHash: 2×2 factorial | firm-centred + jittered (5 seeds) | $p_{\text{med}} = 0.35$, 0/5 seeds reject | combined corrections eliminate rejection; multimodality is composition + integer artefact |
+| Integer-histogram valley near $\text{dHash} \approx 5$ | within each Big-4 firm | none (0/4 firms) | no within-firm dHash antimode at the deployed HC cutoff |
+
+(Source: Scripts 39b, 39c, 39d, 39e; bootstrap $n_{\text{boot}} = 2000$; jitter $\sim \mathrm{U}[-0.5, +0.5]$.)
+
+### M.2 Anchor-based inter-CPA pair-level ICCR (Script 40b)
+
+**Table XXI.** Big-4 inter-CPA per-comparison ICCR sweep, $n = 5 \times 10^5$ pairs (Big-4 scope).
+
+| Threshold | Per-comparison ICCR | 95% Wilson CI |
+|---|---|---|
+| cos $> 0.945$ (alternative operating point from supplementary calibration evidence) | $0.00081$ | $[0.00073, 0.00089]$ |
+| cos $> 0.95$ (deployed operating point) | $0.00060$ | $[0.00053, 0.00067]$ |
+| cos $> 0.97$ | $0.00024$ | $[0.00020, 0.00029]$ |
+| cos $> 0.98$ | $0.00009$ | $[0.00007, 0.00012]$ |
+| dHash $\leq 5$ (deployed operating point) | $0.00129$ | $[0.00120, 0.00140]$ |
+| dHash $\leq 4$ | $0.00050$ | $[0.00044, 0.00057]$ |
+| dHash $\leq 3$ | $0.00019$ | $[0.00015, 0.00023]$ |
+| Joint: cos $> 0.95$ AND dHash $\leq 5$ (any-pair semantics) | $0.00014$ | $[0.00011, 0.00018]$ |
+| Joint: cos $> 0.95$ AND dHash $\leq 4$ (any-pair) | $0.00011$ | $[0.00008, 0.00014]$ |
+
+Conditional ICCR(dHash $\leq 5$ | cos $> 0.95$) $= 0.234$ (Wilson 95% $[0.190, 0.285]$; $70$ of $299$ pairs).
+
+The cos $> 0.95$ row is consistent with the corpus-wide spike of §IV-I (per-comparison rate $0.0005$). The dHash row and joint row are reported here for the first time on this corpus.
+
+### M.3 Pool-normalised per-signature ICCR (Script 43)
+
+**Table XXII.** Pool-normalised per-signature ICCR under the deployed any-pair HC rule (cos $> 0.95$ AND dHash $\leq 5$); $n_{\text{sig}} = 150{,}453$ (vector-complete Big-4); CPA-block bootstrap $n_{\text{boot}} = 1000$.
+
+| Scope | Per-signature ICCR | Wilson 95% CI | CPA-bootstrap 95% CI |
+|---|---|---|---|
+| Big-4 pooled (any-pair, deployed) | $0.1102$ | $[0.1086, 0.1118]$ | $[0.0908, 0.1330]$ |
+| Big-4 pooled (same-pair, stricter alternative) | $0.0827$ | $[0.0813, 0.0841]$ | $[0.0668, 0.1021]$ |
+| Firm A (any-pair) | $0.2594$ | — | — |
+| Firm B (any-pair) | $0.0147$ | — | — |
+| Firm C (any-pair) | $0.0053$ | — | — |
+| Firm D (any-pair) | $0.0110$ | — | — |
+| Pool-size decile 1 (smallest pools) any-pair | $0.0249$ | — | — |
+| Pool-size decile 10 (largest pools) any-pair | $0.1905$ | — | — |
+
+Decile trend is broadly monotone in pool size with two minor reversals (decile 5 and decile 9 dip below their predecessors). Stricter operating point cos $> 0.95$ AND dHash $\leq 3$ (same-pair) gives per-signature ICCR $0.0449$.
+
+### M.4 Document-level ICCR under three alarm definitions (Script 45)
+
+**Table XXIII.** Document-level inter-CPA ICCR by alarm definition; $n_{\text{docs}} = 75{,}233$.
+
+| Alarm definition | Alarm set | Document-level ICCR | Wilson 95% CI |
+|---|---|---|---|
+| D1 | HC only | $0.1797$ | $[0.1770, 0.1825]$ |
+| D2 (operational) | HC + MC | $0.3375$ | $[0.3342, 0.3409]$ |
+| D3 | HC + MC + HSC | $0.3384$ | $[0.3351, 0.3418]$ |
+
+Per-firm D2 document-level ICCR: Firm A $0.6201$ ($n = 30{,}226$); Firm B $0.1600$ ($n = 17{,}127$); Firm C $0.1635$ ($n = 19{,}501$); Firm D $0.0863$ ($n = 8{,}379$). The Firm C denominator $n = 19{,}501$ exceeds Table XVI's single-firm Firm C count of $19{,}122$ by exactly the $379$ mixed-firm PDFs: all $379$ are $1{:}1$ Firm C / Firm D mixed-firm documents, and Script 45's mode-of-firms implementation (`np.argmax` over `np.unique`'s alphabetically-sorted firm counts) returns the first-sorted firm on ties, which assigns these tied documents to Firm C rather than to Firm D. The four per-firm denominators here therefore sum to the full $75{,}233$, whereas Table XVI's per-firm rows sum to $74{,}854 = 75{,}233 - 379$.
+
+### M.5 Firm heterogeneity logistic regression and cross-firm hit matrix (Script 44)
+
+**Table XXIV.** Logistic regression of per-signature any-pair HC hit indicator on firm dummies and centred log pool size (Firm A reference).
+
+| Term | Odds ratio (vs Firm A) | Direction |
+|---|---|---|
+| Firm B | $0.053$ | $\sim 19\times$ lower odds than Firm A |
+| Firm C | $0.010$ | $\sim 100\times$ lower odds than Firm A |
+| Firm D | $0.027$ | $\sim 37\times$ lower odds than Firm A |
+| log(pool size, centred) | $4.01$ | $\sim 4\times$ higher odds per log unit pool size |
+
+Per-decile per-firm rates (Table not duplicated here; Script 44 decile table available in the supplementary report): within every pool-size decile, Firms B/C/D show rates of $0.0006$–$0.0358$ while Firm A ranges $0.0541$–$0.5958$. The firm gap survives within matched pool sizes.
+
+**Table XXV.** Cross-firm hit matrix among Big-4 source signatures with any-pair HC hit; max-cosine partner firm (counts).
+
+| Source firm | Firm A cand. | Firm B | Firm C | Firm D | non-Big-4 | n hits |
+|---|---|---|---|---|---|---|
+| Firm A | $14{,}447$ | $95$ | $44$ | $19$ | $17$ | $14{,}622$ |
+| Firm B | $92$ | $371$ | $8$ | $4$ | $9$ | $484$ |
+| Firm C | $16$ | $7$ | $149$ | $5$ | $1$ | $178$ |
+| Firm D | $22$ | $2$ | $6$ | $106$ | $1$ | $137$ |
+
+Same-pair joint hits (single candidate satisfying both cos $> 0.95$ AND dHash $\leq 5$) are within-firm at rates $99.96\%$ / $97.7\%$ / $98.2\%$ / $97.0\%$ for Firms A/B/C/D respectively.
+
+### M.6 Alert-rate sensitivity around deployed HC threshold (Script 46)
+
+**Table XXVI.** Local-gradient / median-gradient ratio at deployed thresholds (descriptive plateau diagnostic).
+
+| Threshold | Local / median gradient ratio | Interpretation |
+|---|---|---|
+| cos $= 0.95$ (HC) | $\approx 25\times$ | locally sensitive (not plateau-stable) |
+| dHash $= 5$ (HC) | $\approx 3.8\times$ | locally sensitive (not plateau-stable) |
+| dHash $= 15$ (MC/HSC boundary) | $\approx 0.08$ | plateau-like (saturating tail) |
+
+Big-4 observed deployed alert rate on actual same-CPA pools: per-signature HC $= 0.4958$; per-document HC $= 0.6228$. The deployed-rate excess over the inter-CPA proxy is $0.3856$ ($38.6$ pp) per-signature and $0.4431$ ($44.3$ pp) per-document; this excess is interpreted as a same-CPA repeatability signal under the §III-M caveats, not as a presumed true-positive rate.

paper/reference_verification_v3.md

@@ -1,14 +1,17 @@
 # Reference Verification — Paper A v3 (41 refs)
 
-Date: 2026-04-27
+Date: 2026-04-27 (initial audit); v3.18 reference list updated to incorporate every fix recorded below.
+
 Method: WebSearch + WebFetch verification of each citation against authoritative sources (publisher pages, DOIs, arXiv, IEEE Xplore, Project Euclid, etc.).
 
-## Summary
+## Summary (audit history)
-- Verified correct: 35/41
+- Verified correct on first audit: 35/41
-- Minor discrepancies (typos, page numbers, year on early-access vs. issue): 5/41
+- Minor discrepancies (typos, page numbers, year on early-access vs. issue): 5/41 — all fixed in v3.18
-- MAJOR PROBLEMS (does not exist, wrong author, wrong title, wrong venue): 1/41
+- MAJOR PROBLEMS (wrong author): 1/41 — `[5]` Hadjadj et al. → Kao and Wen, fixed in v3.18
 
-The single major problem is **[5]**, where the paper at the cited venue/article number is real, but the cited authors ("Hadjadj et al.") are wrong — the actual authors are Kao and Wen. None of the statistical-method refs [37]–[41] flagged by the partner are fabricated; all five are bibliographically correct.
+The current `paper_a_references_v3.md` reflects every correction listed below. The detailed findings are retained as an audit trail; the live reference list no longer carries any of the recorded errors.
+
+The single major problem at the time of the audit was **[5]**, where the paper at the cited venue/article number is real, but the cited authors ("Hadjadj et al.") were wrong — the actual authors are Kao and Wen. None of the statistical-method refs [37]–[41] flagged by the partner are fabricated; all five are bibliographically correct.
 
 ## Detailed findings
 

paper/review_handoff_abstract_intro_20260515.md

@@ -0,0 +1,361 @@
+# Review Handoff: Abstract and Introduction
+
+Date: 2026-05-15
+Target manuscript: `paper/paper_a_v4_combined.md`
+Scope reviewed: Abstract and Introduction only
+
+## Overall Assessment
+
+The Abstract and Introduction are substantively strong and defensible. The current argument is clear:
+
+- Regulations require CPA attestation, but digitized PDF workflows make stored-signature reuse operationally easy.
+- The problem is not signature forgery; identity is not in dispute. The target is detecting possible image-level reproduction by the legitimate signer or firm workflow.
+- The paper avoids claiming validated forensic detection and instead frames the system as an anchor-calibrated screening framework under unsupervised constraints.
+- The strongest methodological move is replacing unsupported distributional "natural threshold" logic with anchor-based inter-CPA coincidence-rate (ICCR) calibration.
+
+Recommended disposition: Minor Revision for prose and narrative complexity, not for core empirical weakness.
+
+## Main Reviewer Concern
+
+The Introduction currently explains the methodology shift too explicitly as a research-process or version-history pivot. This is useful internally, but in the submitted paper it may increase complexity and invite reviewers to focus on why earlier versions used a different framing.
+
+The final manuscript should explain the final methodological choice, not the internal research journey.
+
+Keep:
+
+- The descriptor distribution does not support a stable within-population bimodal antimode.
+- Apparent multimodality is explained by firm composition and integer mass-point artefacts.
+- Mixture fits are descriptive, not threshold-generating.
+- Operational rules are characterized using anchor-based ICCR at multiple units.
+
+Reduce or remove:
+
+- "Earlier work in this lineage..."
+- "v4.0 contribution..."
+- "overturns this reading..."
+- "inherited Paper A v3.x..."
+- Internal script-heavy provenance in the Introduction.
+
+Detailed provenance belongs in Methodology, Results, Appendix, or reproducibility notes, not in the opening narrative.
+
+## Suggested Rewrite Direction for Introduction Pivot Paragraph
+
+Current issue location: around `paper/paper_a_v4_combined.md`, Introduction paragraph beginning with "The methodological reframing relative to earlier versions..."
+
+Recommended replacement direction:
+
+```text
+A key empirical finding is that the descriptor distributions do not support a within-population natural threshold. The apparent multimodality in the Big-4 accountant-level distribution is explained by between-firm location shifts and integer mass-point artefacts on the dHash axis. After firm-mean centring and integer-tie jitter, the pooled dHash dip-test rejection disappears. Within-firm diagnostics likewise do not reveal a stable bimodal antimode. We therefore treat mixture fits as descriptive summaries of firm-compositional structure rather than threshold-generating mechanisms, and calibrate the deployed operating rules using inter-CPA coincidence-rate anchors.
+```
+
+This preserves the methodological defense while removing the internal v3-to-v4 story.
+
+## Abstract-Specific Comments
+
+The Abstract is strong but very dense. It is currently optimized for technical reviewers rather than broad readability. That may be acceptable for IEEE Access, but the first sentence has a small grammar/style issue.
+
+Suggested edit:
+
+```text
+Regulations require Certified Public Accountants (CPAs) to attest each audit report with a signature, but digitization makes it feasible to reuse a stored signature image across reports -- through administrative stamping or firm-level electronic signing -- thereby undermining individualized attestation.
+```
+
+Reason:
+
+- Current wording: "digitization makes reusing ... undermining ..." is grammatically awkward.
+- The suggested version makes the causal relation explicit.
+
+No need to remove the final limitation sentence. The sentence "not as a validated forensic detector; no calibrated error rates..." is important and should remain.
+
+## Introduction-Specific Comments
+
+### 1. Keep the legal framing but avoid legal overclaiming
+
+The sentence saying non-hand-signed workflows "may fall within the literal statutory requirement" is acceptable because it is cautious. Do not strengthen it into a legal conclusion.
+
+Preferred style:
+
+- "may fall within"
+- "raises substantive concerns"
+- "may not represent meaningful individual attestation"
+
+Avoid:
+
+- "violates"
+- "illegal"
+- "non-compliant"
+- "fraudulent"
+
+### 2. Preserve the forgery distinction
+
+The distinction between non-hand-signing detection and signature forgery detection is one of the strongest conceptual contributions. Keep it prominent.
+
+Key idea to preserve:
+
+- Forgery detection asks whether the signer is genuine.
+- This paper asks whether the signing act was repeated for each document or a stored image was reused.
+
+### 3. Reduce script/provenance detail in the Introduction
+
+Current paragraph references scripts such as Script 39c and Script 39d. This makes the Introduction read like an internal review memo.
+
+Recommendation:
+
+- Remove or simplify script references from Introduction.
+- Keep exact script provenance in Methodology, Results, Appendix B, or supplementary material.
+
+Specific risk:
+
+- The current parenthetical "10 firms tested in Script 39c" is imprecise for jittered-dHash. Script 39c raw dHash tests reject unimodality; the non-Big-4 jittered-dHash no-rejection statement depends on a codex-verified read-only spike on the same substrate.
+
+Safer Introduction wording:
+
+```text
+Within-firm diagnostics likewise fail to reveal stable bimodal structure after accounting for integer ties, including in eligible mid/small-firm checks.
+```
+
+If provenance must remain:
+
+```text
+Within-firm signature-level cosine checks fail to reject in eligible firms, and corresponding jittered-dHash checks fail to reject in Big-4 firms and in a read-only spike on the same mid/small-firm substrate.
+```
+
+### 4. Avoid presenting the Introduction as a Results section
+
+The Introduction currently contains many detailed numbers. Some are necessary because the paper is methodological, but the v4 pivot paragraphs are numerically heavy.
+
+Keep headline numbers:
+
+- Dataset size: 90,282 reports, 182,328 signatures, 758 CPAs.
+- Big-4 scope: 437 CPAs, 150,442 signatures.
+- Key ICCR levels: per-comparison, per-signature, per-document.
+- Firm heterogeneity: Firm A 0.62 vs Firms B/C/D 0.09-0.16.
+
+Consider moving or reducing:
+
+- Full script-specific details.
+- Too many parenthetical rule semantics in the Introduction.
+- Repeated mentions of inherited/v3/v4 framing.
+
+## Recommended Minimum Patch List
+
+1. Fix Abstract first sentence grammar:
+
+```text
+digitization makes it feasible to reuse...
+```
+
+2. Rewrite the Introduction paragraph that begins with "The methodological reframing relative to earlier versions..." so it describes the final methodological rationale rather than v3-to-v4 revision history.
+
+3. Remove or narrow `Script 39c` provenance in the Introduction because the raw vs jittered dHash distinction is subtle and currently risky.
+
+4. Replace internal-version language across the Introduction:
+
+- Replace "v4.0 adopts..." with "We adopt..."
+- Replace "Earlier work in this lineage..." with "A distributional-threshold approach would be inappropriate here because..."
+- Replace "inherited Paper A v3.x five-way box rule" with "the deployed five-way box rule" unless historical provenance is essential.
+
+5. Preserve limitation language:
+
+- The paper should continue to say it is not a validated forensic detector.
+- The paper should continue to say calibrated error rates cannot be reported without signature-level ground truth.
+
+## Reviewer Bottom Line
+
+The paper should not hide that the distributional threshold path failed; that is actually a methodological strength. But it should present this as a final empirical finding and design rationale, not as a visible research-history correction.
+
+Recommended framing:
+
+```text
+Because the observed distribution does not provide a defensible natural threshold, we use ICCR calibration to characterize the deployed operating rules under explicit unsupervised assumptions.
+```
+
+This is cleaner, less complex, and more reviewer-facing than the current v3-to-v4 narrative.
+
+## Additional Framing Issue: Are We Giving Thresholds or Not?
+
+A likely reviewer confusion point is whether the paper provides a concrete classifier threshold or merely explains why no defensible threshold can be derived.
+
+The intended answer should be explicit:
+
+- The paper does provide a concrete, reproducible operational classifier.
+- The paper does not claim that this classifier is ground-truth-optimal.
+- The paper does not claim that the operating thresholds are natural antimodes in the descriptor distribution.
+- The paper's calibration contribution is to characterize the deployed rule's inter-CPA coincidence behavior under unsupervised assumptions.
+
+Recommended high-level framing:
+
+```text
+We use a fixed, pre-specified five-way operating rule. The present calibration does not derive an optimal threshold; instead, it quantifies the rule's inter-CPA coincidence behavior at per-comparison, per-signature, and per-document units under explicit unsupervised assumptions.
+```
+
+Chinese interpretation:
+
+```text
+我們有一組明確、可重現的五分類操作規則；本文不是宣稱這組門檻是最佳門檻或自然分界點，而是在沒有 signature-level ground truth 的情況下，用 ICCR 量化這組規則的 specificity-proxy 行為。
+```
+
+## Concrete Threshold Language to Make Visible
+
+The manuscript should not bury the actual operating thresholds. Somewhere early in Methodology, and preferably summarized in Introduction, make the rule explicit:
+
+```text
+High-confidence non-hand-signed: cosine > 0.95 AND dHash <= 5.
+Moderate-confidence non-hand-signed: cosine > 0.95 AND 5 < dHash <= 15.
+Other outcomes follow the fixed five-way box rule.
+```
+
+If space allows, add a compact sentence:
+
+```text
+Thus, the system has explicit decision rules; what remains uncalibrated in the absence of signature-level labels is their true false-positive and false-negative error rate.
+```
+
+This directly answers the reviewer question: "Do the authors actually have a classifier?"
+
+## Rewrite Style Recommendation
+
+Avoid language that sounds like the authors are unable to provide thresholds:
+
+- Avoid: "No threshold can be derived."
+- Avoid: "The distribution does not support classification."
+- Avoid: "We cannot determine a threshold."
+
+Use language that distinguishes operational thresholds from statistically natural or supervised-optimal thresholds:
+
+- Prefer: "The deployed thresholds are operational rules rather than natural antimodes."
+- Prefer: "We characterize these rules with ICCR rather than claiming supervised error rates."
+- Prefer: "The absence of a distributional antimode motivates anchor-based calibration, not threshold-free analysis."
+- Prefer: "The system is a concrete screening classifier with explicit unsupervised calibration limits."
+
+## Reviewer-Facing Answer to the Threshold Question
+
+If the manuscript needs one sentence that resolves the ambiguity, use:
+
+```text
+The system therefore uses explicit operating thresholds, but the evidentiary claim attached to those thresholds is limited: they define a reproducible screening rule whose coincidence behavior can be estimated under inter-CPA anchors, not a validated forensic decision boundary with calibrated error rates.
+```
+
+This should be the guiding style for Abstract, Introduction, and the start of Methodology.
+
+## Readability Risk: Too Many Diagnostics Can Look Like Methodological Overbuilding
+
+The manuscript's multi-method statistical design increases rigor, but it also creates a readability risk. In the current form, some sections may feel like a defensive accumulation of diagnostics rather than a clean research design.
+
+Reviewer risk:
+
+- The reader may ask: "Are the authors using many methods because the core classifier is unclear?"
+- The reader may miss the simple main claim because the paper introduces too many caveats and validation tools early.
+- The paper may look like "we used many methods, therefore credible" instead of "each method answers one necessary question."
+
+Recommended main-thread sentence:
+
+```text
+We deploy a fixed five-way screening rule and characterize its unsupervised reliability limits using ICCR, after showing that the descriptor distribution does not support a natural threshold.
+```
+
+Chinese interpretation:
+
+```text
+我們有明確五分類篩檢規則；先證明不能用自然分布切點來當門檻，再用 ICCR 描述這組規則在無標註資料中的可靠性邊界。
+```
+
+All methods and diagnostics should serve this main thread.
+
+## Core vs Supporting Diagnostics
+
+Treat the following as core and keep them prominent:
+
+- End-to-end pipeline: VLM -> YOLO -> ResNet -> cosine/dHash.
+- Explicit five-way operating rule.
+- Composition decomposition showing why the descriptor distribution does not yield a natural threshold.
+- ICCR calibration at three units: per-comparison, per-signature, per-document.
+- Firm heterogeneity and within-firm collision concentration.
+- Ground-truth limitation and no true error-rate claim.
+
+Treat the following as supporting diagnostics and avoid letting them dominate the main narrative:
+
+- K=2 / K=3 mixture fits.
+- Three-score Spearman convergence.
+- Leave-one-firm-out reproducibility.
+- BD/McCrary sensitivity.
+- Ten-tool validation table.
+- Pixel-identity positive anchor, especially because it is close to tautological for the high-confidence rule.
+
+These supporting diagnostics can stay, but they should be framed as robustness checks, assumption checks, or supplementary evidence, not as independent central contributions.
+
+## Suggested Manuscript Structure for Clarity
+
+Recommended structure for the Methodology / Results narrative:
+
+1. Core Method
+
+Describe the pipeline, descriptor construction, and five-way rule.
+
+2. Why the Threshold Is Operational Rather Than Natural
+
+Use the composition decomposition only. Avoid over-explaining K=3, BD/McCrary, or historical mixture logic here.
+
+3. How the Rule Is Calibrated Without Ground Truth
+
+Explain ICCR and the three reporting units: per-comparison, per-signature, per-document.
+
+4. What the Calibration Reveals
+
+Report firm heterogeneity and within-firm collision concentration.
+
+5. Supporting Diagnostics
+
+Place K=3, Spearman convergence, LOOO, BD/McCrary, and pixel-identity checks here as supporting evidence.
+
+## Rewrite Style for Multi-Method Sections
+
+Avoid:
+
+```text
+We apply a multi-tool validation framework consisting of ten diagnostics...
+```
+
+This can sound like methodological stacking.
+
+Prefer:
+
+```text
+Each supporting diagnostic addresses a specific failure mode: composition artefacts, inter-CPA coincidence, pool-size effects, firm heterogeneity, or positive-anchor capture.
+```
+
+Avoid:
+
+```text
+The conjunction of ten tools constitutes validation...
+```
+
+Prefer:
+
+```text
+Together, these diagnostics define the limits of what can be supported without signature-level ground truth.
+```
+
+Avoid presenting auxiliary diagnostics before the reader understands the classifier.
+
+Preferred order:
+
+```text
+Rule first. Then why not natural threshold. Then ICCR calibration. Then robustness.
+```
+
+## Reviewer-Facing Principle
+
+The paper should not read as:
+
+```text
+We used many methods, so the result is credible.
+```
+
+It should read as:
+
+```text
+We use one explicit screening rule. Each statistical diagnostic answers one necessary question about how that rule should be interpreted under unsupervised constraints.
+```
+
+This distinction is important for readability and reviewer trust.

paper/review_handoff_body_20260515.md

@@ -0,0 +1,397 @@
+# Review Handoff: Methodology, Results, Discussion, Conclusion
+
+Date: 2026-05-15
+Target manuscript: `paper/paper_a_v4_combined.md`
+Scope reviewed: §III Methodology, §IV Experiments and Results, §V Discussion, §VI Conclusion
+Companion review: `paper/review_handoff_abstract_intro_20260515.md` (Abstract + Introduction)
+
+This handoff continues the same framing principle established for Abstract + Introduction:
+
+> *"One explicit screening rule. Each statistical diagnostic answers one necessary question about how that rule should be interpreted under unsupervised constraints."*
+
+If only the Abstract and Introduction are revised, the manuscript will exhibit tonal mismatch when the reader drops into the body sections, which currently retain internal-version language and a defensive-accumulation framing for the supporting diagnostics. The body must be brought into the same register.
+
+## Overall Assessment
+
+The body sections are substantively defensible. The core empirical results — composition decomposition, anchor-based ICCR at three units, firm heterogeneity logistic, cross-firm hit matrix, alert-rate sensitivity — are presented in adequate quantitative detail with explicit unsupervised-validation caveats. The Discussion correctly distinguishes positive and negative anchors. The Conclusion lists eight methodological contributions that map onto the v4 contribution set.
+
+The recurring weakness across §III / §IV / §V / §VI is *not* empirical. It is two intertwined narrative tendencies:
+
+1. The body is still written as a *revision history* relative to v3.x in many paragraphs — "v4.0 strengthens", "v4.0 retroactively reframes", "v4.0 adopts", "inherited from v3.x", "the v3.x role of Firm A". This is internally honest but, in a submitted paper, signals to the reviewer that the authors are arguing with themselves.
+
+2. The supporting diagnostics are repeatedly presented as a *collection* ("multi-tool framework", "ten-tool unsupervised-validation collection", "Table XXVII"). This collection framing is precisely the readability risk identified in the Abstract / Introduction handoff under "Readability Risk: Too Many Diagnostics Can Look Like Methodological Overbuilding." It currently appears unmodified in §III-M.
+
+Recommended disposition: Minor Revision for narrative voice and structural emphasis, not for empirical weakness.
+
+## Main Reviewer Concerns
+
+### 1. The v3-to-v4 revision narrative is pervasive in the body and must be removed
+
+The Abstract / Introduction handoff identified "v4.0 adopts", "Earlier work in this lineage", and "inherited Paper A v3.x five-way box rule" as patterns to strip. The same patterns occur throughout the body sections. Representative instances (not exhaustive):
+
+- §III-G: "We earlier (v4.0 first draft) listed 'statistical multimodality at the accountant level' among the scope justifications..."
+- §III-H opening: "v4.0 distinguishes two reference populations in its calibration, replacing v3.x's single-anchor framing."
+- §III-I.5 closing sentence: "§III-L develops the v4.0 anchor-based threshold calibration framework..."
+- §III-L.0 "Why retained without v4.0 recalibration" subsection title.
+- §III-L.7 closing: "The operational classifier of §III-L.0 is the inherited v3.x five-way box rule..."
+- §IV opening paragraph: "The v4.0 primary analyses (§IV-D through §IV-J) are scoped to..." and "§IV-A through §IV-C report inherited corpus-wide v3.x material; §IV-L (feature backbone ablation) is also inherited. §IV-M consolidates the v4-new anchor-based ICCR calibration tables."
+- §IV-I: "v4.0 retroactively reframes the metric as inter-CPA pair-level coincidence rate (ICCR) rather than 'False Acceptance Rate'..."
+- §IV-J: "v4.0 does not change this aggregation rule; only the population over which it is computed changes (Big-4 subset)."
+- §IV-M opening: "v4-new empirical results that support..."
+- §V-B: "A central empirical finding of v3.x was that per-signature similarity does not admit a clean two-mechanism mixture... v4.0 strengthens and extends this signature-level reading."
+- §V-C: "In v4.0 we treat Firm A as a templated-end case study rather than as the calibration anchor for the operational threshold."
+- §V-H opening: "The first nine are v4.0-specific; the last five are inherited from v3.20.0 §V-G and still apply to the v4.0 pipeline."
+
+The remediation principle is the same as for the Introduction pivot paragraph. The final manuscript should describe the *final methodological state* and its rationale, not the trajectory by which that state was reached. Internal provenance — "this analysis is reproduced from v3.x §IV-F.1 / Script 28" — belongs in an Appendix B reproducibility table or supplementary material, not in the main narrative arc.
+
+A safe rewriting heuristic: every sentence that begins with "v4.0", "v3.x", "v4-new", "inherited", or "earlier work" should be candidated for either deletion or rewriting in the present tense without version labels.
+
+### 2. The "Ten-Tool Unsupervised-Validation Collection" frame must be retired
+
+§III-M Table XXVII is the canonical instance of the readability risk that the Abstract / Introduction handoff flagged. The current frame is:
+
+> "v4.0 adopts a multi-tool collection of partial-evidence diagnostics (Table XXVII), each with an explicitly disclosed assumption..."
+> "No single tool in this collection provides ground-truth validation. Their conjunction constitutes the unsupervised validation ceiling that the v4.0 corpus admits."
+
+This is exactly the language the Abstract / Introduction handoff identified as risky ("We used many methods, so the result is credible"). It reappears verbatim in the §VI Conclusion as "a multi-tool framework for characterising and disclosing its operational behaviour at the Big-4 sub-corpus scope" and "(8) a ten-tool unsupervised-validation collection (§III-M Table XXVII) that explicitly discloses each tool's untested assumption."
+
+The recommended reframe is:
+
+```text
+The corpus does not admit standard supervised classifier validation: no signature-level
+ground truth exists for hand-signed versus replicated classes, so False Rejection Rate,
+sensitivity, recall, EER, ROC-AUC, precision, and positive predictive value are not
+reportable. Each diagnostic in this section therefore addresses one specific
+failure mode of an unsupervised screening classifier: composition artefacts,
+inter-CPA coincidence, pool-size confounding, firm heterogeneity, threshold
+sensitivity, or positive-anchor capture. Together they characterise the limits of
+what can be claimed without signature-level ground truth.
+```
+
+Keep Table XXVII as a reference table if useful, but retitle it as "Diagnostic — failure mode addressed — disclosed assumption" rather than "Ten-tool collection". The word "ten" should not appear in the manuscript.
+
+### 3. The §V-H Limitations list is correct but defensively ordered
+
+§V-H lists fourteen limitations. The first one — "No signature-level ground truth; no true error rates reportable" — is the load-bearing limitation that everything else in v4.0 hinges on. The next two — "Inter-CPA negative-anchor assumption is partially violated" and "Scope" — are also major. The other eleven are real but secondary. The current presentation gives every item roughly equal visual weight as a flat list.
+
+Recommended reorganisation:
+
+- *Primary limitations (3 items):* (a) no signature-level ground truth, (b) inter-CPA negative-anchor assumption partially violated and firm-dependent, (c) Big-4 scope (full-dataset robustness is light).
+- *Secondary limitations (4 items):* pixel-identity conservative subset; inherited rule components not separately v4-validated; deployed-rate excess not a true-positive rate; A1 pair-detectability stipulation.
+- *Documented features rather than limitations (2 items):* K=3 hard-posterior composition sensitivity; no partner-level mechanism attribution.
+- *Inherited engineering limitations (5 items):* ImageNet features, red-stamp HSV preprocessing, longitudinal scan / PDF / compression, source-exemplar misattribution, legal interpretation.
+
+This preserves the disclosures but signals to the reviewer which limitations carry the methodological weight and which are routine engineering caveats.
+
+### 4. §III-F SSIM and pixel-comparison justification is too long for Methodology
+
+§III-F currently dedicates roughly 15 lines (lines 112–127 in `paper_a_methodology_v3.md`) to justifying *why* SSIM and pixel-level comparison are not used as primary descriptors. The argument is correct (design-level mismatch between SSIM's natural-image quality factors and signature-crop artefacts; sub-pixel alignment fragility of pixel L1/L2), but in its current form it reads as a defensive response to an anticipated reviewer objection rather than as forward Methodology exposition.
+
+Recommended reduction: collapse the argument to one short paragraph (3–4 sentences) and move the full design-level discussion to Appendix B. The Methodology body should state the choice (cosine on deep features + dHash) and briefly justify it (both stable across print-scan cycles by design), with the SSIM / pixel-comparison rebuttal in an appendix or a single citation footnote.
+
+### 5. §IV's section opener still encodes provenance not appropriate to a Results section opener
+
+The current §IV opener:
+
+> "The v4.0 primary analyses (§IV-D through §IV-J) are scoped to the Big-4 sub-corpus (Firms A–D, n = 437 CPAs with n_sig ≥ 10, totalling 150,442 signatures with both descriptors available) per the methodology choice articulated in §III-G. The §IV-K Full-Dataset Robustness section reports the full-dataset (686 CPAs) variant of the K=3 mixture + Paper A box-rule Spearman analysis as a cross-scope robustness check. §IV-A through §IV-C report inherited corpus-wide v3.x material; §IV-L (feature backbone ablation) is also inherited. §IV-M consolidates the v4-new anchor-based ICCR calibration tables."
+
+Recommended replacement direction:
+
+```text
+Section IV reports the empirical results that calibrate and characterise the
+operational classifier of §III-L. The primary analyses (§IV-D through §IV-J,
+§IV-M) are scoped to the Big-4 sub-corpus (Firms A–D, 437 CPAs, 150,442
+signatures); §IV-K reports a full-dataset (686 CPAs) robustness check on the K=3
+mixture and per-CPA score-rank convergence; §IV-A through §IV-C and §IV-L
+report the corpus-wide pipeline performance and feature-backbone ablation that
+support the descriptor choice of §III-F.
+```
+
+This preserves the scope information while removing the v3-to-v4 inheritance labels and the "v4-new" prefix on §IV-M.
+
+## Section-by-Section Comments
+
+### §III-A Pipeline Overview
+
+The pipeline diagram caption (lines 12–20) describes the classifier as "Firm A P7.5-anchored", which is residual v3 language that conflicts with the v4 reframe. v4 explicitly abandons Firm A as the calibration anchor in favour of inter-CPA ICCR (§III-H, §III-L). The figure caption should be updated to read "Anchor-Calibrated Five-Way Classifier" or similar, consistent with the §III-L title "Anchor-Based Threshold Calibration and Operational Classifier".
+
+The §III-A second paragraph ("Throughout this paper we use the term non-hand-signed rather than 'digitally replicated'...") is well-positioned and should be kept.
+
+### §III-B Data Collection
+
+No issues identified.
+
+### §III-C Signature Page Identification
+
+No issues identified. The 98.8% VLM-YOLO agreement footnote is appropriately scoped ("we do not attempt to attribute the residual").
+
+### §III-D Signature Detection
+
+No issues identified.
+
+### §III-E Feature Extraction
+
+No issues identified.
+
+### §III-F Dual-Method Similarity Descriptors
+
+As noted in Main Concern 4: shorten the SSIM and pixel-comparison rebuttal to ~3–4 sentences and move full design-level argument to Appendix B.
+
+### §III-G Unit of Analysis and Scope
+
+This section is currently long and contains the "We earlier (v4.0 first draft) listed..." paragraph that explicitly walks through the methodological revision. That paragraph (currently at the end of §III-G, before the sample-size reconciliation) should be deleted. The four-item scope rationale list above it is good and should be kept.
+
+The sample-size reconciliation paragraph (n=150,442 vs n=150,453) is technically necessary but is repeated almost verbatim in §IV-J as a parenthetical. Consider centralising it in §III-G with a forward reference, or in an Appendix B reproducibility note.
+
+### §III-H Reference Populations
+
+Replace the opening sentence:
+
+> "v4.0 distinguishes two reference populations in its calibration, replacing v3.x's single-anchor framing."
+
+with:
+
+```text
+The calibration distinguishes two reference populations: Firm A as a within-Big-4
+templated-end case study, and the 249 non-Big-4 CPAs as an out-of-target reference
+for internal-consistency checking.
+```
+
+The remainder of §III-H is well-written; the descriptive content is fine. The "v3.x's single-anchor framing" phrase is the only internal-version language that needs removal.
+
+### §III-I Distributional Diagnostics
+
+This is the strongest single section in the body. The four sub-diagnostics (dip test, mixture, BD/McCrary, composition decomposition) are tightly organised around one claim: the descriptor distribution does not provide a within-population bimodal antimode. The 2x2 factorial table at §III-I.4 is the empirical centrepiece of the v4 reframe.
+
+One small narrative issue: §III-I.5 ("Conclusion") closes with "§III-L develops the v4.0 anchor-based threshold calibration framework, which derives operational rates from inter-CPA pair-level negative-anchor coincidences rather than from a distributional antimode." Remove "v4.0" — write "§III-L develops the anchor-based threshold calibration framework..."
+
+### §III-J K=3 as a Descriptive Partition of Firm-Composition Contrast
+
+The section header is clear and the framing ("Both fits are descriptive partitions... not within-population mechanism modes") is correct.
+
+The current closing paragraph references "§III-K" for cross-checks between the box rule and K=3, but §III-K is the next subsection — this is a within-Methodology forward reference and reads slightly oddly. Consider rephrasing as "Cross-checks between the inherited five-way box rule and the K=3 partition appear in §III-K below."
+
+### §III-K Convergent Internal-Consistency Checks
+
+This section is well-handled. The opening caveat — "the three scores are not statistically independent measurements... so their high pairwise rank correlations are partly a mechanical consequence of shared inputs" — is exactly the methodological honesty the v4 reframe needs.
+
+One narrative issue: §III-K.4 (positive-anchor miss rate) and §III-K.3 (LOOO reproducibility) are *summarised* in §III-K but also reported in detail in §III-J and §IV-G respectively. Consider whether the §III-K subsections add narrative value beyond cross-referencing — if not, §III-K could shrink to just the three-score Spearman block (§III-K.1) and a one-line cross-reference to LOOO and pixel-identity, with the detail living in §III-J and §IV-G / §IV-H.
+
+### §III-L Anchor-Based Threshold Calibration and Operational Classifier
+
+This section has the operating-rule text that the Abstract / Introduction handoff explicitly asked for ("Cosine > 0.95 AND dHash ≤ 5" etc., §III-L.0 item 1). Good.
+
+The "Terminological note on FAR" at the end of §III-L.0 is explicit and reviewer-facing. Keep it.
+
+Issues:
+
+- "Why retained without v4.0 recalibration" — replace subsection title and contents to remove v4 references. The argument ("the inherited thresholds preserve continuity with prior reporting; §III-I.4 establishes that recalibration cannot be anchored on distributional antimodes; §III-L.1 confirms the cosine threshold's specificity at the inter-CPA pair level is reproducible") is intact without the v4 label.
+- §III-L.7 ("K=3 not used as classifier") restates content already in §III-J. Consider deleting §III-L.7 and adding a one-line note inside §III-L.0 ("The K=3 mixture of §III-J is used as an accountant-level descriptive summary alongside the per-signature five-way classifier; K=3 hard-posterior membership is not used to assign signature-level or document-level labels in any result table").
+
+### §III-M Validation Strategy and Limitations under Unsupervised Setting
+
+Replace the framing as described in Main Concern 2. Keep the underlying disclosure content. Consider whether Table XXVII is best presented as a numbered methodological table or as an Appendix B reproducibility-and-assumption summary; in either case retitle and reframe so that "ten" does not appear and the unifying principle is "each diagnostic addresses one specific unsupervised failure mode."
+
+The "What v4.0 does not claim" and "What v4.0 does claim" subsections at the end of §III-M are strong but the framing tag "v4.0 does not claim" / "v4.0 does claim" is the problematic version-language pattern. Replace with "Limits of the present analysis" and "Scope of the present analysis."
+
+### §III-N Data Source and Firm Anonymization
+
+No issues. The residual-identifiability disclosure is appropriately framed.
+
+### §IV-A Experimental Setup
+
+No issues identified.
+
+### §IV-B Signature Detection Performance
+
+No issues identified.
+
+### §IV-C All-Pairs Intra-vs-Inter Class Distribution Analysis
+
+The pairwise-non-independence caveat ("we therefore rely primarily on Cohen's d... A Cohen's d of 0.669 indicates a medium effect size, confirming that the distributional difference is practically meaningful, not merely an artifact of the large sample count") is well-positioned. Keep.
+
+### §IV-D Big-4 Accountant-Level Distributional Characterisation
+
+The Table V dip-test row labels are clear. The "v4-new composition-decomposition diagnostics that establish this finding are tabulated in §IV-M below alongside the anchor-based ICCR calibration" should drop the "v4-new" — just write "...are tabulated in §IV-M below alongside the anchor-based ICCR calibration."
+
+### §IV-E Big-4 K=2 / K=3 Mixture Fits
+
+The "descriptive partition; not mechanism clusters per §III-J" labels in Tables VII and VIII are consistent with the v4 reframe. Keep. Drop "(v3.x role)" anywhere it appears.
+
+### §IV-F Convergent Internal-Consistency Checks
+
+This is duplicate Results-side reporting of §III-K. Consider whether the duplication adds value or is redundant. If both sections must remain, then §III-K should describe the *method* (three scores, why they are not independent) and §IV-F should report the *numbers*; currently §III-K reports both the method and the numbers, leaving §IV-F as a near-duplicate. Recommendation: trim §IV-F to just the per-firm summary table and the Cohen-kappa block, with the method description living in §III-K.
+
+### §IV-G Leave-One-Firm-Out Reproducibility
+
+Tables XII and XIII are well-organised. The interpretation paragraph following Table XIII correctly identifies the K=2 vs K=3 contrast (K=2 unstable; K=3 component shape reproducible but hard-posterior membership composition-sensitive). Keep.
+
+### §IV-H Pixel-Identity Positive-Anchor Miss Rate
+
+The "close to tautological" caveat is appropriately positioned. Keep. The reverse-anchor cut by prevalence calibration disclosure is also appropriate.
+
+### §IV-I Inter-CPA Pair-Level Coincidence Rate
+
+Replace:
+
+> "v4.0 retroactively reframes the metric as inter-CPA pair-level coincidence rate (ICCR) rather than 'False Acceptance Rate' because..."
+
+with:
+
+```text
+The metric reported here is the inter-CPA pair-level coincidence rate (ICCR). It
+is the per-pair rate at which two signatures from different CPAs satisfy the
+deployed rule. We do not label it as a False Acceptance Rate because (a) FAR has
+a biometric-verification meaning that requires ground-truth negative labels, and
+(b) the inter-CPA negative-anchor assumption is partially violated by within-firm
+cross-CPA template-like collision structures (§III-L.4 cross-firm hit matrix).
+```
+
+### §IV-J Five-Way Per-Signature + Document-Level Classification Output
+
+The sample-size reconciliation parenthetical ("11 of 150,453 loaded Big-4 signatures lacked one or both descriptors and were excluded") is repeated from §III-G. Centralise once and forward-reference.
+
+"v4.0 does not change this aggregation rule; only the population over which it is computed changes" should be "The aggregation rule is the inherited worst-case rule (HC > MC > HSC > UN > LH); we apply it to the Big-4 sub-corpus."
+
+The MC band capture-rate inheritance disclosure is appropriately framed but should drop the "v4.0 does not re-derive" phrasing; rewrite as "The moderate-confidence band's calibration and capture-rate evidence is reported in [Appendix B / v3.20.0 Tables IX, XI, XII, XII-B] and is not regenerated on the Big-4 subset."
+
+### §IV-K Full-Dataset Robustness
+
+The scope-of-§IV-K paragraph ("The scope of §IV-K is deliberately narrow: we re-run only the K=3 mixture + Paper A operational-rule per-CPA less-replication-dominated rate analysis...") is defensively framed but the substance is correct. Consider shortening the "what we do not do" enumeration and emphasising the "what we do show" finding (K=3 + Paper A box-rule Spearman convergence preserved at full scope; ρ drift = 0.007).
+
+### §IV-L Feature Backbone Comparison
+
+This is inherited v3.x content. The "inherited unchanged from the v3.20.0 backbone-ablation table" framing is acceptable here because it is a methodological choice (do not re-run the ablation at the Big-4 scope) rather than a narrative pivot. Keep.
+
+### §IV-M v4-New Anchor-Based ICCR Calibration Results
+
+Drop the "v4-new" from the section heading. Recommended replacement heading: "Anchor-Based ICCR Calibration Results".
+
+The section is empirically dense and methodologically sound. Tables XXI–XXVI cover the four units (per-comparison, per-signature, per-document, firm logistic + hit matrix) and the alert-rate sensitivity. Keep all tables. Drop "v4 new" / "v4-new" wherever it appears as a row qualifier or section subheading.
+
+### §V-A Non-Hand-Signing Detection as a Distinct Problem
+
+Keep. This section preserves the forgery distinction (Main concern #2 in the Abstract / Introduction handoff).
+
+### §V-B Per-Signature Similarity is a Continuous Quality Spectrum
+
+Replace the v3-to-v4 opening:
+
+> "A central empirical finding of v3.x was that per-signature similarity does not admit a clean two-mechanism mixture: dip-test fails to reject unimodality at the signature level for Firm A, BIC prefers a 3-component fit, and BD/McCrary candidate transitions lie inside the high-similarity mode rather than between modes. v4.0 strengthens and extends this signature-level reading."
+
+with:
+
+```text
+The Big-4 accountant-level descriptor distribution rejects unimodality on both
+marginals at p < 5 × 10⁻⁴ (§IV-D Table V). The composition decomposition of
+§III-I.4 (Scripts 39b–39e) shows this rejection is fully attributable to two
+non-mechanistic sources...
+```
+
+This preserves the §V-B content while removing the v3.x lineage statement.
+
+### §V-C Firm A as the Templated End of Big-4
+
+Replace "In v4.0 we treat Firm A as a templated-end case study rather than as the calibration anchor for the operational threshold" with "We treat Firm A as a templated-end case study within the Big-4 sub-corpus rather than as the calibration anchor for the operational threshold."
+
+Drop the "the v3.x role of Firm A" historical sub-clause that appears in §III-G item 2.
+
+The Firm A byte-level pixel-identity reference (145 signatures across ~50 distinct partners; 35 byte-identical matches across fiscal years) is inherited from v3.x §IV-F.1 / Script 28 — this byte-level granularity is the strongest single piece of v3.x evidence that *should* survive into v4 because it directly supports the §V-C templated-end characterisation. Keep the reference but recast as "Byte-level decomposition of these 145 signatures (Appendix B) shows..." rather than the current "The additional v3.x finding... is inherited from v3.20.0 §IV-F.1 / Script 28..."
+
+### §V-D K=2 / K=3 as Descriptive Firm-Compositional Partitions
+
+Keep. The contrast between K=2 instability and K=3 reproducible-component-shape-but-composition-sensitive-membership is one of the cleanest narrative arcs in the paper.
+
+### §V-E Three-Score Convergent Internal-Consistency
+
+Keep. The "not statistically independent" caveat is correctly positioned. The within-Big-4 non-Firm-A disagreement between Score 2 and Scores 1/3 is correctly disclosed.
+
+### §V-F Anchor-Based Multi-Level Calibration
+
+Keep. This is the v4 contribution. Drop any residual "v4" labels.
+
+### §V-G Pixel-Identity as a Hard Positive Anchor; Inherited Inter-CPA Negative Anchor
+
+Keep. The "positive necessary but not sufficient" caveat and the "specificity proxy under a partially-violated assumption" framing are exactly right.
+
+Drop "Inherited" from the §V-G section heading — the heading currently reads "Pixel-Identity as a Hard Positive Anchor; Inherited Inter-CPA Negative Anchor Reframed as Coincidence Rate", which encodes the v3-to-v4 history in the section title itself. Recommended: "Pixel-Identity Positive Anchor and Inter-CPA Coincidence-Rate Negative Anchor".
+
+### §V-H Limitations
+
+Reorganise as described in Main Concern 3: primary (3) / secondary (4) / documented features (2) / inherited engineering (5).
+
+Drop "inherited from v3.20.0 §V-G" qualifiers — the limitation either applies to the pipeline or it does not; the version source is reproducibility metadata that belongs in Appendix B.
+
+### §VI Conclusion
+
+Replace the opening framing:
+
+> "We present a fully automated pipeline for detecting non-hand-signed CPA signatures in Taiwan-listed financial audit reports and a multi-tool framework for characterising and disclosing its operational behaviour at the Big-4 sub-corpus scope."
+
+with:
+
+```text
+We present a fully automated pipeline for detecting non-hand-signed CPA
+signatures in Taiwan-listed financial audit reports, together with an
+anchor-calibrated screening framework that characterises the pipeline's
+operational behaviour at the Big-4 sub-corpus scope under explicit unsupervised
+assumptions.
+```
+
+The eight numbered contributions are content-correct but presented in flat-list form. Consider grouping into three thematic clusters:
+
+- *Why the descriptor distribution does not anchor a natural threshold* (contributions 1, 5).
+- *How the deployed rule is calibrated under unsupervised constraints* (contributions 2, 6, 7).
+- *What the calibration reveals about firm heterogeneity* (contributions 3, 4).
+- *Methodological positioning* (contribution 8 — but reframe per Main Concern 2).
+
+The Future Work block (four items) is fine; consider trimming the second item ("a separate study to distinguish deliberate template sharing from passive firm-level production artefacts") which is the only item that involves additional fieldwork rather than methodological extension.
+
+## Recommended Minimum Patch List
+
+1. Strip v3-to-v4 revision language throughout §III, §IV, §V, §VI. Mechanical pass on "v4.0", "v3.x", "v4-new", "inherited", "earlier work in this lineage". Replace with present-tense descriptions of the final methodological choice and forward references to Appendix B for reproducibility provenance.
+
+2. Retire the "ten-tool unsupervised-validation collection" framing in §III-M and the "multi-tool framework" phrase in §VI Conclusion. Replace with "each diagnostic addresses one specific unsupervised failure mode" framing. Retitle Table XXVII so that "ten" does not appear.
+
+3. Reorganise §V-H Limitations into primary / secondary / documented-features / inherited-engineering groupings.
+
+4. Shorten §III-F SSIM and pixel-comparison rebuttal to ~3–4 sentences; move design-level discussion to Appendix B.
+
+5. Update Figure 1 caption (currently in §III-A commented HTML) to remove "Firm A P7.5-anchored" residual v3 language.
+
+6. Rewrite the §IV opener paragraph to remove the inherited-vs-v4-new section labels.
+
+7. Rewrite the §IV-I opening paragraph to remove "v4.0 retroactively reframes the metric...".
+
+8. Drop "v4-new" from the §IV-M section heading; replace with "Anchor-Based ICCR Calibration Results".
+
+9. Centralise the n=150,442 vs n=150,453 sample-size reconciliation in §III-G; remove the duplicate parenthetical from §IV-J.
+
+10. Consider trimming §IV-F to numbers-only (per-firm summary table + Cohen kappa), with the method description living in §III-K.
+
+11. Consider deleting §III-L.7 (duplicate of §III-J K=3-not-used-as-classifier claim) and adding a one-line note in §III-L.0.
+
+## Reviewer Bottom Line
+
+The body sections of v4 are empirically defensible and methodologically internally consistent. The required revisions are stylistic and structural rather than substantive:
+
+- Remove the v3-to-v4 revision narrative from the present-tense exposition.
+- Reframe the supporting diagnostics from "ten-tool collection" to "each diagnostic addresses one unsupervised failure mode."
+- Reorganise the limitations list so that the load-bearing limitations are visibly more prominent than the routine engineering caveats.
+- Move provenance and reproducibility detail to Appendix B / supplementary material.
+
+These changes preserve every quantitative claim and every disclosure currently in the manuscript. They tighten the narrative voice so that the reader experiences the v4 methodological choices as the final state of the design rather than as an ongoing argument with an earlier version. Combined with the Abstract / Introduction patches in the companion handoff, the manuscript should read as a single coherent submission rather than as a layered revision document.
+
+## Additional Cross-Cutting Observation: Script Provenance in Tables
+
+Across §III, §IV, §V, and the Conclusion, tables are annotated with `(Source: Script 32 / 34 / 35 / 38 / 40b / 43 / 44 / 45 / 46)` parentheticals. This is appropriate for reproducibility but heavy at the visual level — every table footer in §IV-D through §IV-M carries one of these annotations.
+
+Recommended consolidation: move the script-to-table mapping to a single Appendix B reproducibility table ("Table B-1. Script-to-table provenance map"), and replace the inline annotations with a single one-line note at the start of §IV ("Script-to-table provenance is summarised in Appendix B Table B-1; raw outputs are available in the supplementary repository").
+
+This is a minor change but it materially reduces the visual signal that the paper is built on a large number of separate scripts.
+
+## Closing Note
+
+This review covers the body sections only. The Abstract / Introduction handoff (`paper/review_handoff_abstract_intro_20260515.md`) covers the front matter. The two handoffs should be applied together; applying only one of them will produce tonal mismatch as the reader moves from the front matter into the body.
+
+The References and the Appendix have not been reviewed and may benefit from a separate handoff if the Appendix is to absorb the SSIM / pixel-comparison material and the reproducibility-provenance table recommended above.

paper/v4/paper_a_methodology_v4_section_iii.md

@@ -0,0 +1,450 @@
+# Section III. Methodology — v4.0 Draft v7 (post codex rounds 21–34)
+
+> **Draft note (2026-05-13, v7; internal — remove before submission).** This file replaces the §III-G through §III-M block of `paper/paper_a_methodology_v3.md` (v3.20.0). Sub-sections III-A through III-F (Pipeline / Data Collection / Page Identification / Detection / Feature Extraction / Dual-Method Descriptors) are unchanged from v3.20.0 and not reproduced here. The §III-G through §III-M block has been substantially restructured between v6 and v7 (2026-05-13): codex round-29 demolished the distributional path to thresholds (Scripts 39b–39e prove (cos, dHash) multimodality is composition + integer artefact); v4.0 pivots to anchor-based multi-level inter-CPA coincidence-rate calibration (Scripts 40b, 43, 44, 45, 46); §III-I is rewritten as the no-natural-threshold diagnostic; §III-J is recast as a firm-compositional descriptive partition (not three mechanism clusters); §III-L is a new major sub-section on anchor-based threshold calibration; §III-M is a new sub-section on validation strategy and limitations under the unsupervised setting. Prior internal draft notes (v2–v6 changelog) have been moved to `paper/v4/CHANGELOG.md`.
+>
+> Empirical anchors throughout reference Scripts 32–46 on branch `paper-a-v4-big4`; a curated provenance table appears at the end of this section listing the principal numerical claims with their script and report path.
+
+## G. Unit of Analysis and Scope
+
+We analyse signatures at two units of resolution. The **signature** — one signature image extracted from one report — is the operational unit of classification (§III-L) and of the signature-level analyses in §IV (notably §IV-J for the five-way per-signature category counts and the inherited inter-CPA negative-anchor coincidence-rate analysis referenced in §IV-I; reported under prior "FAR" terminology in v3.x). The **accountant** — one CPA aggregated over all of their signatures in the corpus — is the unit of mixture-model characterisation (§III-J), of per-CPA internal-consistency analysis (§III-K), and of the leave-one-firm-out reproducibility check (§III-K). At the accountant level we compute, for each CPA with $n_{\text{sig}} \geq 10$ signatures, the per-CPA mean of the per-signature best-match cosine ($\overline{\text{cos}}_a$) and the per-CPA mean of the independent-minimum dHash ($\overline{\text{dHash}}_a$). The minimum threshold of 10 signatures per CPA is required for the per-CPA mean to be a stable summary; CPAs below this threshold are excluded from the accountant-level analyses but remain in the per-signature analyses.
+
+We make no within-year or across-year uniformity assumption about CPA signing mechanisms. Per-signature labels are signature-level quantities throughout this paper; we do not translate them to per-report or per-partner mechanism assignments, and we abstain from partner-level frequency inferences (such as "X% of CPAs hand-sign") that would require such a translation. A CPA's per-CPA mean is a *summary statistic* of their observed signatures, not a claim that all of their signatures share a single mechanism.
+
+We adopt one stipulation about same-CPA pair detectability:
+
+> **(A1) Pair-detectability.** *If a CPA uses image replication anywhere in the corpus, then at least one same-CPA signature pair is near-identical (after reproduction noise) within the cross-year same-CPA pool used by the max-cosine / min-dHash computation.*
+
+A1 is plausible for high-volume stamping or firm-level electronic signing workflows but is not guaranteed when (i) the corpus contains only one observed replicated report for a CPA, (ii) multiple template variants are used in parallel, or (iii) scan-stage noise pushes a replicated pair outside the detection regime. A1 is the only assumption the per-signature detector requires to be sensitive to replication.
+
+**Scope: the Big-4 sub-corpus.** v4.0's primary analyses (§III-I, §III-J, §III-K, §III-L, and the v4-new analyses in §IV-D through §IV-J) are restricted to the four largest accounting firms in Taiwan, pseudonymously labelled Firm A through Firm D throughout the manuscript. §IV-A through §IV-C, §IV-I (inter-CPA negative-anchor coincidence rate), and §IV-L (feature-backbone ablation) report inherited corpus-wide v3.x material that v4.0 does not re-scope to Big-4. §IV-K reports a deliberately narrow full-dataset cross-check at $n = 686$ CPAs. The Big-4 sub-corpus comprises 437 CPAs (171 / 112 / 102 / 52 across Firms A through D) with $n_{\text{sig}} \geq 10$ — the threshold for accountant-level analyses (Scripts 36, 38) — totalling 150,442 Big-4 signatures with both pre-computed descriptors available. Restricting the v4-new analyses to Big-4 is a methodological choice driven by four considerations:
+
+1. **Leave-one-firm-out fold feasibility.** §III-K reports leave-one-firm-out (LOOO) cross-validation of the Big-4 K=3 fit. The Big-4 sub-corpus permits a four-fold LOOO at the firm level (one fold per Big-4 firm). No analogous firm-level fold is available outside Big-4 because mid/small firms have CPA counts of $O(1)$–$O(30)$ per firm.
+
+2. **Firm A as templated-end case study.** Firm A is empirically the firm whose CPAs are most concentrated in the high-cosine, low-dHash corner of the descriptor plane (§III-J K=3 component cross-tab; v3.x byte-level pair analysis referenced in §III-H). v4.0 retains Firm A within the Big-4 scope as a descriptive case study of the templated end, rather than treating Firm A as the calibration anchor for thresholds (the v3.x role of Firm A).
+
+3. **Within-firm cross-CPA collision structure analysis.** §III-L.4 reports a Big-4 cross-firm hit-matrix analysis (Script 44) that quantifies the within-firm cross-CPA template-like collision pattern. The four-firm setting affords the cleanest signal for this analysis; replicating the same matrix structure on the heterogeneous mid/small-firm tail is left as future work.
+
+4. **Restricted generalisability claim.** v4.0's primary claims are scoped to the Big-4 audit-report context; we do not assert that the same descriptive mixture structure or operational alert behaviour extends to mid/small firms. The 249 non-Big-4 CPAs enter only (a) as an external reference population in §III-H's reverse-anchor internal-consistency check, (b) as a robustness comparison in §IV-K, and (c) as a corroborating-population check on the dHash discrete-mass-point artefact in §III-I.4 (Script 39c). Generalisation beyond Big-4 is left as future work.
+
+We earlier (v4.0 first draft) listed "statistical multimodality at the accountant level" among the scope justifications, on the basis that the Hartigan dip test rejects unimodality on the Big-4 accountant-level marginals. §III-I.4 reports diagnostics (Scripts 39b–39e) that explain the rejection as a joint effect of between-firm composition shift and dHash integer mass points, not as evidence of within-population continuous bimodality. We therefore no longer list dip-test multimodality among the Big-4 scope rationales; the K=3 mixture is retained as a descriptive partition (§III-J), not as inferential evidence for two mechanism modes.
+
+**Sample-size reconciliation.** Two Big-4 signature counts appear in this section and §IV: $n = 150{,}442$ for analyses using the pre-computed per-signature descriptors $\text{cos}_s$ (`max_similarity_to_same_accountant`) and $\text{dHash}_s$ (`min_dhash_independent`), and $n = 150{,}453$ for analyses recomputing pair-level metrics directly from the stored feature and dHash byte vectors (Scripts 40b, 43, 44). The $11$-signature difference reflects descriptor-completion status: $11$ signatures have feature vectors and dHash byte vectors stored but lack the pre-computed extrema. The $11$ signatures are negligible at population scale and do not affect any reported coincidence rate within $0.01$ percentage point. The CPA counts $468$ (all Big-4 CPAs with both vectors stored) and $437$ (Big-4 CPAs with $n_{\text{sig}} \geq 10$ for accountant-level stability) likewise reflect a single uniform exclusion rule rather than analysis-specific subsetting.
+
+## H. Reference Populations
+
+v4.0 distinguishes two reference populations in its calibration, replacing v3.x's single-anchor framing.
+
+**Internal reference: Firm A as the templated-end case study.** Firm A is empirically the firm whose CPAs are most concentrated in the high-cosine, low-dHash corner of the Big-4 descriptor plane. In the Big-4 K=3 descriptive partition (§III-J; Scripts 35, 38), Firm A accounts for 0% of the C1 component (low-cos / high-dHash corner; cos $\approx 0.946$, dHash $\approx 9.17$, weight $\approx 0.143$), 17.5% of the C2 component (central region), and 82.5% of the C3 component (high-cos / low-dHash corner); the opposite pattern holds at Firm C (Script 35: 23.5% C1, 75.5% C2, 1.0% C3, hereafter referred to as "the Firm whose CPAs are most concentrated in C1"). The byte-level pair analysis reported in v3.x §IV-F.1 identifies 145 Firm A pixel-identical signatures at the signature level (Script 40 verifies the 145/262 split among Big-4 pixel-identical signatures); the additional details that v3.x attributes to this analysis (50 distinct Firm A partners of 180 registered; 35 byte-identical matches spanning different fiscal years) are inherited from the Script 28 / Appendix B byte-decomposition output and were not regenerated in the v4.0 spike scripts. We retain those v3.x details by reference and mark them in the provenance table as "inherited from v3 §IV-F.1 / Script 28."
+
+In v4.0, Firm A is *not* the calibration anchor for the operational threshold. Firm A enters the Big-4 mixture on equal footing with Firms B through D; the K=3 components are derived from the joint Big-4 distribution (§III-J), not from Firm A alone. Firm A's role in the methodology is descriptive: it is the Big-4 firm whose CPAs are most concentrated in the high-cosine, low-dHash corner of the descriptor plane, and the byte-level pair evidence above provides the firm-level signature-reuse evidence that anchors §III-K's pixel-identity positive-anchor miss rate.
+
+**External reference: non-Big-4 as the reverse-anchor reference for internal-consistency checking.** The 249 non-Big-4 CPAs ($n_{\text{sig}} \geq 10$, drawn from $\sim$30 mid- and small-firms) constitute a population strictly outside the Big-4 target. Their per-CPA $(\overline{\text{cos}}_a, \overline{\text{dHash}}_a)$ distribution defines a 2D Gaussian reference (fit by Minimum Covariance Determinant with support fraction 0.85 for robustness; Script 38). This reference is used in §III-K's reverse-anchor internal-consistency check: each Big-4 CPA's location relative to the reference centre, measured as the marginal cosine cumulative-distribution-function value under the reference, is one of three feature-derived scores that v4.0 uses as a cross-check on the inherited per-signature classifier. The reverse-anchor reference is *not* a positive or negative anchor for threshold derivation — its role is to provide a strictly out-of-target benchmark against which the within-Big-4 mixture-derived ranking can be internally cross-checked.
+
+The reverse-anchor reference centre is at $\overline{\text{cos}} = 0.935$, $\overline{\text{dHash}} = 9.77$ (Script 38). The reference sits at a lower cosine and higher dHash than the Big-4 K=3 low-cos / high-dHash component (cos $= 0.946$, dHash $= 9.17$; §III-J); compared to the Big-4 high-cos / low-dHash component (cos $= 0.983$, dHash $= 2.41$; §III-J) the reference is markedly less replication-dominated. The reverse-anchor metric for a given Big-4 CPA is the percentile of $\overline{\text{cos}}_a$ within the reference marginal cosine distribution, sign-flipped so that lower percentile (further into the left tail of the reference) corresponds to a Big-4 CPA whose mean cosine sits further from the templated end of the descriptor plane. This is a "deviation in the less-replication-dominated descriptor-position direction" measure, not a "deviation toward the templated descriptor-position" measure; the reference is the less-replication-dominated population.
+
+## I. Distributional Diagnostics: Why the Composition Path Does Not Yield a Natural Threshold
+
+This section characterises the joint distribution of accountant-level descriptor means $(\overline{\text{cos}}_a, \overline{\text{dHash}}_a)$ across the 437 Big-4 CPAs of §III-G and tests whether the distribution provides distributional support — in the form of within-population bimodality — for the operational thresholds inherited from v3.x. We apply four diagnostic procedures in turn: a univariate unimodality test on each accountant-level marginal; a 2D Gaussian mixture fit (developed in §III-J); a density-smoothness diagnostic; and a composition decomposition that distinguishes within-population multimodality from between-firm location-shift artefacts (the v4-new diagnostic battery). The four diagnostics jointly imply that the operational thresholds are *not* anchored by distributional bimodality: §III-L develops an anchor-based calibration framework that does not require this assumption.
+
+**1. Hartigan dip test on each accountant-level marginal.** We apply the Hartigan & Hartigan dip test [37] to each of the two marginal distributions $\{\overline{\text{cos}}_a\}_{a=1}^{437}$ and $\{\overline{\text{dHash}}_a\}_{a=1}^{437}$, with bootstrap-based $p$-value estimation ($n_{\text{boot}} = 2000$). In both cases no bootstrap replicate exceeded the observed dip statistic, so the empirical $p$-value is bounded above by $5 \times 10^{-4}$; we report this in tables as $p < 5 \times 10^{-4}$ rather than $p = 0$ to reflect the bootstrap resolution (Script 34). For comparison, no rejection of unimodality holds in the comparison scopes tested in Script 32: Firm A pooled alone ($p_{\text{cos}} = 0.992$, $p_{\text{dHash}} = 0.924$, $n = 171$); Firms B + C + D pooled ($p_{\text{cos}} = 0.998$, $p_{\text{dHash}} = 0.906$, $n = 266$); all non-Firm-A CPAs pooled ($p_{\text{cos}} = 0.998$, $p_{\text{dHash}} = 0.907$, $n = 515$). Single-firm dip tests for Firms B, C, and D were not separately computed; the comparison scopes above sufficed to establish that no narrower-than-Big-4 *tested* scope at the accountant level rejected unimodality. The accountant-level Big-4 rejection is a descriptive observation; §III-I.4 below shows that the rejection is fully explained by between-firm location-shift effects rather than within-population bimodality.
+
+**2. K=2 / K=3 Gaussian mixture fits (descriptive partition).** A 2-component 2D Gaussian Mixture Model (full covariance, $n_{\text{init}} = 15$, fixed seed 42; Script 34) recovers components at $(\overline{\text{cos}}, \overline{\text{dHash}}) = (0.954, 7.14)$, weight $0.689$, and $(0.983, 2.41)$, weight $0.311$. The marginal crossings of the K=2 fit are $\overline{\text{cos}}^* = 0.9755$ and $\overline{\text{dHash}}^* = 3.755$, with bootstrap 95% confidence intervals $[0.9742, 0.9772]$ and $[3.48, 3.97]$ over $n_{\text{boot}} = 500$ resamples. The 3-component fit (§III-J) is BIC-preferred — using the convention that lower BIC is preferred, $\text{BIC}(K{=}3) - \text{BIC}(K{=}2) = -3.48$ (Script 36). The $\Delta$BIC magnitude is small in absolute terms; we do not treat $\Delta\text{BIC} = 3.5$ alone as decisive evidence for K=3 as a population mixture. Following §III-I.4 we treat both K=2 and K=3 fits as *descriptive partitions* of the joint Big-4 distribution that reflect firm-composition structure (Firm A vs others; §III-J) rather than as inferential evidence for two or three latent population modes.
+
+**3. Burgstahler-Dichev / McCrary density-smoothness diagnostic.** We apply the discontinuity test of [38, 39] as a *density-smoothness diagnostic* (rather than as a threshold estimator) on each accountant-level marginal axis (cosine in bins of $0.002$, dHash in integer bins). At the Big-4 scope, the diagnostic identifies no significant transition on either marginal at $\alpha = 0.05$ (Script 34). Outside Big-4, the diagnostic does flag dHash transitions in some subsets (Script 32: `big4_non_A` dHash transition at $10.8$; `all_non_A` dHash transition at $6.6$; pre-2018 and post-2020 time-stratified variants also exhibit one or more dHash transitions), but no cosine transition is identified in any subset. The Big-4-scope null on both axes is consistent with §III-I.4 below: under the composition decomposition the Big-4 marginals are unimodal once between-firm and integer-tie confounds are removed, so a local-discontinuity test correctly fails to flag a within-population transition.
+
+**4. Composition decomposition (Scripts 39b–39e).** §III-I.1 establishes that the accountant-level marginals reject unimodality at the Big-4 sub-corpus. The remaining question is whether the rejection reflects (a) genuine within-population bimodality at the signature or accountant level, (b) between-firm location-shift artefacts (firms with different mean descriptor positions pool to a multi-peaked distribution), or (c) integer mass-point artefacts on the integer-valued dHash axis (the dHash dip statistic is sensitive to spikes at integer values). We apply four diagnostics that decompose the rejection into these candidate sources:
+
+*Within-firm signature-level dip (Scripts 39b, 39c).* Repeating the dip test at the signature level inside each individual Big-4 firm (Script 39b) and inside each individual non-Big-4 firm with $\geq 500$ signatures (Script 39c) yields a consistent picture. The cosine marginal *fails* to reject unimodality in every single firm tested — all four Big-4 firms ($p_{\text{cos}} \in \{0.176, 0.991, 0.551, 0.976\}$ for Firms A through D; Script 39b) and ten non-Big-4 firms with $\geq 500$ signatures ($p_{\text{cos}} \in [0.59, 0.99]$; Script 39c). The raw dHash marginal *does* reject unimodality in every firm tested ($p < 5 \times 10^{-4}$ in all $14$ firms), but the raw dHash values are integer-valued in $\{0, 1, \ldots, 64\}$, leaving open the possibility of an integer-tie artefact.
+
+*Integer-jitter robustness (Scripts 39d, 39e).* Adding independent uniform jitter $\sim \mathrm{U}[-0.5, +0.5]$ to break exact dHash ties and re-running the dip test on the perturbed signature cloud (5 seeds, $n_{\text{boot}} = 2000$; Script 39d) eliminates the dHash within-firm rejection in every Big-4 firm tested (Firm A jittered $p_{\text{median}} = 0.999$; B $0.996$; C $0.999$; D $0.9995$; $0$/$5$ seeds reject at $\alpha = 0.05$ in any firm). A codex-verified read-only spike applying the same jitter procedure to the ten non-Big-4 firms with $\geq 500$ signatures (Script 39c substrate) likewise yields no rejection ($0$/$10$ firms reject at $\alpha = 0.05$; per-firm median-$p$ range $[0.38, 1.00]$). The pooled-Big-4 dHash dip *does* survive jitter alone ($p_{\text{median}} = 0$, $5$/$5$ seeds reject), but Firm A's mean dHash ($2.73$) is substantially below Firms B/C/D's ($6.46$, $7.39$, $7.21$) — a between-firm location shift. Script 39e applies a 2 \times 2 factorial correction (firm-mean centring $\times$ integer jitter) on the Big-4 pooled dHash:
+
+| Condition | Firm-mean centred | Integer jitter | Median dip $p$ |  Reject at $\alpha = 0.05$ |
+|---|---|---|---|---|
+| 1 raw | — | — | $< 5 \times 10^{-4}$ | $5/5$ |
+| 2 centred only | $\checkmark$ | — | $< 5 \times 10^{-4}$ | $5/5$ |
+| 3 jittered only | — | $\checkmark$ | $< 5 \times 10^{-4}$ | $5/5$ |
+| 4 centred and jittered | $\checkmark$ | $\checkmark$ | $\mathbf{0.35}$ | $\mathbf{0/5}$ |
+
+Removing *both* the between-firm location shift *and* the integer mass points eliminates the Big-4 dHash rejection. The Big-4 pooled dHash multimodality is therefore fully attributable to firm-composition contrast (primarily Firm A's mean $\text{dHash} = 2.73$ versus Firms B/C/D $\approx 6.5$–$7.4$) and integer-density artefacts, with no residual continuous within-firm bimodality.
+
+*Cosine analogue.* The cosine axis exhibits the same pattern. Codex-verified read-only spike on the Big-4 pooled signature cloud yields signature-level cosine dip $p < 5 \times 10^{-4}$ on the raw data, but $p = 0.597$ after firm-mean centring; accountant-level cosine $p = 1.0$ after firm-mean centring. The cosine multimodality is therefore between-firm composition-driven, not within-population bimodality.
+
+*Integer-histogram valleys (Script 39d).* A genuine within-firm dHash antimode would appear as a strict local minimum in the count histogram with deep relative depth. Within each of the four Big-4 firms, the dHash histogram on bins $0$–$20$ exhibits no strict local minimum; the Big-4 pooled histogram exhibits one shallow valley at $\text{dHash} = 4$ with relative depth $0.021$ (a $2.1\%$ count drop). No valley near the inherited $\text{dHash} = 5$ operational boundary appears within any individual firm. The hypothesised dHash antimode near $\text{dHash} \approx 5$ is not empirically supported by the histogram analysis.
+
+**5. Conclusion: no natural threshold from the descriptor distribution.** §III-I.4 jointly establishes that (a) the Big-4 accountant-level dip rejection is fully attributable to between-firm composition and integer mass-point artefacts; (b) within any individual firm, the descriptor marginals at the signature level are unimodal once integer ties are broken; and (c) no integer-histogram valley near the inherited $\text{dHash} = 5$ operational boundary exists within any firm. The descriptor distributions therefore do not contain a within-population bimodal antimode that could anchor an operational threshold. The K=2 / K=3 mixture fits of §III-I.2 and §III-J are retained as *descriptive partitions* that reflect firm-composition contrast, not as inferential evidence for two or three population modes. §III-L develops the v4.0 anchor-based threshold calibration framework, which derives operational rates from inter-CPA pair-level negative-anchor coincidences rather than from a distributional antimode.
+
+## J. K=3 as a Descriptive Partition of Firm-Composition Contrast
+
+This section develops the K=2 and K=3 Gaussian mixture fits to the Big-4 accountant-level distribution and clarifies their role. **Both fits are descriptive partitions of the joint Big-4 distribution; they reflect firm-composition contrast — primarily Firm A versus Firms B, C, D — rather than within-population mechanism modes.** §III-I.4 demonstrates that the apparent multimodality of the accountant-level marginals is fully explained by between-firm location shifts and integer mass-point artefacts, leaving no residual evidence for two or three latent within-population mechanism classes. Neither mixture is used to assign signature-level or document-level labels in the v4.0 primary analysis. The operational classifier of §III-L is calibrated via inter-CPA negative-anchor coincidence rates, not via mixture-derived antimodes.
+
+**K=2 fit.** Two components at $(\overline{\text{cos}}, \overline{\text{dHash}}) = (0.954, 7.14)$ (weight $0.689$) and $(0.983, 2.41)$ (weight $0.311$) (Script 34). $\text{BIC}(K{=}2) = -1108.45$. Marginal crossings: $\overline{\text{cos}}^* = 0.9755$, $\overline{\text{dHash}}^* = 3.755$. We refer to the components by index rather than by mechanism labels, since §III-I.4 establishes that the K=2 separation is firm-compositional rather than mechanistic.
+
+**K=3 fit.** Three components, sorted by ascending cosine mean (Script 35; Script 38 reproduces):
+
+| Component | $\overline{\text{cos}}$ | $\overline{\text{dHash}}$ | weight | descriptive position |
+|---|---|---|---|---|
+| C1 | 0.9457 | 9.17 | 0.143 | low-cos / high-dHash corner |
+| C2 | 0.9558 | 6.66 | 0.536 | central region |
+| C3 | 0.9826 | 2.41 | 0.321 | high-cos / low-dHash corner |
+
+$\text{BIC}(K{=}3) = -1111.93$, lower than $K{=}2$ by $3.48$ (mild numerical preference for K=3 under standard BIC interpretation, but not by itself decisive). The "descriptive position" column replaces v3.x's "hand-leaning / mixed / replicated" mechanism labels: §III-I.4 establishes that the cosine and dHash axes both lack within-population bimodality, so component centres are best interpreted as locations in a continuous descriptor space rather than as latent mechanism modes.
+
+**Per-firm component composition (Script 35 firm × cluster cross-tab).** The K=3 partition is dominated by firm membership:
+
+- Firm A: $0\%$ C1, $17.5\%$ C2, $82.5\%$ C3
+- Firm B: $8.9\%$ C1, $\sim 78\%$ C2, $\sim 13\%$ C3
+- Firm C: $23.5\%$ C1, $75.5\%$ C2, $1.0\%$ C3
+- Firm D: $11.5\%$ C1, $\sim 84\%$ C2, $\sim 4.5\%$ C3
+
+Firm A accounts for $141$ of the $143$ C3-assigned CPAs; Firm C accounts for $24$ of the $40$ C1-assigned CPAs. The K=3 partition is therefore well-described as a firm-compositional decomposition: C3 is essentially "Firm A and any non-Firm-A CPA whose mean descriptors happen to land in the high-cos / low-dHash corner"; C1 is essentially "non-Firm-A CPAs whose mean descriptors land in the low-cos / high-dHash corner." The composition contrast that K=3 captures at the accountant level reappears at the deployment level in the cross-firm hit matrix of §III-L.4 (Script 44): under the deployed any-pair rule, within-firm collision concentration is $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms). The K=3 partition and the cross-firm hit matrix therefore describe the same underlying firm-compositional structure at two different units of analysis.
+
+**Leave-one-firm-out stability (Scripts 36, 37).** Leave-one-firm-out cross-validation shows that K=2 is unstable across folds: holding Firm A out gives a fold rule cos $> 0.938$ AND dHash $\leq 8.79$, while holding any single non-Firm-A Big-4 firm out gives a fold rule near cos $> 0.975$ AND dHash $\leq 3.76$ (Script 36). The maximum absolute deviation of the four fold cosine crossings from their across-fold mean is $0.028$ (the corresponding pairwise across-fold range is $0.0376$, from $0.9380$ for the held-out-Firm-A fold to $0.9756$ for the held-out-Firm-D fold; Script 36 stability summary). The $0.028$ value is $5.6\times$ the report's $0.005$ across-fold stability tolerance. K=3 in contrast has a *reproducible component shape*: across the four folds the C1 cosine mean varies by at most $0.005$, the C1 dHash mean by at most $0.96$, and the C1 weight by at most $0.023$ (Script 37). K=3 hard-posterior membership for the held-out firm is composition-sensitive — for Firm C the held-out C1 rate is $36.3\%$ vs the full-Big-4 baseline of $23.5\%$, an absolute difference of $12.8$ pp; for Firm A the held-out C1 rate is $4.7\%$ vs baseline $0.0\%$; the report's own legend classifies this pattern as `P2_PARTIAL` ("the C1 cluster exists but membership is not well-predicted by the held-out fit"). We accordingly do not use K=3 hard-posterior membership as an operational label.
+
+We take the joint K=2 / K=3 LOOO evidence as supporting the following descriptive claims, all of which are used in §III-K and §V but none of which underwrites the v4.0 operational classifier:
+
+- The Big-4 K=2 marginal crossing $(0.975, 3.76)$ is essentially a firm-mass separator between Firm A and Firms B + C + D, not a within-Big-4 mechanism boundary.
+- The Big-4 K=3 mixture exhibits a reproducible three-component component shape across LOOO folds at the descriptor-position level, with C1 reproducibly located at $\overline{\text{cos}} \approx 0.946$, $\overline{\text{dHash}} \approx 9.17$.
+- Hard-posterior K=3 membership is composition-sensitive across folds (max absolute deviation $12.8$ pp); K=3 is therefore not used to assign operational labels to CPAs in v4.0.
+
+The operational signature-level classifier of §III-L is calibrated against inter-CPA pair-level negative-anchor coincidence rates, not against mixture-derived antimodes. Cross-checks between the inherited five-way box rule and the K=3 partition appear in §III-K.
+
+## K. Convergent Internal-Consistency Checks
+
+The descriptive partition of §III-J is supported by three feature-derived per-CPA scores and a hard-ground-truth subset analysis. We caution at the outset that the three scores are **not statistically independent measurements** — all three are deterministic functions of the same per-CPA descriptor means $(\overline{\text{cos}}_a, \overline{\text{dHash}}_a)$ — so their high pairwise rank correlations are partly a mechanical consequence of shared inputs. Per §III-I.4, none of the three scores has a within-population bimodality interpretation; they are firm-compositional position scores at the accountant level. The checks below therefore document **internal consistency among feature-derived ranks**, not external validation against an independent hand-signed ground truth (which the corpus does not provide).
+
+**1. Three feature-derived per-CPA scores (Script 38).** For each Big-4 CPA we compute:
+
+- **Score 1 (K=3 posterior on the low-cos / high-dHash component):** $P(\text{C1})$ from the K=3 fit of §III-J. Per §III-J this is a firm-compositional position score on the (cos, dHash) plane (not a probability of any latent "hand-signing mechanism") — a function of both descriptor means.
+- **Score 2 (reverse-anchor cosine percentile):** the marginal cosine CDF value of $\overline{\text{cos}}_a$ under the non-Big-4 reference Gaussian of §III-H, sign-flipped so that lower percentile (further into the reference's left tail) corresponds to a Big-4 CPA whose mean cosine sits further from the templated end. This is a function of $\overline{\text{cos}}_a$ alone.
+- **Score 3 (inherited binary high-confidence box rule rate):** the per-CPA fraction of signatures that do **not** satisfy the inherited binary high-confidence box rule (cos $> 0.95$ AND dHash $\leq 5$). This is a per-signature-aggregated function of the same descriptors.
+
+Pairwise Spearman rank correlations among the three scores, $n = 437$ Big-4 CPAs (Script 38):
+
+| Pair | Spearman $\rho$ | $p$-value |
+|---|---|---|
+| Score 1 vs Score 3 | $+0.9627$ | $< 10^{-248}$ |
+| Score 2 vs Score 3 | $+0.8890$ | $< 10^{-149}$ |
+| Score 1 vs Score 2 | $+0.8794$ | $< 10^{-142}$ |
+
+We read this as the strongest internal-consistency signal in v4.0: three different summarisations of the same descriptor pair agree on the per-CPA descriptor-position ranking with $\rho > 0.87$. The three scores agree on placing Firm A as the most replication-dominated descriptor position and the three non-Firm-A Big-4 firms further from the templated end, but they do not all rank the non-Firm-A firms identically: the K=3 posterior P(C1) and the box-rule less-replication-dominated rate (Scores 1 and 3) place Firm C at the less-replication-dominated end of Big-4 (mean P(C1) $= 0.311$; mean box-rule less-replication-dominated rate $= 0.790$), while the reverse-anchor cosine percentile (Score 2) places Firm D fractionally higher than Firm C (mean reverse-anchor score $-0.7125$ vs Firm C $-0.7672$, with higher value indicating deeper into the reference left tail). The mean values for Firms B and D sit between Firms A and C on Scores 1 and 3 (Script 38 per-firm summary). We do not claim this constitutes external validation of any operational classifier; the inherited box rule is calibrated separately (§III-L), and the convergence above shows that a mixture-derived score and a reverse-anchor score concur with the box rule's per-CPA-aggregated outputs on the directional ordering, with a modest disagreement at the less-replication-dominated end between the three non-A Big-4 firms.
+
+**2. Per-signature consistency (Script 39).** Per-CPA aggregation could in principle reflect averaging across within-CPA heterogeneity rather than coherent within-CPA behaviour. We test this by repeating the K=3 fit at the signature level — fitting a fresh K=3 GMM to the 150,442 Big-4 signature-level $(\text{cos}, \text{dHash}_{\text{indep}})$ points (Script 39) — and comparing labels. The per-CPA and per-signature K=3 fits recover a broadly similar three-component ordering; per-CPA C1 is at $\overline{\text{cos}} = 0.946$, $\overline{\text{dHash}} = 9.17$ vs per-signature C1 at $\overline{\text{cos}} = 0.928$, $\overline{\text{dHash}} = 9.75$ (an absolute cosine drift of $0.018$). Cohen $\kappa$ on the binary collapse (replication-dominated vs less-replication-dominated):
+
+| Pair | Cohen $\kappa$ |
+|---|---|
+| Paper A binary high-confidence box rule vs per-CPA K=3 hard label | $0.662$ |
+| Paper A binary high-confidence box rule vs per-signature K=3 hard label | $0.559$ |
+| Per-CPA K=3 vs per-signature K=3 | $0.870$ |
+
+The Script 39 report verdict is `SIG_CONVERGENCE_MODERATE`. The $\kappa = 0.870$ between per-CPA-fit and per-signature-fit K=3 binary labels indicates that per-CPA aggregation does not collapse the broad three-component ordering. The lower $\kappa = 0.56\text{–}0.66$ between the binary box rule and either K=3 fit is consistent with two factors: different decision geometries (rectangular box vs Gaussian-mixture posterior boundary), and the fact that the binary box rule is a strict subset of the inherited five-way rule. We note that this comparison validates only the binary high-confidence rule (cos $> 0.95$ AND dHash $\leq 5$); §III-K does not directly validate the five-way rule's `5 < \text{dHash} \leq 15` moderate-confidence band, which retains its v3.20.0 calibration and capture-rate evaluation (v3.20.0 Tables IX, XI, XII, XII-B; documented as inherited in §IV-J).
+
+**3. Leave-one-firm-out reproducibility (Scripts 36, 37).** Discussed in §III-J above. We summarise the joint result for cross-reference:
+
+- *K=2 LOOO is unstable.* The maximum absolute deviation of the four fold cosine crossings from their across-fold mean is $0.028$, against the report's $0.005$ across-fold stability tolerance (Script 36; pairwise fold range $0.0376$, from $0.9380$ to $0.9756$). When Firm A is held out, the fold rule classifies $171/171$ of held-out Firm A CPAs as templated; when any non-Firm-A Big-4 firm is held out, the fold rule classifies $0$ of the held-out firm's CPAs as templated. This pattern indicates the K=2 boundary is essentially a Firm-A-vs-others separator rather than a within-Big-4 mechanism boundary.
+
+- *K=3 LOOO is partially stable.* The C1 (low-cos / high-dHash) component shape is reproducible across folds: max deviation from the full-Big-4 baseline is $0.005$ in cosine, $0.96$ in dHash, and $0.023$ in mixture weight (Script 37). Hard-posterior membership remains composition-sensitive — observed absolute differences are $1.8$–$12.8$ pp across the four folds, with the Firm C fold exceeding the report's $5$ pp viability bar; the report's own verdict is `P2_PARTIAL` ("K=3 is not predictively useful as an operational classifier"). We accordingly do not use K=3 hard-posterior membership as an operational label.
+
+**4. Positive-anchor miss rate on byte-identical signatures (Script 40).** The corpus provides one hard ground-truth subset: signatures whose nearest same-CPA match is byte-identical after crop and normalisation. Independent hand-signing cannot produce pixel-identical images, so byte-identical signatures are conservative-subset ground truth for the *replicated* class. The Big-4 byte-identical subset comprises $n = 262$ signatures ($145 / 8 / 107 / 2$ across Firms A through D; Script 40).
+
+We report each candidate check's *positive-anchor miss rate* — the fraction of byte-identical signatures classified as belonging to the less-replication-dominated descriptor positions. This is a one-sided check against a conservative positive subset, **not a paired specificity metric in the usual two-class sense**; we do not report a paired negative-anchor metric here because no signature-level hand-signed ground truth exists. The corresponding signature-level inter-CPA negative-anchor ICCR evidence is developed in §III-L.1 (Big-4 v4 sample) and the inherited corpus-wide v3.x version cited at §IV-I (reported under prior "FAR" terminology):
+
+| Candidate check | Pixel-identity miss rate (Wilson 95% CI) |
+|---|---|
+| Inherited Paper A binary high-confidence box rule (cos $> 0.95$ AND dHash $\leq 5$) | $0\%$ $[0\%, 1.45\%]$ |
+| K=3 per-CPA hard label (C3 high-cos / low-dHash corner; descriptive only) | $0\%$ $[0\%, 1.45\%]$ |
+| Reverse-anchor with prevalence-calibrated cut | $0\%$ $[0\%, 1.45\%]$ |
+
+All three candidate scores correctly assign every byte-identical signature to the replicated class. We caution that for the inherited box rule this result is close to tautological: byte-identical nearest-neighbour signatures have cosine $\approx 1$ and dHash $\approx 0$ by construction, so any threshold strictly below cos $= 1$ and strictly above dHash $= 0$ will capture them; v3.x discussed this conservative-subset caveat at length (v3 §III-J item 1, V-F). The positive-anchor miss rate is therefore a necessary check (a classifier that *failed* this check would be disqualified), not a sufficient validation of the classifier's behaviour on the non-byte-identical replicated population. The reverse-anchor cut here is chosen by prevalence calibration against the inherited box rule's overall replicated rate ($49.58\%$ of Big-4 signatures; Script 40); this is a documented v4.0 limitation since no signature-level hand-signed ground truth exists to permit direct ROC optimisation.
+
+## L. Anchor-Based Threshold Calibration and Operational Classifier
+
+§III-I.4 established that the descriptor distributions do not contain a within-population bimodal antimode that could anchor an operational threshold; the K=3 mixture of §III-J is a descriptive firm-compositional partition, not a mechanism-cluster model. This section develops v4.0's anchor-based threshold calibration: the operational thresholds inherited from v3.x are characterised by their inter-CPA pair-level negative-anchor coincidence behaviour and their pool-normalised per-signature and per-document alert behaviour, at multiple units of analysis. Throughout this section we report **inter-CPA coincidence rates** rather than "False Acceptance Rates"; we explain the terminological choice in §III-L.0.
+
+### L.0. Calibration methodology
+
+**Operational classifier (inherited from v3.20.0 §III-K, retained unchanged).** Each Big-4 signature is assigned to one of five categories using the per-signature descriptor pair $(\text{cos}_s, \text{dHash}_s)$ where $\text{cos}_s$ is the maximum cosine similarity to another signature by the same CPA and $\text{dHash}_s$ is the minimum independent dHash to another signature by the same CPA:
+
+1. **High-confidence non-hand-signed (HC):** Cosine $> 0.95$ AND $\text{dHash}_{\text{indep}} \leq 5$. Both descriptors converge on strong replication evidence.
+2. **Moderate-confidence non-hand-signed (MC):** Cosine $> 0.95$ AND $5 < \text{dHash}_{\text{indep}} \leq 15$. Feature-level evidence is strong; structural similarity is present but below the high-confidence cutoff.
+3. **High style consistency (HSC):** Cosine $> 0.95$ AND $\text{dHash}_{\text{indep}} > 15$. High feature-level similarity without structural corroboration — consistent with a CPA who signs very consistently but not via image reproduction.
+4. **Uncertain (UN):** Cosine between the all-pairs intra/inter KDE crossover ($0.837$) and $0.95$.
+5. **Likely hand-signed (LH):** Cosine $\leq 0.837$.
+
+The thresholds ($\text{cos} = 0.95$ as the cosine operating point, $\text{cos} = 0.837$ as the all-pairs KDE crossover, $\text{dHash} = 5$ and $15$ as structural-similarity sub-band cutoffs) are inherited from v3.x §III-K and retain their v3.x calibration provenance. Document-level labels are aggregated via the v3.x worst-case rule: each audit report inherits the most-replication-consistent category among its certifying-CPA signatures (rank order HC > MC > HSC > UN > LH).
+
+**Why retained without v4.0 recalibration.** The inherited thresholds preserve continuity with v3.x reporting and with the existing literature. §III-I.4 establishes that a v4.0 recalibration cannot be anchored on distributional antimodes (no within-population bimodality exists); §III-L.1 confirms that the cosine threshold's specificity behaviour at the inter-CPA pair level (the v3.x calibration anchor) is reproducible on the v4 spike sample, and §III-L.1 newly characterises the structural-dimension threshold $\text{dHash} \leq 5$'s pair-level coincidence behaviour. Sub-band thresholds ($\text{dHash} = 15$, $\text{cos} = 0.837$) retain v3.x's inherited calibration; v4.0 does not provide independent calibration for those sub-bands.
+
+**Three units of analysis.** We report inter-CPA negative-anchor coincidence behaviour at three units, each addressing a different operational question:
+
+- *Per comparison.* For a randomly drawn pair of signatures from different CPAs, what fraction satisfies the rule (cos $>$ cos\_threshold and / or dHash $\leq$ dHash\_threshold)? This is the unit at which v3.x §IV-I characterised the cosine threshold's specificity behaviour and at which threshold-derivation in biometric verification is conventionally calibrated. We report it for both the cosine and dHash dimensions, marginally and jointly (§III-L.1).
+- *Per signature pool.* For a Big-4 source signature $s$ with same-CPA pool of size $n_{\text{pool}}(s)$, what is the probability that the deployed rule fires *under the counterfactual* of replacing the source's same-CPA pool with $n_{\text{pool}}(s)$ random non-same-CPA candidates? This addresses the standard concern that a per-pair rate computed on independent pairs is not the deployed-rule rate at the per-signature classifier level: the deployed rule takes max-cosine and min-dHash over a pool of size $n_{\text{pool}}(s)$, so its effective coincidence rate is approximately $1 - (1 - p_{\text{pair}})^{n_{\text{pool}}}$ in the independence limit (§III-L.2).
+- *Per document.* For an audit report aggregated via the worst-case rule, what fraction of documents have at least one signature whose deployed pool-normalised rule fires under the same inter-CPA candidate-replacement counterfactual? This is the operational alarm-rate unit (§III-L.3).
+
+**Any-pair vs same-pair semantics.** The deployed rule uses independent extrema: a signature satisfies the HC rule if $\max_{\text{pool}} \text{cos} > 0.95$ AND $\min_{\text{pool}} \text{dHash} \leq 5$, *not* if a single candidate in the pool satisfies both. We refer to this as the **any-pair** rule. A stricter alternative — the **same-pair** rule — requires a single candidate to satisfy both inequalities; the deployed v3/v4 rule is any-pair, but we report same-pair as a stricter alternative classifier where useful (§III-L.2, §III-L.4).
+
+**Terminological note on "FAR".** The v3.x and biometric-verification literature speak of "False Acceptance Rate" (FAR) for a per-pair rate computed on independent inter-CPA pairs. We adopt **inter-CPA coincidence rate (ICCR)** as the v4.0 metric name and *do not* use "FAR" in the manuscript prose, for two reasons: (a) FAR has a specific biometric-verification meaning that requires ground-truth negative labels (which the corpus does not provide at the signature level); (b) §III-L.4 shows that the inter-CPA negative-anchor assumption — that inter-CPA pairs are negative — is partially violated by within-firm cross-CPA template-like collision structures. Reading "inter-CPA coincidence rate" as a *specificity proxy* under an explicitly disclosed assumption is faithful to the evidence; reading it as a true biometric FAR would overstate the evidence. We retain the v3.x numerical results (which are quantitatively reproduced in §III-L.1) under the new terminology.
+
+### L.1. Per-comparison inter-CPA coincidence rate (Script 40b)
+
+We sample $5 \times 10^5$ inter-CPA pairs uniformly at random from Big-4 signatures, computing for each pair the cosine similarity (feature dot product) and Hamming distance between the dHash byte vectors. Marginal and joint rates at threshold $k$ are reported with Wilson 95% confidence intervals (Script 40b).
+
+| Threshold | Per-comparison inter-CPA coincidence rate | 95% Wilson CI |
+|---|---|---|
+| Cosine $> 0.95$ | $0.00060$ | $[0.00053, 0.00067]$ |
+| Cosine $> 0.945$ (v3.x published "natural threshold") | $0.00081$ | $[0.00073, 0.00089]$ |
+| Cosine $> 0.97$ | $0.00024$ | $[0.00020, 0.00029]$ |
+| Cosine $> 0.98$ | $0.00009$ | $[0.00007, 0.00012]$ |
+| dHash $\leq 5$ | $0.00129$ | $[0.00120, 0.00140]$ |
+| dHash $\leq 4$ | $0.00050$ | $[0.00044, 0.00057]$ |
+| dHash $\leq 3$ | $0.00019$ | $[0.00015, 0.00023]$ |
+| dHash $\leq 2$ | $0.00006$ | $[0.00004, 0.00008]$ |
+| Joint: cos $> 0.95$ AND dHash $\leq 5$ | $0.00014$ | (any-pair semantics) |
+| Joint: cos $> 0.95$ AND dHash $\leq 4$ | $0.00011$ | (any-pair semantics) |
+
+The cosine row at $\text{cos} > 0.95$ replicates the v3.x §IV-I Table X result (v3.x reported the per-comparison rate as $0.0005$ under prior "FAR" terminology from a similarly-sized inter-CPA negative anchor; the v4 spike on a $5 \times 10^5$-pair sample yields $0.00060$, within the v3.x reported precision). The dHash and joint rows are v4-new: v3.x calibration did not provide an inter-CPA pair-level coincidence rate for the structural dimension or the joint rule.
+
+The all-firms-scope sample yields slightly lower per-comparison coincidence rates (cos $> 0.95$: $0.00031$; dHash $\leq 5$: $0.00073$; joint: $0.00007$); the all-firms sample weights small CPAs more heavily under CPA-uniform pair sampling, so we treat the Big-4 sample as the primary calibration scope and report all-firms as a corroborating-scope robustness check.
+
+**Conditional inter-CPA coincidence rate.** A natural follow-up question is whether the dHash dimension provides marginal specificity beyond the cosine gate. For pairs with cos $> 0.95$, the conditional rate of dHash $\leq 5$ is $0.234$ (Wilson 95% CI $[0.190, 0.285]$; $70$ of $299$ pairs in the Big-4 sample). At cos $> 0.95$, dHash provides $\sim 4.3\times$ further per-comparison specificity (joint $0.00014$ vs cos-only $0.00060$).
+
+The per-comparison rate is a useful *specificity-proxy calibration* for the deployed rule's pair-level behaviour. It does *not* directly translate to the deployed-rule specificity at the per-signature classifier level, because the deployed classifier takes extrema over a same-CPA pool of size $n_{\text{pool}}$. The pool-normalised inter-CPA alert rate is reported in §III-L.2.
+
+### L.2. Pool-normalised inter-CPA alert rate (Script 43)
+
+The deployed rule uses $\max_{\text{pool}} \text{cos}$ and $\min_{\text{pool}} \text{dHash}$ over the same-CPA pool of size $n_{\text{pool}}(s)$ for each signature $s$. A per-comparison rate is therefore not the rate at which the deployed classifier fires per signature. To compute the per-signature inter-CPA-equivalent rate, for each Big-4 source signature $s$ we simulate one realisation of an inter-CPA candidate pool of the same size $n_{\text{pool}}(s)$, drawn uniformly from non-same-CPA signatures across all firms, compute the deployed extrema and rule indicator, and aggregate (Script 43; $n_{\text{sig}} = 150{,}453$ vector-complete in this analysis; CPA-block bootstrap 95% CIs reported below).
+
+**Headline rates (deployed any-pair rule, HC = cos $> 0.95$ AND dHash $\leq 5$).** Wilson 95% CIs on the point estimate, CPA-block bootstrap 95% CI on $n_{\text{boot}} = 1000$ replicates:
+
+| Rule semantics | Per-signature ICCR | Wilson 95% CI | CPA-bootstrap 95% CI |
+|---|---|---|---|
+| Any-pair (deployed) | $0.1102$ | $[0.1086, 0.1118]$ | $[0.0908, 0.1330]$ |
+| Same-pair (stricter alternative) | $0.0827$ | $[0.0813, 0.0841]$ | $[0.0668, 0.1021]$ |
+
+Per-firm any-pair rates (no bootstrap; descriptive):
+
+| Firm | $n_{\text{sig}}$ | Any-pair ICCR | Same-pair ICCR |
+|---|---|---|---|
+| Firm A | $60{,}450$ | $0.2594$ | $0.2018$ |
+| Firm B | $34{,}254$ | $0.0147$ | $0.0023$ |
+| Firm C | $38{,}616$ | $0.0053$ | $0.0019$ |
+| Firm D | $17{,}133$ | $0.0110$ | $0.0051$ |
+
+**Pool-size decile dependence.** The deployed rule's pool-normalised rate is monotonically (broadly) increasing in $n_{\text{pool}}$, consistent with the $1 - (1 - p_{\text{pair}})^{n_{\text{pool}}}$ form expected under inter-CPA independence (Script 43 decile table). Decile 1 (smallest pools, $n_{\text{pool}} \in [0, 201]$): any-pair ICCR $= 0.0249$. Decile 10 (largest, $n_{\text{pool}} \in [846, 1115]$): any-pair ICCR $= 0.1905$. The trend is broadly monotonic with two minor non-monotone reversals (decile 5 and decile 9 dip below their predecessors).
+
+**Threshold sensitivity at per-signature unit.** Tightening the HC rule from $\text{dHash} \leq 5$ to $\text{dHash} \leq 3$ (same-pair) reduces the per-signature ICCR from $0.0827$ to $0.0449$ (Big-4 pooled); tightening to $\text{dHash} \leq 4$ gives $0.0639$ (same-pair). Stakeholders requiring a per-signature ICCR of $\leq 0.05$ at HC can adopt dHash $\leq 3$ same-pair as a stricter operating point; at $\leq 0.10$ the inherited HC any-pair rule with $\text{dHash} \leq 5$ at $0.1102$ is within tolerance.
+
+### L.3. Document-level inter-CPA proxy alert rate (Script 45)
+
+The deployed worst-case aggregation classifies each document by the most-replication-consistent category among its constituent signatures (§III-L.0). Three operationally meaningful document-level alarm definitions are reported, each as the fraction of documents whose worst-case signature category falls in the alarm set under the same inter-CPA candidate-pool counterfactual as §III-L.2 (Script 45; $n_{\text{docs}} = 75{,}233$ Big-4 documents):
+
+| Alarm definition | Alarm set | Document-level ICCR | Wilson 95% CI |
+|---|---|---|---|
+| D1 | HC only | $0.1797$ | $[0.1770, 0.1825]$ |
+| D2 | HC + MC ("any non-hand-signed verdict") | $0.3375$ | $[0.3342, 0.3409]$ |
+| D3 | HC + MC + HSC | $0.3384$ | $[0.3351, 0.3418]$ |
+
+Per-firm D2 document-level rates:
+
+| Firm | $n_{\text{docs}}$ | D2 (HC + MC) ICCR |
+|---|---|---|
+| Firm A | $30{,}226$ | $0.6201$ |
+| Firm B | $17{,}127$ | $0.1600$ |
+| Firm C | $19{,}501$ | $0.1635$ |
+| Firm D | $8{,}379$ | $0.0863$ |
+
+The document-level D2 rate of $33.75\%$ pooled over Big-4 is the most operationally relevant alarm-rate metric: it is the fraction of audit documents that would carry at least one signature flagged HC or MC under the counterfactual of inter-CPA candidate-pool replacement. The non-trivial per-document inter-CPA alarm rate (and its concentration in Firm A at $62\%$) motivates the positioning of the operational system as a **screening framework with human-in-the-loop review**, not as an autonomous forensic classifier (§III-M).
+
+### L.4. Firm heterogeneity (Script 44)
+
+§III-L.2 and §III-L.3 report large per-firm variation in the deployed rule's pool-normalised behaviour: Firm A's any-pair per-signature ICCR is $0.2594$, an order of magnitude larger than Firm B's $0.0147$, Firm C's $0.0053$, Firm D's $0.0110$. A natural alternative explanation is the pool-size confound: Firm A's median pool size ($\sim 285$) is larger than other firms', and pool size monotonically (broadly) increases the per-signature rate (§III-L.2 decile trend). We test the firm-vs-pool confound with a logistic regression of the per-signature hit indicator (any-pair HC) on firm dummies (Firm A = reference) and centred log pool size (Script 44):
+
+| Term | Odds ratio (vs Firm A) | Direction | Magnitude |
+|---|---|---|---|
+| Firm B | $0.053$ | $< 1$ | $\sim 19\times$ lower odds than Firm A |
+| Firm C | $0.010$ | $< 1$ | $\sim 100\times$ lower odds than Firm A |
+| Firm D | $0.027$ | $< 1$ | $\sim 37\times$ lower odds than Firm A |
+| log(pool size, centred) | $4.01$ | $> 1$ | $\sim 4\times$ higher odds per unit log pool size |
+
+The Firm B/C/D odds ratios are very small after controlling for pool size, indicating that firm membership accounts for a large multiplicative effect on the per-signature rate that is *not* explained by pool size alone. (We report odds ratios rather than $z$-scores because per-signature observations are clustered by CPA and firm, and naive standard errors would be inflated by within-cluster correlation; a cluster-robust standard error analysis is left as a robustness check.)
+
+The per-decile per-firm breakdown (Script 44) confirms the pattern: within every pool-size decile, Firms B/C/D have rates of $0.0006$–$0.0358$, while Firm A's rate ranges $0.0541$–$0.5958$ across deciles. The firm gap is large within matched pool sizes, not driven by pool composition.
+
+**Cross-firm hit matrix.** Among Big-4 source signatures whose any-pair rule fires under the inter-CPA candidate-pool counterfactual, the candidate firm of the max-cosine partner is distributed as follows (Script 44):
+
+| Source firm | Firm A candidate | Firm B | Firm C | Firm D | non-Big-4 | hits |
+|---|---|---|---|---|---|---|
+| Firm A | $14{,}447$ | $95$ | $44$ | $19$ | $17$ | $14{,}622$ |
+| Firm B | $92$ | $371$ | $8$ | $4$ | $9$ | $484$ |
+| Firm C | $16$ | $7$ | $149$ | $5$ | $1$ | $178$ |
+| Firm D | $22$ | $2$ | $6$ | $106$ | $1$ | $137$ |
+
+For the same-pair joint event (a single candidate satisfying both $\text{cos} > 0.95$ and $\text{dHash} \leq 5$), the candidate firm is even more strongly concentrated within the source firm: Firm A source $\to$ Firm A candidate in $11{,}314$ of $11{,}319$ same-pair hits ($99.96\%$); Firm B source $\to$ Firm B candidate in $85$ of $87$ ($97.7\%$); Firm C source $\to$ Firm C candidate in $54$ of $55$ ($98.2\%$); Firm D source $\to$ Firm D candidate in $64$ of $66$ ($97.0\%$).
+
+**Interpretation.** Under the deployed any-pair rule, the within-firm collision concentration is $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D — Firm A's pattern is markedly more within-firm-concentrated than the other three firms', though every Big-4 firm still has more than three quarters of its any-pair collisions falling on candidates within the same firm. The stricter same-pair joint event — a single candidate satisfying both cos $> 0.95$ and dHash $\leq 5$ — saturates at $97.0$–$99.96\%$ within-firm across all four firms. This pattern is consistent with — but not by itself diagnostic of — firm-specific template, stamp, or document-production reuse: within-firm scanning workflows, common form templates, and shared report-generation infrastructure could produce visually similar signature crops across different CPAs within the same firm. The byte-level evidence of v3.x §IV-F.1 (Firm A's $145$ pixel-identical signatures across $\sim 50$ distinct certifying partners) provides direct evidence that firm-level template reuse does occur at Firm A; the broader inter-CPA collision pattern in §III-L.4 is consistent with that mechanism extending in milder form to Firms B/C/D. We report this as "inter-CPA collision concentration is within-firm" — a descriptive observation about deployed-rule behaviour — and refrain from inferring that the within-firm hits constitute deliberate or systematic template sharing.
+
+This connects back to §III-J: the K=3 firm-composition contrast at the accountant level (Firm A dominating C3; Firm C dominating C1) reappears at the deployment level in the cross-firm hit matrix, where the within-firm collision concentration is the dominant pattern at all four Big-4 firms — most strongly at Firm A ($98.8\%$ any-pair, $99.96\%$ same-pair) and at materially lower but still majority levels at Firms B/C/D ($76.7$–$83.7\%$ any-pair; $97.0$–$98.2\%$ same-pair). The K=3 partition and the cross-firm hit matrix describe the same underlying firm-compositional structure at two different units of analysis.
+
+### L.5. Alert-rate sensitivity around inherited thresholds (Script 46)
+
+To test whether the inherited cosine threshold $0.95$ and dHash threshold $5$ coincide with a low-gradient (plateau-stable) region of the deployed-rule alert-rate surface — which would be weak distributional evidence that the inherited thresholds are stable operating points — we sweep each threshold across a range and report the per-signature alert rate on actual observed Big-4 same-CPA pools (not inter-CPA-replaced pools), comparing the local gradient at the inherited threshold to the median gradient across the sweep (Script 46).
+
+At the inherited HC operating point cos $> 0.95$ AND dHash $\leq 5$, the local gradient of the per-signature alert rate is substantially larger than the median gradient across the sweep (cosine: ratio $\approx 25\times$ at the $0.95$ point relative to median; dHash: ratio $\approx 3.8\times$ at the $5$ point relative to median; both Script 46). Reading these ratios descriptively, the inherited HC threshold is *locally sensitive* rather than plateau-stable: small threshold perturbations materially change the deployed alert rate (cosine sweep at dHash $\leq 5$ yields rates of $0.5091$ at cos $> 0.945$ vs $0.4789$ at cos $> 0.955$, a $3.0$ pp swing across a $0.01$ cosine perturbation; dHash sweep at cos $> 0.95$ yields rates of $0.4207$ at dHash $\leq 4$ vs $0.5639$ at dHash $\leq 6$, a $14.3$ pp swing across a single integer step). The local-gradient-to-median-gradient ratios are descriptive diagnostics, not formal plateau tests; the primary evidence for "no within-population bimodal antimode at these thresholds" comes from §III-I.4's composition decomposition, not from §III-L.5.
+
+The MC/HSC boundary at dHash $= 15$, by contrast, *is* in a low-gradient region (ratio $\approx 0.08$ to the median); the plateau-like behaviour around dHash $= 15$ is corroborating evidence that the high-end structural threshold lies in a regime where the rule's alert rate is approximately saturated, consistent with the high-dHash tail behaviour expected once near-identical pairs have been exhausted. The §III-L.5 non-plateau / local-sensitivity finding therefore applies specifically to the HC cutoff (cos $= 0.95$, dHash $= 5$); the MC/HSC sub-band boundary at dHash $= 15$ exhibits the opposite behaviour and is plateau-like.
+
+We interpret the inherited HC thresholds as **specificity-anchored operating points** chosen for the specificity-vs-alert-yield tradeoff (§III-L.1), *not* as distributional antimodes. Stakeholders requiring different operating points on the tradeoff curve can derive thresholds by inverting the per-comparison or pool-normalised ICCR curves (§III-L.1, §III-L.2) at their preferred specificity target.
+
+### L.6. Observed deployed alert rate on actual same-CPA pools
+
+The pool-normalised inter-CPA rates of §III-L.2 and §III-L.3 use the counterfactual of replacing the source signature's same-CPA pool with random non-same-CPA candidates. The **observed deployed alert rate** uses the source's actual same-CPA pool, i.e., the rate at which the deployed rule fires on the real corpus. For Big-4, the inherited HC any-pair rule fires on $49.58\%$ of signatures and $62.28\%$ of documents (Script 46; Script 42 reproduces the per-signature rate at $49.58\%$).
+
+The per-signature observed-deployed rate is $\sim 4.5\times$ the pool-normalised inter-CPA rate ($0.4958$ vs $0.1102$); the per-document observed-deployed rate is $\sim 3.5\times$ the pool-normalised inter-CPA D1 (HC) rate ($0.6228$ vs $0.1797$). We refer to this multiplicative gap as the **deployed-rate excess over the inter-CPA proxy**:
+
+- Per-signature: $0.4958 - 0.1102 = 0.3856$ ($38.6$ pp excess)
+- Per-document HC: $0.6228 - 0.1797 = 0.4431$ ($44.3$ pp excess)
+
+We *do not* interpret the deployed-rate excess as a presumed true-positive rate; the inferential limits of this interpretation are developed in §III-M. The deployed-rate excess is best read as a *same-CPA repeatability signal* — a quantity that exceeds what random inter-CPA candidate replacement would produce — rather than as an estimate of true replication prevalence.
+
+### L.7. K=3 not used as classifier
+
+The K=3 mixture of §III-J is reported in §IV as an accountant-level descriptive summary alongside the per-signature five-way classifier. We do not assign signature-level or document-level labels from the K=3 mixture in any v4.0 result table; the K=3 hard label is used only for the accountant-level firm × cluster cross-tabulation (§III-J; Script 35), and the K=3 *posterior* P(C1) is used (as the continuous Score 1) in the internal-consistency Spearman correlations of §III-K. The operational classifier of §III-L.0 is the inherited v3.x five-way box rule; the calibration evidence in §III-L.1 through §III-L.6 characterises its multi-level coincidence behaviour against the inter-CPA negative anchor.
+
+## M. Validation Strategy and Limitations under Unsupervised Setting
+
+The v4.0 corpus lacks signature-level ground-truth replication labels: no signature is annotated as definitively hand-signed or definitively templated. The conservative positive anchor (pixel-identical same-CPA signatures; §III-K.4 and v3.x §IV-F.1) is by construction near $\text{cos} = 1$ and $\text{dHash} = 0$, providing a tautological capture-check rather than a sensitivity estimate for the non-byte-identical replicated class. The corpus therefore does not admit standard supervised classifier validation: we cannot report False Rejection Rate, sensitivity, recall, Equal Error Rate, ROC-AUC, or precision against ground truth.
+
+In place of supervised validation, v4.0 adopts a **multi-tool collection of partial-evidence diagnostics** (Table XXVII), each with an explicitly disclosed assumption:
+
+**Table XXVII.** Ten-tool unsupervised-validation collection with disclosed untested assumptions.
+
+| Tool | What it measures | Untested assumption |
+|---|---|---|
+| Composition decomposition (§III-I.4; Scripts 39b–39e) | Whether descriptor multimodality is within-population (mechanism) or between-group (composition + integer artefact); $p_{\text{median}} = 0.35$ under joint firm-mean centring + integer-tie jitter | Integer-tie jitter and firm-mean centring are unbiased over the descriptor support; corroborated by Big-4 per-firm jitter (Script 39d; per-firm dHash rejection disappears under jitter at every Big-4 firm) and Big-4 pooled centred + jittered ($n_{\text{seeds}} = 5$; Script 39e) |
+| Per-comparison inter-CPA coincidence rate (§III-L.1; Script 40b) | Pair-level specificity proxy under a random-pair negative anchor | Inter-CPA pairs are negative (i.e., not template-related); partially violated by within-firm sharing (§III-L.4) |
+| Pool-normalised per-signature ICCR (§III-L.2; Script 43) | Deployed-rule specificity proxy at per-signature unit, accounting for pool size | Same as above + that pool replacement preserves the negative-anchor property |
+| Document-level ICCR (§III-L.3; Script 45) | Operational alarm rate proxy at per-document unit under three alarm definitions | Same as above |
+| Firm-heterogeneity logistic regression (§III-L.4; Script 44) | Multiplicative effect of firm membership on per-signature rate, controlling for pool size | Per-signature observations are clustered by CPA/firm; naïve standard errors inflated; cluster-robust analysis is a future check |
+| Cross-firm hit matrix (§III-L.4; Script 44) | Concentration of inter-CPA collisions within source firm | Concentration depends on deployed-rule semantics (the stricter same-pair joint event yields $97.0$–$99.96\%$ within-firm at all four firms versus $76.7$–$98.8\%$ under any-pair; §III-L.4); per-document per-firm assignment uses Script 45's mode-of-firms tie-break (§IV-M.4 footnote) |
+| Alert-rate sensitivity sweep (§III-L.5; Script 46) | Local sensitivity of deployed rule to threshold perturbation | Gradient comparison is descriptive, not a formal plateau test |
+| Convergent score Spearman ranking (§III-K.1; Script 38) | Internal-consistency of three feature-derived per-CPA scores | Scores share underlying inputs and are not statistically independent |
+| Pixel-identical conservative positive capture (§III-K.4; v3.x; Script 40) | Trivial sanity check on the conservative positive anchor | Anchor is tautologically captured by any reasonable threshold |
+| LOOO firm-level reproducibility (§III-K.3; Scripts 36, 37) | Algorithmic stability of K=2 / K=3 partition across firm folds | Stability is necessary but not sufficient for classification validity |
+
+No single tool in this collection provides ground-truth validation. Their conjunction constitutes the unsupervised validation ceiling that the v4.0 corpus admits.
+
+**What v4.0 does not claim.** We do not claim a validated forensic detector or an autonomous classification system. We do not report False Rejection Rate, sensitivity, recall, EER, ROC-AUC, precision, or positive predictive value against ground truth, because no ground truth exists at the signature level. We do not interpret the deployed-rate excess of §III-L.6 as a presumed true-positive rate: that interpretation would require assuming that the within-firm same-CPA pool's collision rate equals the inter-CPA proxy rate in the absence of replication (i.e., that genuine same-CPA hand-signing would produce a collision rate no higher than random inter-CPA pairs). Two factors make the assumption unsafe: (a) a CPA who signs consistently can produce stylistically similar signatures across years that exceed inter-CPA similarity at the cosine axis; (b) within-firm template sharing (§III-L.4 cross-firm hit matrix; v3.x byte-level evidence of Firm A's pixel-identical signatures across partners) places a substantial inter-CPA collision floor that itself reflects template-like reuse rather than independent inter-CPA random matching. We do not infer that the within-firm collision concentration of §III-L.4 constitutes deliberate template sharing; we describe it as "inter-CPA collision concentration is within-firm" and treat the mechanism as an open empirical question.
+
+**What v4.0 does claim.** The deployed signature-replication screening rule is characterised at three units of analysis (per-comparison, per-signature pool, per-document) against an inter-CPA negative-anchor coincidence-rate calibration. The per-comparison rates ($\leq 0.0006$ at cos $> 0.95$; $\leq 0.0013$ at dHash $\leq 5$; $\leq 0.00014$ jointly) are specificity-proxy-anchored operating points consistent with biometric-verification convention, with the proxy nature recorded in §III-L.0 and §III-M. The per-signature and per-document rates ($0.11$ and $0.34$ respectively under the deployed any-pair HC + MC alarm) are operationally meaningful **alarm-yield** indicators rather than true error rates. Per-firm rates show substantial heterogeneity (Firm A's per-document HC + MC alarm at $0.62$ vs Firm B/C/D at $0.09$–$0.16$), driven by firm-level rather than pool-size effects, and concentrated in within-firm cross-CPA candidate matching. The framework is positioned as a **specificity-proxy-anchored screening tool with human-in-the-loop review**, not as a validated forensic classifier.
+
+**Specificity-alert-yield tradeoff.** Because sensitivity is unobservable, stakeholders cannot derive an operating point by optimising a ROC criterion. Instead, the specificity-proxy-anchored framework offers a *specificity-alert-yield tradeoff*: tighter operating points (e.g., cos $> 0.98$ AND dHash $\leq 3$) reduce both per-comparison ICCR (to $\approx 5 \times 10^{-5}$; §III-L.1 inversion) and per-signature alert yield (to $\approx 0.05$; §III-L.2), with an unknown effect on actual replication-detection recall. Tighter operating points are not necessarily preferable: any tightening reduces the alert rate but may also miss true replicated signatures whose noise has pushed them outside the tighter envelope. The deployment decision depends on the relative cost of manual review (per alarm) and missed-replication risk (per false negative) — neither directly observable from corpus data.
+
+---
+
+## Provenance table for key numerical claims in §III-G through §III-L
+
+The table below lists the principal numerical claims and their data-source scripts. The table is curated for primary results; supporting numbers used illustratively in prose (e.g., all-firms-scope corroborating rates, per-decile fold values, illustrative threshold-inversion examples) are documented in the corresponding spike-script JSON outputs at `reports/v4_big4/*/` and are not individually tabled here.
+
+| Claim | Value | Source | Notes |
+|---|---|---|---|
+| Big-4 CPA count, $n_{\text{sig}} \geq 10$ | $437$ ($171/112/102/52$) | Script 36 sample sizes; Script 38 per-firm summary | direct |
+| Big-4 signature count (descriptor-complete) | $150{,}442$ | Script 39 / 39b / 39d / 39e / 45 / 46 | analyses using pre-computed descriptors |
+| Big-4 signature count (vector-complete) | $150{,}453$ | Script 40b / 43 / 44 | analyses recomputing from feature + dHash vectors |
+| Non-Big-4 reference CPA count | $249$ | Script 38 reference population | direct |
+| Big-4 K=2 marginal crossings $(0.9755, 3.755)$ | direct | Script 34; Script 36 §A | direct |
+| Bootstrap 95% CI cosine $[0.9742, 0.9772]$ | direct | Script 34; Script 36 §A | $n_{\text{boot}} = 500$ |
+| Bootstrap 95% CI dHash $[3.48, 3.97]$ | direct | Script 34; Script 36 §A | $n_{\text{boot}} = 500$ |
+| Bootstrap CI half-width $0.0015$ (cos) | direct | Script 36 (mean of CI half-widths) | direct |
+| Dip-test Big-4 cosine $p < 5 \times 10^{-4}$ | direct | Script 34 reports $p = 0.0000$; we bound by bootstrap resolution $n_{\text{boot}} = 2000$ | reporting convention |
+| Dip-test Big-4 dHash $p < 5 \times 10^{-4}$ | direct | Script 34 | reporting convention |
+| Dip-test Firm A $(p_{\text{cos}} = 0.992, p_{\text{dHash}} = 0.924)$ | direct | Script 32 §`firm_A` | direct |
+| Dip-test `big4_non_A` $(0.998, 0.906)$ | direct | Script 32 §`big4_non_A` | direct |
+| Dip-test `all_non_A` $(0.998, 0.907)$ | direct | Script 32 §`all_non_A` | direct |
+| K=3 component centers / weights | $(0.9457, 9.17, 0.143)$ / $(0.9558, 6.66, 0.536)$ / $(0.9826, 2.41, 0.321)$ | Script 35 / Script 38 | direct |
+| $\Delta\text{BIC}(K{=}3, K{=}2) = -3.48$ | direct | Script 34 (BIC K=2 = $-1108.45$; Script 36 reports BIC K=3 = $-1111.93$) | direct (arithmetic) |
+| K=2 LOOO max cosine deviation $0.028$ | direct | Script 36 stability summary | direct |
+| K=2 LOOO Firm A held-out $171/171$ replicated | direct | Script 36 fold table | direct |
+| K=3 C1 component shape drift (cos $0.005$, dHash $0.96$, weight $0.023$) | direct | Script 37 stability summary | direct |
+| K=3 LOOO held-out C1 absolute differences $1.8$–$12.8$ pp | direct | Script 37 held-out prediction check | direct |
+| Three-score pairwise Spearman ($0.963$, $0.889$, $0.879$) | direct | Script 38 correlations | direct |
+| Per-CPA / per-signature K=3 Cohen $\kappa$ ($0.662$, $0.559$, $0.870$) | direct | Script 39 kappa table | direct |
+| Per-CPA / per-signature K=3 C1 center drift $0.018$ (cosine) | derived | $\lvert 0.9457 - 0.9280 \rvert$; Script 39 components | direct |
+| Pixel-identity Big-4 subset $n = 262$ ($145/8/107/2$) | direct | Script 40 sample | direct |
+| Full-dataset accountant count $n = 686$ | direct | Script 41 (`fulldataset_report.md`) | direct |
+| Positive-anchor miss rate $0\%$ on $n = 262$ (Wilson upper $1.45\%$) | direct | Script 40 results table | direct |
+| Inter-CPA cos $> 0.95$ ICCR $0.0005$ (Wilson 95% $[0.0003, 0.0007]$) | inherited | v3 §IV-F.1 / Table X | v3 reported this as "FAR"; v4.0 reframes as inter-CPA coincidence rate per §III-L.0 |
+| Firm A byte-identical $145$ pixel-identical signatures in Big-4 subset | direct | Script 40 sample breakdown | direct |
+| Firm A byte-identical "50 distinct partners of 180; 35 cross-year" | inherited | v3 §IV-F.1 / Script 28 / Appendix B byte-decomposition output | **inherited from v3; not regenerated in v4.0 spike scripts** |
+| Big-4 K=3 per-firm C1 hard-assignment ($0\%$ / $8.9\%$ / $23.5\%$ / $11.5\%$) | direct | Script 35 firm × cluster cross-tab | direct |
+| **Composition decomposition (§III-I.4):** | | | |
+| Within-firm signature-level dip $p_{\text{cos}}$ Big-4 (A/B/C/D) | $0.176 / 0.991 / 0.551 / 0.976$ | Script 39b per-firm | direct, $n_{\text{boot}} = 2000$ |
+| Within-firm signature-level dip $p_{\text{cos}}$ non-Big-4 (10 firms, range) | $[0.59, 0.99]$ | Script 39c per-firm | direct, firms with $\geq 500$ signatures |
+| Within-firm jittered-dHash dip $p$ Big-4 (5 seeds, median) A/B/C/D | $0.999 / 0.996 / 0.999 / 0.9995$ | Script 39d multi-seed | uniform jitter $[-0.5, +0.5]$ |
+| Within-firm jittered-dHash dip $p$ non-Big-4 (5 seeds, range across 10 firms) | $[0.38, 1.00]$ ($0/10$ firms reject) | codex-verified read-only spike on Script 39c substrate | uniform jitter $[-0.5, +0.5]$ |
+| Big-4 pooled dHash dip $p$ raw / jittered (seed median) | $< 5 \times 10^{-4}$ / $< 5 \times 10^{-4}$ | Script 39d | jitter alone does not eliminate Big-4 pooled rejection |
+| Big-4 pooled dHash dip $p$ firm-centred + jittered (5-seed median) | $0.35$ | Script 39e 2×2 factorial | both corrections eliminate rejection ($0/5$ seeds at $\alpha = 0.05$) |
+| Big-4 firm-centred signature-level cos dip $p$ | $0.597$ | codex round-30 verification on Script 43 substrate | independent verification |
+| Big-4 firm-centred accountant-level cos\_mean dip $p$ | $1.0$ | codex round-30 verification | independent verification |
+| Per-firm Big-4 dHash mean (A/B/C/D) | $2.73 / 6.46 / 7.39 / 7.21$ | Script 39e per-firm summary | direct |
+| Big-4 integer-histogram valley near $\text{dHash} \approx 5$ within any firm | none in any of A/B/C/D | Script 39d valley analysis | bins $0$–$20$ |
+| **Anchor-based calibration (§III-L.1):** | | | |
+| Per-comparison ICCR cos $> 0.95$ Big-4 | $0.00060$ (Wilson 95% $[0.00053, 0.00067]$) | Script 40b | $5 \times 10^5$ inter-CPA pairs, Big-4 scope |
+| Per-comparison ICCR cos $> 0.945$ Big-4 | $0.00081$ (Wilson 95% $[0.00073, 0.00089]$) | Script 40b | direct |
+| Per-comparison ICCR cos $> 0.97$ / cos $> 0.98$ Big-4 | $0.00024$ / $0.00009$ | Script 40b | direct |
+| Per-comparison ICCR dHash $\leq 5$ Big-4 | $0.00129$ (Wilson 95% $[0.00120, 0.00140]$) | Script 40b | direct, v4 new |
+| Per-comparison ICCR dHash $\leq 4 / 3 / 2$ Big-4 | $0.00050 / 0.00019 / 0.00006$ | Script 40b | direct |
+| Per-comparison joint ICCR cos $> 0.95$ AND dHash $\leq 5$ Big-4 | $0.00014$ | Script 40b | any-pair semantics |
+| Per-comparison joint ICCR cos $> 0.95$ AND dHash $\leq 4$ Big-4 | $0.00011$ | Script 40b | any-pair semantics |
+| Conditional ICCR dHash $\leq 5$ given cos $> 0.95$ Big-4 | $0.234$ (Wilson 95% $[0.190, 0.285]$) | Script 40b | $70 / 299$ pairs |
+| All-firms per-comparison joint ICCR | $0.00007$ | Script 40b | corroborating scope |
+| **Pool-normalised per-signature alert rate (§III-L.2):** | | | |
+| Per-signature any-pair ICCR HC Big-4 | $0.1102$ (Wilson 95% $[0.1086, 0.1118]$; CPA-bootstrap 95% $[0.0908, 0.1330]$) | Script 43 | $n_{\text{sig}} = 150{,}453$ (vector-complete) |
+| Per-signature same-pair ICCR HC Big-4 | $0.0827$ (Wilson 95% $[0.0813, 0.0841]$; CPA-bootstrap 95% $[0.0668, 0.1021]$) | Script 43 | stricter alternative |
+| Per-firm any-pair ICCR HC (A/B/C/D) | $0.2594 / 0.0147 / 0.0053 / 0.0110$ | Script 43 per-firm | direct |
+| Per-firm same-pair ICCR HC (A/B/C/D) | $0.2018 / 0.0023 / 0.0019 / 0.0051$ | Script 43 per-firm | direct |
+| Pool-size decile 1 / decile 10 any-pair ICCR | $0.0249 / 0.1905$ | Script 43 decile table | broadly monotone with two minor reversals |
+| Per-signature tighter ICCR cos $> 0.95$ AND dHash $\leq 3$ same-pair Big-4 | $0.0449$ | Script 43 | optional stricter operating point |
+| **Document-level alert rate (§III-L.3):** | | | |
+| Document-level ICCR D1 (HC only) Big-4 | $0.1797$ (Wilson 95% $[0.1770, 0.1825]$) | Script 45 | $n_{\text{docs}} = 75{,}233$ |
+| Document-level ICCR D2 (HC + MC) Big-4 | $0.3375$ (Wilson 95% $[0.3342, 0.3409]$) | Script 45 | operational alarm definition |
+| Document-level ICCR D3 (HC + MC + HSC) Big-4 | $0.3384$ (Wilson 95% $[0.3351, 0.3418]$) | Script 45 | descriptive |
+| Per-firm document-level D2 ICCR (A/B/C/D) | $0.6201 / 0.1600 / 0.1635 / 0.0863$ | Script 45 per-firm | direct |
+| **Firm-heterogeneity logistic regression (§III-L.4):** | | | |
+| Logistic OR (Firm B / C / D vs A) | $0.053 / 0.010 / 0.027$ | Script 44 regression | controlling for log pool size; reference $=$ Firm A |
+| Logistic OR log(pool size, centred) | $4.01$ | Script 44 regression | pool-size effect after firm adjustment |
+| Cross-firm hit matrix Firm A source $\to$ Firm A candidate (any-pair) | $14{,}447 / 14{,}622$ | Script 44 cross-firm matrix | $98.8\%$ within-firm |
+| Cross-firm hit matrix same-pair within-firm rate (A/B/C/D) | $99.96\% / 97.7\% / 98.2\% / 97.0\%$ | Script 44 same-pair section | direct |
+| **Threshold-sensitivity (§III-L.5):** | | | |
+| Local / median gradient ratio cos $= 0.95$ | $\approx 25\times$ | Script 46 plateau diagnostic | descriptive, not formal plateau test |
+| Local / median gradient ratio dHash $= 5$ | $\approx 3.8\times$ | Script 46 plateau diagnostic | descriptive |
+| Local / median gradient ratio dHash $= 15$ | $\approx 0.08$ | Script 46 plateau diagnostic | MC/HSC boundary plateau-like |
+| **Observed deployed alert rate (§III-L.6):** | | | |
+| Per-signature observed-deployed HC rate Big-4 | $0.4958$ | Script 46 / Script 42 | actual same-CPA pools |
+| Per-document observed-deployed HC rate Big-4 | $0.6228$ | Script 46 | actual same-CPA pools |
+| Deployed-rate excess over inter-CPA proxy (per-sig HC) | $0.3856$ pp | derived | $0.4958 - 0.1102$ |
+| Deployed-rate excess over inter-CPA proxy (per-doc HC) | $0.4431$ pp | derived | $0.6228 - 0.1797$ |
+| **Sample-size reconciliation:** | | | |
+| Big-4 signatures with pre-computed descriptors | $150{,}442$ | Script 39 / 39b / 39d / 39e / 45 / 46 | descriptor-complete subset |
+| Big-4 signatures with feature + dHash vectors stored | $150{,}453$ | Script 40b / 43 / 44 | vector-complete subset |
+| Difference between the two counts | $11$ signatures | direct (descriptor-completion lag) | negligible at population scale |
+| Big-4 CPAs all (any signature count) | $468$ | Script 40b / 43 / 44 | direct |
+| Big-4 CPAs with $n_{\text{sig}} \geq 10$ for accountant-level stability | $437$ | Scripts 36 / 38 / 39 | accountant-level analysis threshold |
+
+---
+
+## Cross-reference index (author working checklist; remove before submission)
+
+- **Big-4 sub-corpus definition** (§III-G) — 437 CPAs / $n_{\text{sig}} \geq 10$ at accountant-level, 468 CPAs / 150,442–150,453 signatures at signature-level (sample-size reconciliation in §III-G).
+- **Reference populations** (§III-H) — Firm A as templated-end case study; non-Big-4 ($n = 249$) as reverse-anchor reference (less-replicated population).
+- **Distributional diagnostics + composition decomposition** (§III-I) — Big-4 accountant-level dip-test rejection ($p < 5 \times 10^{-4}$); §III-I.4's 2×2 factorial decomposition (firm centring × integer jitter) shows the rejection is fully explained by between-firm location shift + integer mass-point artefacts; **no within-population bimodality and no natural threshold**.
+- **K=3 as descriptive firm-compositional partition** (§III-J) — C1/C2/C3 are descriptive positions on the descriptor plane reflecting Firm A vs others composition; not mechanism clusters; not used as operational classifier.
+- **Convergent internal-consistency** (§III-K) — three feature-derived scores ($\rho \geq 0.879$, not independent measurements); per-signature K=3 ($\kappa = 0.87$ vs per-CPA fit); K=2 LOOO unstable, K=3 LOOO partial; pixel-identity miss rate $0\%$ on $n = 262$.
+- **Anchor-based threshold calibration + operational classifier** (§III-L) — inherited five-way rule retained; characterised by inter-CPA negative-anchor coincidence rates at per-comparison (§III-L.1: cos $> 0.95$ at $0.0006$, dHash $\leq 5$ at $0.0013$, joint at $0.00014$), per-signature pool (§III-L.2: $0.11$ any-pair HC), per-document (§III-L.3: HC $0.18$; HC+MC $0.34$); firm heterogeneity (§III-L.4) decisive after pool-size adjustment; within-firm cross-CPA collision concentration $\geq 97\%$; threshold-sensitivity analysis (§III-L.5) confirms HC threshold is locally sensitive, not plateau-stable; deployed-rate excess over proxy (§III-L.6) $\approx 38$ pp per-signature and $\approx 44$ pp per-document.
+- **Validation strategy and limitations** (§III-M) — multi-tool diagnostic collection (9 tools, each with disclosed untested assumption); positioning as anchor-calibrated screening framework with human-in-the-loop review, not as validated forensic detector; no FRR / sensitivity / EER / ROC-AUC reportable.
+
+## Open questions remaining for partner / reviewer
+
+1. **Five-way rule validation against the moderate-confidence band.** §III-K's $\kappa$ evidence covers only the binary high-confidence rule (cos $> 0.95$ AND dHash $\leq 5$). The moderate-confidence band (cos $> 0.95$ AND $5 < \text{dHash} \leq 15$) inherits its v3.x calibration and capture-rate evidence (v3.20.0 Tables IX, XI, XII, XII-B). Is this inheritance sufficient (Big-4 per-firm MC proportions are reported descriptively in §IV-J's Table XV), or should v4.0 add a Big-4-specific MC-band capture-rate analysis as an additional sub-section?
+
+2. **Anonymisation of within-Big-4 firm contrasts.** §III-H states that Firm C is the firm most concentrated in C1 hand-leaning at $23.5\%$ (Script 35). The within-Big-4 ordering by hand-leaning concentration is informative for the §V discussion. v3.x reports under pseudonyms throughout. Confirm that we maintain pseudonyms consistently in §IV–V even when discussing the specific Firm C / Firm B / Firm D hand-leaning rates.
+
+3. **Section IV table numbering.** Defer until §III final accepted by partner / reviewer; results numbering should mirror §III flow (sample/scope → mixture characterisation → convergent checks → LOOO → pixel-identity → signature/document classification → full-dataset robustness).

paper/v4/paper_a_prose_v4_phase4.md

@@ -0,0 +1,162 @@
+# Paper A v4.0 Phase 4 Prose Draft v3 (post codex rounds 26–34)
+
+> **Draft note (2026-05-13, Phase 4 v3; internal — remove before submission).** This file replaces the v3.20.0 Abstract, §I Introduction, §II Related Work, §V Discussion, and §VI Conclusion blocks with the v4.0 prose. The methodology and results sections (§III v7 and §IV v3.2 on this branch) are the technical foundation; Phase 4 prose aligns the narrative with the post-codex-round-34 framing. v3 (2026-05-13) reflects the major restructuring driven by codex rounds 29–34: distributional path to thresholds demolished (Scripts 39b–39e); anchor-based multi-level inter-CPA coincidence-rate calibration adopted (Scripts 40b, 43, 44, 45, 46); K=3 demoted to descriptive firm-compositional partition; "FAR" terminology replaced by "inter-CPA coincidence rate (ICCR)" throughout; nine-tool unsupervised validation strategy disclosed; positioning as anchor-calibrated screening framework with human-in-the-loop review (not validated forensic detector). Empirical anchors cite Scripts 32–46 on branch `paper-a-v4-big4`. Prior Phase 4 v2 changelog has been moved to `paper/v4/CHANGELOG.md`.
+
+---
+
+# Abstract
+
+> *IEEE Access target: <= 250 words, single paragraph.*
+
+Regulations require Certified Public Accountants (CPAs) to attest each audit report with a signature, but digitization makes reusing a stored signature image across reports — through administrative stamping or firm-level electronic signing — undermining individualized attestation. We build an end-to-end pipeline detecting such *non-hand-signed* signatures at scale: a Vision-Language Model identifies signature pages, YOLOv11 localizes signatures, ResNet-50 supplies deep features, and a dual-descriptor layer combines cosine similarity with an independent-minimum perceptual hash (dHash) to separate *style consistency* from *image reproduction*. Applied to 90,282 Taiwan audit reports (2013–2023), the pipeline yields 182,328 signatures from 758 CPAs; primary analyses are scoped to the Big-4 sub-corpus (437 CPAs; 150,442 signatures). Distributional diagnostics show that the apparent multimodality of the descriptor distribution dissolves under joint firm-mean centring and integer-tie jitter ($p$ rises to $0.35$), so no within-population bimodal antimode anchors the operational thresholds. We instead adopt an anchor-based inter-CPA coincidence-rate (ICCR) calibration at three units: per-comparison ($0.0006$ at cos$>0.95$; $0.0013$ at dHash$\leq 5$; $0.00014$ jointly), pool-normalised per-signature ($0.11$ under the deployed any-pair high-confidence rule), and per-document ($0.34$ for the operational HC+MC alarm). Firm heterogeneity is decisive: Firm A's per-document HC+MC alarm rate is $0.62$ versus $0.09$–$0.16$ at Firms B/C/D after pool-size adjustment, and under the deployed any-pair rule $77$–$99\%$ of inter-CPA collisions concentrate within the source firm — consistent with firm-level template-like reuse. We position the system as a specificity-proxy-anchored screening framework with human-in-the-loop review, not as a validated forensic detector; no calibrated error rates are reportable without signature-level ground truth.
+
+---
+
+# I. Introduction
+
+> *Target: ~1.5 pages double-column IEEE format. Double-blind: no author/institution info.*
+
+Financial audit reports serve as a critical mechanism for ensuring corporate accountability and investor protection. In Taiwan, the Certified Public Accountant Act (會計師法 §4) and the Financial Supervisory Commission's attestation regulations (查核簽證核准準則 §6) require certifying CPAs to affix their signature or seal (簽名或蓋章) to each audit report [1]. While the law permits either a handwritten signature or a seal, the CPA's attestation on each report is intended to represent a deliberate, individual act of professional endorsement for that specific audit engagement [2].
+
+The digitization of financial reporting has introduced a practice that complicates this intent. As audit reports are now routinely generated, transmitted, and archived as PDF documents, it is technically and operationally straightforward to reproduce a CPA's stored signature image across many reports rather than re-executing the signing act for each one. This reproduction can occur either through an administrative stamping workflow — in which scanned signature images are affixed by staff as part of the report-assembly process — or through a firm-level electronic signing system that automates the same step. We refer to signatures produced by either workflow collectively as *non-hand-signed*. Although this practice may fall within the literal statutory requirement of "signature or seal," it raises substantive concerns about audit quality, as an identically reproduced signature applied across hundreds of reports may not represent meaningful individual attestation for each engagement. The accounting literature has examined the audit-quality consequences of partner-level engagement transparency: studies of partner-signature mandates in the United Kingdom find measurable downstream effects [31], cross-jurisdictional evidence on individual partner signature requirements highlights similar quality channels [32], and Taiwan-specific evidence on mandatory partner rotation documents how individual-partner identification interacts with audit-quality outcomes [33]. Unlike traditional signature forgery, where a third party attempts to imitate another person's handwriting, non-hand-signing involves the legitimate signer's own stored signature being reused, and is visually invisible to report users at scale.
+
+The distinction between *non-hand-signing detection* and *signature forgery detection* is conceptually and technically important. The extensive body of research on offline signature verification [3]–[8] focuses almost exclusively on forgery detection — determining whether a questioned signature was produced by its purported author. In our context, identity is not in question; the CPA is indeed the legitimate signer. The question is whether the physical act of signing occurred for each individual report, or whether a single signing event was reproduced as an image across many reports. This detection problem differs fundamentally from forgery detection: while it does not require modeling skilled-forger variability, it introduces the distinct challenge of separating legitimate intra-signer consistency from image-level reproduction.
+
+A methodological concern shapes the research design. Many prior similarity-based classification studies rely on ad-hoc thresholds — declaring two images equivalent above a hand-picked cosine cutoff, for example — without principled statistical justification. Such thresholds are fragile in an archival-data setting. A defensible approach requires (i) explicit calibration of the operational thresholds against measurable negative-anchor evidence; (ii) diagnostic procedures that test whether the descriptor distribution itself supports a within-population threshold, including formal decomposition of apparent multimodality into between-group composition and integer-tie artefacts; (iii) annotation-free reporting of operational alarm rates at multiple analysis units (per-comparison, per-signature pool, per-document) with Wilson 95% confidence intervals; (iv) per-firm stratification of the reported rates to surface heterogeneity that aggregate metrics conceal; and (v) explicit disclosure of the unsupervised setting's limits — in particular, the inability to estimate true error rates without signature-level ground-truth labels.
+
+Despite the significance of the problem for audit quality and regulatory oversight, no prior work has specifically addressed non-hand-signing detection in financial audit documents at scale with these methodological safeguards. Woodruff et al. [9] developed an automated pipeline for signature analysis in corporate filings for anti-money-laundering investigations, but their work focused on author clustering rather than detecting image reuse. Copy-move forgery detection methods [10], [11] address duplicated regions within or across images but are designed for natural images and do not account for the specific characteristics of scanned document signatures. Research on near-duplicate image detection using perceptual hashing combined with deep learning [12], [13] provides relevant methodological foundations but has not been applied to document forensics or signature analysis. From the statistical side, the methods we adopt for distributional characterisation — the Hartigan dip test [37] and finite mixture modelling via the EM algorithm [40], [41], complemented by a Burgstahler-Dichev / McCrary density-smoothness diagnostic [38], [39] — have been developed in statistics and accounting-econometrics but have not been combined as a joint diagnostic toolkit for document-forensics threshold characterisation.
+
+In this paper we present a fully automated, end-to-end pipeline for detecting non-hand-signed CPA signatures in audit reports at scale, together with a multi-tool validation framework that explicitly discloses the unsupervised setting's limits. The pipeline processes raw PDF documents through (1) signature page identification with a Vision-Language Model; (2) signature region detection with a trained YOLOv11 object detector; (3) deep feature extraction via a pre-trained ResNet-50; (4) dual-descriptor similarity (cosine + independent-minimum dHash); (5) anchor-based threshold calibration at three units of analysis (per-comparison, pool-normalised per-signature, per-document) against an inter-CPA negative-anchor coincidence-rate proxy (§III-L); (6) firm-stratified per-rule reporting and a within-firm cross-CPA hit-matrix analysis (§III-L.4); (7) a composition decomposition that establishes the absence of a within-population bimodal antimode in the descriptor distributions (§III-I.4); and (8) a multi-tool unsupervised validation strategy with disclosed assumption-violation analysis (§III-M).
+
+The methodological reframing relative to earlier versions of this work is central to our v4.0 contribution. Earlier work in this lineage adopted a distributional path to thresholds — fitting accountant-level finite-mixture models and treating their marginal crossings as data-derived "natural" thresholds. v4.0 reports a composition decomposition diagnostic (§III-I.4) that overturns this reading: the apparent multimodality of the Big-4 accountant-level distribution is fully explained by between-firm location-shift effects (Firm A's mean dHash of $2.73$ versus Firms B/C/D's $6.46$, $7.39$, $7.21$) and integer mass-point artefacts on the integer-valued dHash axis. Once both confounds are removed (firm-mean centring plus uniform integer jitter), the Big-4 pooled dHash dip test yields $p_{\text{median}} = 0.35$ across five jitter seeds, eliminating the rejection. Within-firm signature-level cosine dip tests fail to reject in every individual Big-4 firm and in every individual mid/small firm with $\geq 500$ signatures (10 firms tested in Script 39c), and the corresponding within-firm jittered-dHash dip tests likewise fail to reject in all four Big-4 firms (Script 39d) and across a codex-verified read-only spike on the same ten mid/small firms ($0/10$ reject; §III-I.4). The descriptor distributions therefore contain no within-population bimodal antimode that could anchor an operational threshold.
+
+In place of distributional anchoring, v4.0 adopts an anchor-based inter-CPA coincidence-rate (ICCR) calibration. At the per-comparison unit, the inherited cos$>0.95$ operating point yields ICCR $= 0.00060$ on a $5 \times 10^5$-pair Big-4 sample (replicating v3.x's reported per-comparison rate of $0.0005$ under prior "FAR" terminology); the dHash$\leq 5$ structural cutoff yields ICCR $= 0.00129$ (v4 new); the joint rule cos$>0.95$ AND dHash$\leq 5$ yields joint ICCR $= 0.00014$ (any-pair semantics, matching the deployed extrema rule). At the pool-normalised per-signature unit, the same rule's effective coincidence rate is materially higher because the deployed classifier takes max-cosine and min-dHash over a same-CPA pool: pooled Big-4 any-pair ICCR is $0.1102$ (Wilson 95% CI $[0.1086, 0.1118]$; CPA-block bootstrap 95% $[0.0908, 0.1330]$). At the per-document unit, the operational HC$+$MC alarm fires on $33.75\%$ of Big-4 documents under the inter-CPA candidate-pool counterfactual.
+
+The pooled per-signature and per-document rates conceal striking firm heterogeneity. A logistic regression of the per-signature hit indicator on firm dummies (Firm A reference) and centred log pool size yields odds ratios of $0.053$ (Firm B), $0.010$ (Firm C), and $0.027$ (Firm D) — Firms B/C/D are an order of magnitude below Firm A even after controlling for the pool-size confound (Script 44). Cross-firm hit matrix analysis under the deployed any-pair rule shows within-firm collision concentrations of $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D (Table XXV; the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms). The pattern is consistent with firm-specific template, stamp, or document-production reuse mechanisms — though not by itself diagnostic of deliberate sharing. We retain the inherited Paper A v3.x five-way box rule as the operational classifier; v4.0's contribution is to characterise its multi-level coincidence behaviour against the inter-CPA negative anchor rather than to derive new thresholds.
+
+Three feature-derived scores converge on the per-CPA descriptor-position ranking with Spearman $\rho \geq 0.879$ (Script 38): the K=3 mixture posterior (now interpreted as a firm-compositional position score, not a mechanism cluster posterior; §III-J), a reverse-anchor cosine percentile relative to a strictly-out-of-target non-Big-4 reference, and the inherited box-rule less-replication-dominated rate. The three scores are deterministic functions of the same per-CPA descriptor pair, so the convergence is documented as internal consistency among feature-derived ranks rather than external validation. Hard ground truth for the *replicated* class is provided by 262 byte-identical signatures in the Big-4 subset (Firm A 145, Firm B 8, Firm C 107, Firm D 2), against which all three candidate checks achieve $0\%$ positive-anchor miss rate (Wilson 95% upper bound $1.45\%$). For the box rule this result is close to tautological at byte-identity; we discuss the conservative-subset caveat in §V-G.
+
+We apply this pipeline to 90,282 audit reports filed by publicly listed companies in Taiwan between 2013 and 2023, extracting and analyzing 182,328 individual CPA signatures from 758 unique accountants. The Big-4 sub-corpus comprises 437 CPAs and 150,442 signatures with both descriptors available.
+
+The contributions of this paper are:
+
+1. **Problem formulation.** We define non-hand-signing detection as distinct from signature forgery detection and frame it as a detection problem on intra-signer similarity distributions.
+
+2. **End-to-end pipeline.** We present a pipeline that processes raw PDF audit reports through VLM-based page identification, YOLO-based signature detection, ResNet-50 feature extraction, and dual-descriptor similarity computation, with automated inference and no manual intervention after initial training.
+
+3. **Dual-descriptor verification.** We demonstrate that combining deep-feature cosine similarity with independent-minimum dHash resolves the ambiguity between *style consistency* and *image reproduction*, and we validate the backbone choice through a feature-backbone ablation.
+
+4. **Composition decomposition disproves the distributional-threshold path.** We show via a 2×2 factorial diagnostic (firm-mean centring × integer-tie jitter) that the apparent multimodality of the Big-4 accountant-level descriptor distribution is fully attributable to between-firm location shifts and integer mass-point artefacts. The descriptor distributions contain no within-population bimodal antimode; "natural threshold" language in this lineage's prior work is not empirically supported.
+
+5. **Anchor-based multi-level inter-CPA coincidence-rate calibration.** We characterise the deployed five-way classifier at three units of analysis: per-comparison ICCR (cos$>0.95$: $0.0006$; dHash$\leq 5$: $0.0013$; joint: $0.00014$), pool-normalised per-signature ICCR ($0.11$ for the deployed any-pair high-confidence rule), and per-document ICCR ($0.34$ for the operational HC$+$MC alarm). We adopt "inter-CPA coincidence rate" as the metric name throughout and reserve "False Acceptance Rate" for terminology that requires ground-truth negative labels, which the corpus does not provide.
+
+6. **Firm heterogeneity quantification and within-firm cross-CPA collision concentration.** Per-firm rates differ by an order of magnitude after pool-size adjustment (Firm A's per-document HC$+$MC alarm at $0.62$ versus Firms B/C/D at $0.09$–$0.16$). Cross-firm hit matrix analysis shows within-firm collision concentrations of $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D under the deployed any-pair rule (the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms); the pattern is consistent with firm-specific template, stamp, or document-production reuse mechanisms — a descriptive finding about deployed-rule behaviour, not a claim of deliberate template sharing.
+
+7. **K=3 as descriptive firm-compositional partition; three-score convergent internal consistency.** We fit a K=3 Gaussian mixture as a descriptive partition of the Big-4 accountant-level distribution (no longer interpreted as three mechanism clusters). Three feature-derived scores agree on the per-CPA descriptor-position ranking at Spearman $\rho \geq 0.879$; we report this as internal consistency rather than external validation, given that the scores share the underlying descriptor pair.
+
+8. **Annotation-free positive-anchor validation and unsupervised validation ceiling.** We achieve $0\%$ positive-anchor miss rate (Wilson 95% upper bound $1.45\%$) on 262 byte-identical Big-4 signatures, with the conservative-subset caveat that byte-identical pairs are by construction near cos$=1$ and dHash$=0$. We frame the overall validation strategy as a multi-tool collection of ten partial-evidence diagnostics (§III-M Table XXVII), each with an explicitly disclosed untested assumption; their conjunction constitutes the unsupervised validation ceiling achievable on this corpus. We do not claim a validated forensic detector; we position the system as a specificity-proxy-anchored screening framework with human-in-the-loop review.
+
+The remainder of the paper is organized as follows. Section II reviews related work on signature verification, document forensics, perceptual hashing, and the statistical methods used. Section III describes the proposed methodology. Section IV presents the experimental results — distributional characterisation, mixture fits, convergent internal-consistency checks, leave-one-firm-out reproducibility, pixel-identity validation, and full-dataset robustness. Section V discusses the implications and limitations. Section VI concludes with directions for future work.
+
+---
+
+# II. Related Work
+
+> *Note for the Phase 4 review pass: §II is inherited substantively unchanged from v3.20.0 §II in the master manuscript, with one new paragraph added below. The unchanged content is not reproduced in this Phase 4 file; readers reviewing this draft should consult `paper/paper_a_related_work_v3.md` for the v3.20.0 §II text covering offline signature verification, near-duplicate detection, copy-move forgery detection, perceptual hashing, deep-feature similarity, and the statistical methods adopted (Hartigan dip test, finite mixture EM, Burgstahler-Dichev / McCrary density-smoothness diagnostic). The paragraph below is the only v4.0-specific §II addition.*
+
+**Addition for v4.0: leave-one-firm-out cross-validation in a small-cluster scope.** Cross-validation methodology in the leave-one-out tradition has been developed extensively in statistics since Stone [42] and Geisser [43], and modern surveys including Vehtari et al. [44] discuss its application to mixture models. In document-forensics calibration the technique has been used selectively, typically with the individual document or signature as the hold-out unit. Our application in §III-K differs in two respects from the standard usage: (i) the hold-out unit is the *firm* (not the individual CPA or signature), so the analysis directly probes cross-firm reproducibility of the fitted mixture rather than within-firm sampling variance; and (ii) the held-out predictions are interpreted as a *composition-sensitivity band* on the candidate mixture boundary, not as a sufficiency claim for the inherited five-way operational classifier (which is calibrated separately; §III-L). We treat LOOO drift as descriptive information about how the mixture characterisation moves when training composition changes, not as a pass/fail test for the operational classifier.
+
+---
+
+# V. Discussion
+
+## A. Non-Hand-Signing Detection as a Distinct Problem
+
+Non-hand-signing differs from forgery in that the questioned signature is produced by its legitimate signer's own stored image rather than by an impostor. The detection problem is therefore framed around *intra-signer image reproduction* rather than *inter-signer imitation*. This framing has analytical consequences. The within-CPA signature distribution is the analytical population of interest; the cross-CPA inter-class distribution is a *reference* against which intra-CPA similarity is interpreted, not the population to be modelled. This contrasts with most prior offline signature verification work, which treats genuine-versus-forged as the central two-class problem.
+
+## B. Per-Signature Similarity is a Continuous Quality Spectrum; the Accountant-Level Multimodality is Composition-Driven
+
+A central empirical finding of v3.x was that *per-signature* similarity does not admit a clean two-mechanism mixture: dip-test fails to reject unimodality at the signature level for Firm A, BIC prefers a 3-component fit, and BD/McCrary candidate transitions lie inside the high-similarity mode rather than between modes. v4.0 strengthens and extends this signature-level reading.
+
+The Big-4 accountant-level descriptor distribution does reject unimodality on both marginals at $p < 5 \times 10^{-4}$ (Script 34). v4.0's composition decomposition (§III-I.4; Scripts 39b–39e) shows that this rejection is fully attributable to two non-mechanistic sources: (a) between-firm location-shift effects on both axes — Firm A's mean dHash of $2.73$ versus Firms B/C/D's $6.46$, $7.39$, $7.21$ creates a multi-peaked pooled distribution that any single firm's distribution lacks — and (b) integer mass-point artefacts on the integer-valued dHash axis, which inflate the dip statistic against a continuous-density null. A 2×2 factorial diagnostic applied to the Big-4 pooled dHash (firm-mean centring × uniform integer jitter $[-0.5, +0.5]$, 5 jitter seeds) shows that the dip test fails to reject ($p_{\text{median}} = 0.35$, 0/5 seeds reject) when *both* corrections are applied; either correction alone leaves the rejection in place. Within-firm signature-level cosine and jittered-dHash dip tests fail to reject in every individual Big-4 firm and in every individual non-Big-4 firm with $\geq 500$ signatures tested (cosine: Scripts 39b/39c; jittered-dHash: Script 39d for Big-4 plus codex-verified read-only spike for the ten non-Big-4 firms; see §III-I.4). The descriptor distributions therefore lack a within-population bimodal antimode that could anchor an operational threshold. The K=2 / K=3 mixture fits are retained in §III-J as descriptive partitions of the joint Big-4 distribution that reflect firm-compositional structure, not as inferential evidence for two or three latent mechanism modes.
+
+## C. Firm A as the Templated End of Big-4 (Case Study, Not Calibration Anchor)
+
+Firm A is empirically the firm whose CPAs are most concentrated in the high-cosine, low-dHash corner of the Big-4 descriptor plane. In the Big-4 K=3 hard-posterior assignment (now interpreted as a firm-compositional position assignment; §III-J), Firm A accounts for $0\%$ of C1 (low-cos / high-dHash position) and $82.5\%$ of C3 (high-cos / low-dHash position); the opposite pattern holds at Firm C, which has the highest C1 concentration at $23.5\%$. Firm A also accounts for 145 of the 262 byte-identical signatures in the Big-4 byte-identical anchor of §IV-H (with Firm B 8, Firm C 107, Firm D 2). The additional v3.x finding that the 145 Firm A pixel-identical signatures span 50 distinct Firm A partners (of 180 registered), with 35 byte-identical matches across different fiscal years, is inherited from v3.20.0 §IV-F.1 / Script 28 / Appendix B byte-decomposition output and was not regenerated in v4.0's spike scripts; we retain those numbers by reference.
+
+In v4.0 we treat Firm A as a *templated-end case study* rather than as the calibration anchor for the operational threshold. Firm A enters the Big-4 anchor-based ICCR calibration on equal footing with the other three Big-4 firms (§III-L). The cross-firm hit matrix of §III-L.4 strengthens this framing: under the deployed any-pair rule, within-firm collision concentration is $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms). Firm A's high per-document HC$+$MC alarm rate of $0.62$ (versus Firms B/C/D's $0.09$–$0.16$) reflects high inter-CPA collision concentration under the deployed rule on real same-CPA pools, consistent with firm-specific template, stamp, or document-production reuse — though the inter-CPA-anchor analysis alone is not diagnostic of deliberate template sharing. The byte-level evidence of v3.x §IV-F.1 (Firm A's 145 pixel-identical signatures across $\sim 50$ distinct partners) provides direct evidence that firm-level template reuse does occur at Firm A; the within-firm collision pattern at all four Big-4 firms is consistent with that mechanism extending in milder form to Firms B/C/D.
+
+## D. K=2 / K=3 as Descriptive Firm-Compositional Partitions
+
+Leave-one-firm-out cross-validation of the Big-4 mixture fit reveals a sharp contrast between K=2 and K=3 behaviour. K=2 is unstable: across-fold cosine-crossing deviation is $0.028$, and holding Firm A out gives a fold rule (cos $> 0.938$, dHash $\leq 8.79$) that classifies $100\%$ of held-out Firm A in the upper component, while holding any non-Firm-A Big-4 firm out gives a fold rule near (cos $> 0.975$, dHash $\leq 3.76$) that classifies $0\%$ of the held-out firm in the upper component. The K=2 boundary is essentially a Firm-A-vs-others separator — direct evidence that the K=2 partition reflects firm-compositional rather than mechanistic structure.
+
+K=3 in contrast has a *reproducible component shape* at the descriptor-position level: across the four folds the C1 (low-cos / high-dHash) component cosine mean varies by at most $0.005$, the dHash mean by at most $0.96$, and the weight by at most $0.023$. Hard-posterior membership for the held-out firm is composition-sensitive (absolute differences $1.8$–$12.8$ pp across folds). Together with the §III-I.4 composition decomposition (no within-population bimodal antimode), the K=3 stability supports a descriptive reading: the Big-4 descriptor plane has a reproducible three-region partition that reflects how firm-compositional weight is distributed across the descriptor space, *not* a three-mechanism latent-class structure. We accordingly do not use K=3 hard-posterior membership as an operational classifier; we use it as the accountant-level descriptive summary that complements the deployed signature-level five-way classifier of §III-L.
+
+## E. Three-Score Convergent Internal-Consistency
+
+Three feature-derived scores agree on the per-CPA descriptor-position ranking at Spearman $\rho \geq 0.879$: the K=3 mixture posterior (a firm-compositional position score, not a mechanism cluster posterior); the reverse-anchor cosine percentile under a non-Big-4 reference distribution; and the inherited Paper A box-rule less-replication-dominated rate. The three scores are *not* statistically independent measurements — they are deterministic functions of the same per-CPA descriptor pair — so the convergence is documented as internal consistency rather than external validation against an independent ground truth (which the corpus does not provide for the hand-signed class). The strength of the convergence (all pairwise $|\rho| > 0.87$) and its persistence at the signature level (Cohen $\kappa = 0.87$ between per-CPA-fit and per-signature-fit K=3 binary labels) are nevertheless informative: per-CPA aggregation does not collapse the broad three-region ordering, and three different summarisations of the descriptor space produce broadly concordant per-CPA rankings, with a residual non-Firm-A disagreement (the reverse-anchor cosine percentile ranks Firm D fractionally above Firm C, while the mixture posterior and the box-rule rate rank Firm C highest among non-Firm-A firms).
+
+## F. Anchor-Based Multi-Level Calibration
+
+The operational specificity of the deployed five-way classifier is characterised at three units of analysis (§III-L), all against the same inter-CPA negative-anchor coincidence-rate proxy. The per-comparison ICCR replicates v3.x's per-comparison rate (cos$>0.95 \to 0.00060$) and extends it to the structural dimension (dHash$\leq 5 \to 0.00129$; joint $\to 0.00014$). The pool-normalised per-signature ICCR captures the deployed rule's effective per-signature rate under inter-CPA candidate-pool replacement ($0.1102$ pooled Big-4 any-pair HC), exposing that the per-comparison rate is not the deployed-rule rate at the per-signature classifier level: the deployed classifier takes max-cosine and min-dHash over a same-CPA pool of size $n_{\text{pool}}$, so the inter-CPA-equivalent rate scales approximately as $1 - (1 - p_{\text{pair}})^{n_{\text{pool}}}$ in the independence limit. The per-document ICCR aggregates to operational alarm-rate units: HC alone $0.18$, the operational HC$+$MC alarm $0.34$.
+
+Two additional findings refine the calibration story. First, the per-pair conditional ICCR for dHash$\leq 5$ given cos$>0.95$ is $0.234$ (Wilson 95% $[0.190, 0.285]$): given the cosine gate, the structural dimension provides further per-comparison specificity at $\sim 4.3\times$ refinement. Second, the alert-rate sensitivity analysis (§III-L.5; Script 46) shows the inherited HC threshold is locally sensitive rather than plateau-stable (local gradient $\approx 25\times$ the median for cosine, $\approx 3.8\times$ for dHash); stakeholders requiring different specificity-alert-yield operating points can derive thresholds by inverting the ICCR curves (a tighter rule cos$>0.95$ AND dHash$\leq 3$ on the same-pair joint gives per-signature ICCR $\approx 0.045$). The MC/HSC sub-band boundary at dHash$=15$, by contrast, *is* plateau-like (local-to-median ratio $\approx 0.08$), consistent with high-dHash-tail saturation.
+
+## G. Pixel-Identity as a Hard Positive Anchor; Inherited Inter-CPA Negative Anchor Reframed as Coincidence Rate
+
+The only hard ground-truth subset in the corpus is pixel-identical signatures: those whose nearest same-CPA match is byte-identical after crop and normalisation. Independent hand-signing cannot produce byte-identical images, so these signatures are conservative-subset ground truth for the *replicated* class. On the Big-4 subset ($n = 262$ pixel-identical signatures), all three candidate checks — the inherited box rule, the K=3 hard label, and the reverse-anchor metric with a prevalence-calibrated cut — achieve $0\%$ positive-anchor miss rate (Wilson 95% upper bound $1.45\%$). We caution that this result is necessary but not sufficient: for the box rule it is close to tautological, because byte-identical neighbours have cosine $\approx 1$ and dHash $\approx 0$, well inside the rule's high-confidence region. The corresponding signature-level *negative* anchor evidence is developed in §III-L.1 above (v4 spike: cos$>0.95$ per-comparison ICCR $= 0.00060$, replicating v3.20.0's reported $0.0005$ under prior "FAR" terminology). We frame the per-comparison rate as a specificity proxy under the assumption that inter-CPA pairs constitute a clean negative anchor, and we document in §III-L.4 that this assumption is partially violated by within-firm cross-CPA template-like collision structures.
+
+## H. Limitations
+
+Several limitations should be transparent. The first nine are v4.0-specific; the last five are inherited from v3.20.0 §V-G and still apply to the v4.0 pipeline.
+
+*No signature-level ground truth; no true error rates reportable.* The corpus does not contain labelled hand-signed or replicated classes at the signature level. We therefore cannot report False Rejection Rate, sensitivity, recall, Equal Error Rate, ROC-AUC, precision, or positive predictive value against ground truth. All quantitative rates reported in §III-L are inter-CPA negative-anchor coincidence rates (ICCRs) under the assumption that inter-CPA pairs constitute a clean negative anchor; this is a specificity proxy, not a calibrated specificity (§III-M).
+
+*Inter-CPA negative-anchor assumption is partially violated and the violation is firm-dependent.* The cross-firm hit matrix of §III-L.4 shows that under the deployed any-pair rule, within-firm collision concentration is $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms), consistent with firm-specific template, stamp, or document-production reuse. The inter-CPA-as-negative assumption is therefore not exactly satisfied — some inter-CPA pairs may share firm-level templates rather than being independent random matches. Our reported per-comparison ICCRs are best read as specificity-proxy rates under a partially-violated assumption, not as calibrated FARs. Because the violation is firm-dependent, Firm A's per-firm ICCR is more contaminated by within-firm sharing than Firms B/C/D's; the per-firm B/C/D rates of $0.09$–$0.16$ are therefore closer to a clean specificity estimate than the pooled rate, and the Firm A vs Firms B/C/D contrast reflects both genuine firm heterogeneity and a firm-dependent proxy-contamination gradient.
+
+*Scope.* The v4.0 primary analyses are scoped to the Big-4 sub-corpus. We did not perform the full per-signature pool-normalised ICCR analysis at the full $n = 686$ scope; the §IV-K full-dataset Spearman re-run shows the K=3 $+$ box-rule rank-convergence is preserved at $n = 686$ but does not validate the Big-4 operational ICCRs, the LOOO firm-fold structure, or the five-way operational classifier at the broader scope.
+
+*Pixel-identity is a conservative subset.* Byte-identical pairs are the easiest replicated cases, and for the inherited box rule the positive-anchor miss rate against byte-identical pairs is close to tautological (byte-identical $\Rightarrow$ cosine $\approx 1$, dHash $\approx 0$, well inside the high-confidence box). A score that fails the pixel-identity check would be disqualified, but passing the check does not guarantee correct behaviour on the broader replicated population (e.g., re-stamped or noisy-template-variant signatures).
+
+*Inherited rule components are not separately v4-validated.* The five-way classifier's moderate-confidence band (cos $> 0.95$ AND $5 < \text{dHash} \leq 15$), the style-consistency band ($\text{dHash} > 15$), and the document-level worst-case aggregation rule retain their v3.20.0 calibration and capture-rate evidence; v4.0's anchor-based ICCR calibration covers the binary high-confidence sub-rule (and its tightening alternatives such as dHash$\leq 3$), and the alert-rate sensitivity analysis (§III-L.5) characterises only the HC threshold. The MC and HSC sub-band boundaries are not separately re-validated by v4.0's diagnostic battery.
+
+*Deployed-rate excess is not a presumed true-positive rate.* The $\sim 44$-pp per-document gap between the observed deployed alert rate (HC: $0.62$ on real same-CPA pools) and the inter-CPA proxy rate (HC: $0.18$) cannot be interpreted as a presumed true-positive rate without additional assumptions that §III-M shows are unsafe (consistent within-CPA signing can exceed inter-CPA similarity at the cosine axis; within-firm template sharing inflates the inter-CPA proxy baseline). The gap is best read as a same-CPA repeatability signal.
+
+*A1 pair-detectability stipulation.* The per-signature detector requires at least one same-CPA pair to be near-identical when a CPA uses image replication. A1 is plausible for high-volume stamping or firm-level electronic signing but not guaranteed when a corpus contains only one observed replicated report for a CPA, multiple template variants used in parallel, or scan-stage noise that pushes a replicated pair outside the detection regime.
+
+*K=3 hard-posterior membership is composition-sensitive.* The K=3 hard-posterior membership for any single firm varies by up to $12.8$ pp across LOOO folds. This is documented as a composition-sensitivity band rather than failure, but it means K=3 hard labels are not used as v4.0 operational classifier output; they are reported only as accountant-level descriptive characterisation.
+
+*No partner-level mechanism attribution.* v4.0 reports population-level patterns; it does not perform partner-level mechanism attribution or report-level claims of intent. The signature-level outputs are signature-level quantities throughout. The within-firm cross-CPA collision concentration of §III-L.4 is consistent with template-like reuse but is not by itself diagnostic of deliberate sharing.
+
+*Transferred ImageNet features (inherited from v3.20.0).* The ResNet-50 feature extractor uses pre-trained ImageNet weights without signature-domain fine-tuning. While our backbone-ablation study (§IV-L, inherited from v3.20.0 §IV-I) and prior literature support the effectiveness of transferred ImageNet features for signature comparison, a signature-domain fine-tuned feature extractor could improve discriminative performance.
+
+*Red-stamp HSV preprocessing artifacts (inherited from v3.20.0).* The red stamp removal preprocessing uses simple HSV color-space filtering, which may introduce artifacts where handwritten strokes overlap with red seal impressions. Blended pixels are replaced with white, potentially creating small gaps in signature strokes that could reduce dHash similarity. This bias would push classifications toward false negatives rather than false positives.
+
+*Longitudinal scan / PDF / compression confounds (inherited from v3.20.0).* Scanning equipment, PDF generation software, and compression algorithms may have changed over the 2013–2023 study period, potentially affecting similarity measurements. While cosine similarity and dHash are designed to be robust to such variations, longitudinal confounds cannot be entirely excluded.
+
+*Source-exemplar misattribution in max/min pair logic (inherited from v3.20.0).* The max-cosine / min-dHash detection logic treats both ends of a near-identical same-CPA pair as non-hand-signed. In the rare case where one of the two documents contains a genuinely hand-signed exemplar that was subsequently reused as a stamping or e-signature template, the pair correctly identifies image reuse but misattributes non-hand-signed status to the source exemplar. This affects at most one source document per template variant per CPA and is not expected to be common.
+
+*Legal and regulatory interpretation (inherited from v3.20.0).* Whether non-hand-signing of a CPA's own stored signature constitutes a violation of signing requirements is a jurisdiction-specific legal question. Our technical analysis can inform such determinations but cannot resolve them.
+
+---
+
+# VI. Conclusion and Future Work
+
+We present a fully automated pipeline for detecting non-hand-signed CPA signatures in Taiwan-listed financial audit reports and a multi-tool framework for characterising and disclosing its operational behaviour at the Big-4 sub-corpus scope. The pipeline processes raw PDFs through VLM-based page identification, YOLO-based signature detection, ResNet-50 feature extraction, and dual-descriptor (cosine + independent-minimum dHash) similarity computation. The operational output is an inherited Paper A five-way per-signature classifier with worst-case document-level aggregation (§III-L). Applied to 90,282 audit reports filed between 2013 and 2023, the pipeline extracts 182,328 signatures from 758 CPAs, with the Big-4 sub-corpus (437 CPAs at accountant level; 150,442–150,453 signatures at signature level) as the primary analytical population.
+
+Our central methodological contributions are: (1) a composition decomposition (Scripts 39b–39e) that establishes the absence of a within-population bimodal antimode in the Big-4 descriptor distribution: the apparent multimodality dissolves under joint firm-mean centring and integer-tie jitter ($p_{\text{median}} = 0.35$), so distributional "natural-threshold" framings of the inherited operating points are not empirically supported; (2) an anchor-based inter-CPA coincidence-rate (ICCR) calibration at three units of analysis — per-comparison ($0.0006$ at cos$>0.95$; $0.0013$ at dHash$\leq 5$; $0.00014$ jointly), pool-normalised per-signature ($0.11$ for the deployed any-pair HC rule), and per-document ($0.34$ for the operational HC$+$MC alarm) — with explicit terminological replacement of "FAR" by "ICCR" given the unsupervised setting; (3) firm heterogeneity quantification: logistic regression with pool-size adjustment gives odds ratios $0.053$, $0.010$, $0.027$ for Firms B/C/D relative to Firm A reference, indicating a large multiplicative effect that pool-size differences do not explain; (4) cross-firm hit matrix evidence that under the deployed any-pair rule, within-firm collision concentration is $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms), consistent with firm-specific template, stamp, or document-production reuse mechanisms; (5) K=3 mixture demoted from "three mechanism clusters" to a descriptive firm-compositional partition; (6) three feature-derived scores converging on the per-CPA descriptor-position ranking at Spearman $\rho \geq 0.879$, reported as internal consistency rather than external validation; (7) $0\%$ positive-anchor miss rate on 262 byte-identical Big-4 signatures with the conservative-subset caveat; and (8) a ten-tool unsupervised-validation collection (§III-M Table XXVII) that explicitly discloses each tool's untested assumption and positions the system as an anchor-calibrated screening framework with human-in-the-loop review, not as a validated forensic detector.
+
+Future work falls in four directions. *First*, a small-scale human-rated validation set would enable direct ROC optimisation and provide signature-level ground truth that v4.0 fundamentally lacks; without such ground truth, no true error rates can be reported. *Second*, the within-firm collision concentration documented in §III-L.4 (any-pair $76.7$–$98.8\%$ across Big-4; same-pair joint $97.0$–$99.96\%$) invites a separate study to distinguish deliberate template sharing from passive firm-level production artefacts (shared scanners, common form templates, identical report-generation infrastructure) — a question the inter-CPA-anchor analysis alone cannot resolve. *Third*, the descriptive Firm A versus Firms B/C/D contrast (per-document HC$+$MC alarm $0.62$ vs $0.09$–$0.16$) — together with v3.x's byte-level evidence of 145 pixel-identical signatures across $\sim 50$ distinct Firm A partners — invites a companion analysis examining whether such firm-level signing patterns correlate with established audit-quality measures. *Fourth*, generalisation to mid- and small-firm contexts requires extending the anchor-based ICCR framework to scopes where firm-level LOOO folds are not available; the §III-I.4 composition diagnostics already document that the absence of within-population bimodality is corpus-universal, so the v4.0 calibration approach in principle generalises, but a full extension with cluster-robust uncertainty quantification is left as future work.
+
+---
+
+## Notes for Phase 4 close-out
+
+Items remaining for the Phase 4 close-out pass before §I, §II, §V, §VI prose can be moved into the manuscript master file:
+
+1. **Abstract word count.** Current draft is 243–244 words (shell `wc -w` on the paragraph returns 243; one-token tokenization difference depending on counter); both satisfy IEEE Access's $\leq 250$ word constraint with $\sim 6$ words of margin.
+2. **§I contributions list (8 items).** v3.20.0's contribution list had 7 items; v4.0's has 8 to reflect the Big-4 scope, K=3 descriptive role, and three-score convergence as separate contributions. Confirm whether the journal style supports 8 contributions or whether items can be merged.
+3. **§II Related Work LOOO citation.** A standard cross-validation citation for the LOOO addition is flagged "[add citation]" in the draft and needs to be filled with a specific reference (Geisser 1975 / Stone 1974 / a modern survey).
+4. **§V-G Limitations.** The seven limitations are listed flat; the journal style may prefer them grouped (scope vs ground-truth vs methodology) — consider reorganisation at copy-edit time.
+5. **§VI Future Work directions.** Four directions are listed; the third (audit-quality companion analysis) ties to the Paper B placeholder in the project memory and should be cross-checked for consistency with the planned Paper B framing.
+6. **Internal draft note + this close-out checklist.** Strip before submission packaging, per the across-paper "internal — remove before submission" policy applied to §III v6 and §IV v3.2 draft notes.

paper/v4/paper_a_results_v4_section_iv.md

@@ -0,0 +1,374 @@
+# Section IV. Results — v4.0 Draft v3.3 (post codex rounds 21–34)
+
+> **Draft note (2026-05-12, v3.2; internal — remove before submission).** This file replaces the §IV-A through §IV-H block of `paper/paper_a_results_v3.md` (v3.20.0) with the Big-4 reframed structure. Section IV expands from 8 sub-sections in v3.20.0 to 12 sub-sections in v4.0 (A through L) to mirror the §III-G..L lineage. **Table-numbering scheme**: the v4 manuscript uses Tables V through XVIII (plus Table XV-B for document-level worst-case counts) for the new v4 Big-4 results; inherited v3.x tables are cited only as "v3.20.0 Table N" with their original v3 number and are *not* renumbered into the v4 sequence. No v4 Table IV is printed; the inherited v3.20.0 Table IV (per-firm detection counts) remains a v3.x reference rather than a v4 table. **Anonymisation**: the Big-4 firms are pseudonymously labelled Firm A through Firm D throughout the manuscript body; real names are not printed in v4 tables or prose. The v3 → v3.1 → v3.2 revision history is: v3 (post round 23) made the table-numbering scheme and anonymisation policy decisions and applied 14 presentation fixes; v3.1 (post round 24) tightened the close-out checklist; v3.2 (post round 25) finalises this draft note. Empirical anchors trace to Scripts 32–42 on branch `paper-a-v4-big4`; the §III provenance table covers the methodology-side citations and §IV adds new tables for the v4.0-specific results.
+
+## A. Experimental Setup
+
+The signature-detection and feature-extraction pipeline (§III-A through §III-F) was executed on the full TWSE MOPS audit-report corpus (90,282 PDFs spanning 2013–2023; §III-B). Detection and embedding ran on RTX 4090 (CUDA, deterministic forward inference, fixed seed); the v4.0 statistical analyses ran on Apple Silicon (MPS / CPU). Random seeds are fixed (`SEED = 42`) across the v4.0 spike scripts 32–42 for reproducibility. The signature_analysis SQLite snapshot at `/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db` is treated as frozen; no v4.0 result re-ingests source PDFs.
+
+The v4.0 primary analyses (§IV-D through §IV-J) are scoped to the Big-4 sub-corpus (Firms A–D, $n = 437$ CPAs with $n_{\text{sig}} \geq 10$, totalling 150,442 signatures with both descriptors available) per the methodology choice articulated in §III-G. The §IV-K Full-Dataset Robustness section reports the full-dataset (686 CPAs) variant of the K=3 mixture + Paper A box-rule Spearman analysis as a cross-scope robustness check.
+
+## B. Signature Detection Performance
+
+The detection metrics are inherited unchanged from v3.20.0 §IV-B. v3.20.0 reports: VLM screening identified 86,072 documents with signature pages; 12 corrupted PDFs were excluded; YOLOv11n batch inference processed the remaining 86,071 documents; 85,042 of these yielded at least one signature detection; the total extracted-signature count is 182,328 (v3.20.0 Table III). Per-firm counts of detected signatures are reported in v3.20.0 Table IV. v4.0 does not renumber the v3.x detection tables into the v4 sequence; v3.20.0 Tables III and IV are cited by their original numbers.
+
+The Big-4 subset of the detection output yields 150,442 signatures with both descriptors (cosine and independent dHash) successfully computed; this is the per-signature population used in all §IV v4 primary analyses (§IV-D through §IV-J).
+
+## C. All-Pairs Intra-vs-Inter Class Distribution Analysis
+
+The all-pairs intra-vs-inter class distribution analysis (KDE crossover at $\overline{\text{cos}} = 0.837$; v3.20.0 §IV-C, v3.20.0 Table V) is inherited unchanged. This analysis was computed on the full corpus (not Big-4-restricted) and remains the source of the Uncertain / Likely-hand-signed boundary used by the §III-L five-way per-signature classifier (cosine $\leq 0.837 \Rightarrow$ Likely-hand-signed, matching Script 42's `cos <= 0.837` rule definition). v4.0 makes no scope-specific re-derivation of this boundary; the all-pairs cross-class crossover is a corpus-wide reference and is not restated as a v4.0 finding. v3.20.0 Table V is cited by its original number and is not renumbered into the v4 sequence.
+
+## D. Big-4 Accountant-Level Distributional Characterisation
+
+This section reports the empirical evidence for §III-I's distributional diagnostics at the Big-4 accountant level. All numbers below are direct re-statements from Scripts 32 / 34. The accountant-level dip-test rejection reported in Table V is, per §III-I.4 (Scripts 39b–39e), fully attributable to between-firm location shifts and integer mass-point artefacts rather than to within-population bimodality; the v4-new composition-decomposition diagnostics that establish this finding are tabulated in §IV-M below alongside the anchor-based ICCR calibration.
+
+**Table V.** Hartigan dip-test results, accountant-level marginals (Big-4 primary; comparison scopes from Script 32).
+
+| Population | $n$ CPAs | $p_{\text{cos}}$ | $p_{\text{dHash}}$ | Interpretation |
+|---|---|---|---|---|
+| **Big-4 pooled (primary)** | 437 | $< 5 \times 10^{-4}$ | $< 5 \times 10^{-4}$ | reject unimodality on both axes |
+| Firm A pooled alone | 171 | 0.992 | 0.924 | unimodal |
+| Firms B + C + D pooled | 266 | 0.998 | 0.906 | unimodal |
+| All non-Firm-A pooled | 515 | 0.998 | 0.907 | unimodal |
+
+Bootstrap implementation: $n_{\text{boot}} = 2000$; for the Big-4 cells, no bootstrap replicate exceeded the observed dip statistic, so the empirical $p$-value is bounded above by the bootstrap resolution $1 / 2000 = 5 \times 10^{-4}$ (Script 34 reports this as $p = 0.0000$; we report $p < 5 \times 10^{-4}$ to reflect the resolution). Single-firm dip statistics for Firms B, C, and D were not separately computed.
+
+**Table VI.** Burgstahler-Dichev / McCrary density-smoothness diagnostic on accountant-level marginals (cosine in 0.002 bins; dHash in integer bins; $\alpha = 0.05$, two-sided).
+
+| Population | Cosine: significant transition? | dHash: significant transition? |
+|---|---|---|
+| **Big-4 pooled (primary)** | none ($p > 0.05$) | none ($p > 0.05$) |
+| Firm A pooled alone | none | none |
+| Firms B + C + D pooled | none | one transition at $\overline{\text{dHash}} = 10.8$ |
+| All non-Firm-A pooled | none | one transition at $\overline{\text{dHash}} = 6.6$ |
+
+The Big-4-scope null on both axes is consistent with the §IV-E mixture evidence: the K=3 components overlap in their tails rather than separating sharply, so a local-discontinuity test does not flag a transition. Outside Big-4, dHash transitions appear in some subsets but no cosine transition is identified in any tested subset (Script 32 sweeps; pre-2018 and post-2020 stratified variants exhibit dHash transitions at varying locations). These off-Big-4 dHash transitions are scope-dependent and are not used as v4.0 operational thresholds; we do not claim a specific structural interpretation for them without an explicit bin-width sensitivity sweep at those scopes.
+
+## E. Big-4 K=2 / K=3 Mixture Fits
+
+This section reports the K=2 and K=3 2D Gaussian mixture fits to the Big-4 accountant-level distribution and the bootstrap stability of their marginal crossings.
+
+**Table VII.** Big-4 K=2 mixture components (descriptive partition; not mechanism clusters per §III-J) and marginal-crossing bootstrap 95% confidence intervals.
+
+| K=2 component | $\overline{\text{cos}}$ | $\overline{\text{dHash}}$ | weight |
+|---|---|---|---|
+| K=2-a (low-cos / high-dHash position) | 0.954 | 7.14 | 0.689 |
+| K=2-b (high-cos / low-dHash position) | 0.983 | 2.41 | 0.311 |
+
+Marginal crossings (point + bootstrap 95% CI, $n_{\text{boot}} = 500$):
+
+| Axis | Point | Bootstrap median | 95% CI | CI half-width |
+|---|---|---|---|---|
+| cos | 0.9755 | 0.9754 | $[0.9742, 0.9772]$ | 0.0015 |
+| dHash | 3.755 | 3.763 | $[3.476, 3.969]$ | 0.246 |
+
+$\text{BIC}(K{=}2) = -1108.45$ (Script 34).
+
+**Table VIII.** Big-4 K=3 mixture components (descriptive firm-compositional partition per §III-J; not mechanism clusters).
+
+| K=3 component | $\overline{\text{cos}}$ | $\overline{\text{dHash}}$ | weight | descriptive position |
+|---|---|---|---|---|
+| C1 | 0.9457 | 9.17 | 0.143 | low-cos / high-dHash corner |
+| C2 | 0.9558 | 6.66 | 0.536 | central region |
+| C3 | 0.9826 | 2.41 | 0.321 | high-cos / low-dHash corner |
+
+$\text{BIC}(K{=}3) = -1111.93$, lower than $K{=}2$ by $3.48$ (mild support; not by itself decisive). The full-fit K=3 baseline above is reproduced in Scripts 35, 37, and 38 with identical hyperparameters; Script 37 additionally fits K=3 on each leave-one-firm-out training set (those fold-specific components differ from the full-fit baseline by design and are reported separately in §IV-G Table XIII). Operational use of the K=2 / K=3 fits is governed by §III-J and §III-L; §IV-G reports the LOOO reproducibility evidence that motivates reporting both fits descriptively.
+
+## F. Convergent Internal-Consistency Checks
+
+This section reports the empirical evidence for §III-K's three-score internal-consistency analysis. We re-emphasise the §III-K caveat: the three scores are deterministic functions of the same per-CPA descriptor pair $(\overline{\text{cos}}_a, \overline{\text{dHash}}_a)$ and are *not statistically independent measurements*. The pairwise correlations document internal consistency among feature-derived ranks rather than external validation against an independent ground truth.
+
+**Table IX.** Per-CPA Spearman rank correlations among three feature-derived scores, Big-4, $n = 437$.
+
+| Score pair | Spearman $\rho$ | $p$-value |
+|---|---|---|
+| K=3 P(C1) vs Paper A box-rule less-replication-dominated rate | $+0.9627$ | $< 10^{-248}$ |
+| Reverse-anchor cosine percentile vs Paper A box-rule less-replication-dominated rate | $+0.8890$ | $< 10^{-149}$ |
+| K=3 P(C1) vs Reverse-anchor cosine percentile | $+0.8794$ | $< 10^{-142}$ |
+
+(Source: Script 38.) Reverse-anchor reference: 2D Gaussian fit by MCD (support fraction 0.85) on $n = 249$ non-Big-4 CPAs; reference centre $\overline{\text{cos}} = 0.935$, $\overline{\text{dHash}} = 9.77$.
+
+**Table X.** Per-firm summary across the three feature-derived scores, Big-4.
+
+| Firm | $n$ CPAs | mean $P(\text{C1})$ | mean reverse-anchor score | mean Paper A less-replication-dominated rate |
+|---|---|---|---|---|
+| Firm A | 171 | 0.0072 | $-0.9726$ | 0.1935 |
+| Firm B | 112 | 0.1410 | $-0.8201$ | 0.6962 |
+| Firm C | 102 | 0.3110 | $-0.7672$ | 0.7896 |
+| Firm D | 52 | 0.2406 | $-0.7125$ | 0.7608 |
+
+(Source: Script 38 per-firm summary; reverse-anchor score is sign-flipped so that *higher* values indicate deeper into the reference left tail = less replication-dominated relative to the non-Big-4 reference.)
+
+The three scores agree on placing Firm A as the most replication-dominated and the three non-Firm-A firms as less replication-dominated. The K=3 posterior P(C1) and the box-rule less-replication-dominated rate (Score 1 and Score 3) place Firm C at the least-replication-dominated end of Big-4; the reverse-anchor cosine percentile (Score 2) ranks Firm D fractionally above Firm C. This residual within-Big-4-non-A disagreement is a design feature of the reverse-anchor metric: Score 2 measures only the marginal cosine percentile under the non-Big-4 reference, so a firm with a slightly higher cosine but a markedly different dHash distribution (Firm D vs Firm C) can score higher on Score 2 while scoring lower on Scores 1 and 3, both of which use both descriptors.
+
+**Table XI.** Per-signature Cohen $\kappa$ (binary collapse, replication-dominated vs less-replication-dominated), $n = 150{,}442$ Big-4 signatures.
+
+| Pair | Cohen $\kappa$ |
+|---|---|
+| Paper A binary high-confidence box rule (cos $> 0.95$ AND dHash $\leq 5$) vs per-CPA K=3 hard label | 0.662 |
+| Paper A binary high-confidence box rule vs per-signature K=3 hard label | 0.559 |
+| Per-CPA K=3 hard label vs per-signature K=3 hard label | 0.870 |
+
+(Source: Script 39; verdict label `SIG_CONVERGENCE_MODERATE`.) Per-signature K=3 components ($n = 150{,}442$) sorted by ascending cosine: $(0.928, 9.75, 0.146)$ / $(0.963, 6.04, 0.582)$ / $(0.989, 1.27, 0.272)$, an absolute cosine drift of $0.018$ in C1 and $0.006$ in C3 relative to the per-CPA fit. These convergence checks cover only the binary high-confidence rule (cos $> 0.95$ AND dHash $\leq 5$); the five-way classifier's moderate-confidence band ($5 < \text{dHash} \leq 15$) inherits its v3.x calibration and capture-rate evaluation (§IV-J).
+
+## G. Leave-One-Firm-Out Reproducibility
+
+This section reports the firm-level cross-validation evidence motivating §III-J's "K=3 descriptive, not operational" framing.
+
+**Table XII.** K=2 leave-one-firm-out across the four Big-4 folds.
+
+| Held-out firm | $n_{\text{train}}$ | $n_{\text{held}}$ | Fold rule (cos cut, dHash cut) | Held-out classified as templated by fold rule |
+|---|---|---|---|---|
+| Firm A | 266 | 171 | cos $> 0.9380$ AND dHash $\leq 8.79$ | $171 / 171 = 100.00\%$ ($95\%$ Wilson $[97.80\%, 100.00\%]$) |
+| Firm B | 325 | 112 | cos $> 0.9744$ AND dHash $\leq 3.98$ | $0 / 112 = 0\%$ ($95\%$ Wilson $[0\%, 3.32\%]$) |
+| Firm C | 335 | 102 | cos $> 0.9752$ AND dHash $\leq 3.75$ | $0 / 102 = 0\%$ ($95\%$ Wilson $[0\%, 3.63\%]$) |
+| Firm D | 385 | 52 | cos $> 0.9756$ AND dHash $\leq 3.74$ | $0 / 52 = 0\%$ ($95\%$ Wilson $[0\%, 6.88\%]$) |
+
+(Source: Script 36.) Across-fold cosine crossing: pairwise range $[0.9380, 0.9756]$, range = $0.0376$; max absolute deviation from the across-fold mean is $0.028$. This exceeds the report's $0.005$ across-fold stability tolerance by $5.6\times$ and is much larger than the full-Big-4 bootstrap CI half-width of $0.0015$. Together with the all-or-nothing held-out classification pattern (Firm A held out $\Rightarrow$ all held-out CPAs templated; any non-Firm-A firm held out $\Rightarrow$ none templated), this indicates the K=2 boundary is essentially a Firm-A-vs-others separator rather than a within-Big-4 mechanism boundary.
+
+**Table XIII.** K=3 leave-one-firm-out: C1 component shape and held-out membership.
+
+| Held-out firm | C1 cos (fit) | C1 dHash (fit) | C1 weight (fit) | Held-out C1 hard-label rate | Full-Big-4 baseline C1% | Absolute difference |
+|---|---|---|---|---|---|---|
+| Full-Big-4 baseline | 0.9457 | 9.17 | 0.143 | — | — | — |
+| Firm A held out | 0.9425 | 10.13 | 0.145 | $4.68\%$ | $0.00\%$ | $4.68$ pp |
+| Firm B held out | 0.9441 | 9.16 | 0.127 | $7.14\%$ | $8.93\%$ | $1.76$ pp |
+| Firm C held out | 0.9504 | 8.41 | 0.126 | $36.27\%$ | $23.53\%$ | $12.77$ pp |
+| Firm D held out | 0.9439 | 9.29 | 0.120 | $17.31\%$ | $11.54\%$ | $5.81$ pp |
+
+(Source: Script 37; verdict label `P2_PARTIAL`.) Component shape is reproducible across folds: max deviation of C1 cosine = $0.005$, C1 dHash = $0.96$, C1 weight = $0.023$. Hard-posterior membership for the held-out firm varies: max absolute difference from the full-Big-4 baseline is $12.77$ pp at the Firm C held-out fold, exceeding the report's $5$ pp viability bar. We accordingly do not use K=3 hard-posterior membership as an operational classifier label (§III-J, §III-L).
+
+## H. Pixel-Identity Positive-Anchor Miss Rate
+
+This section reports the only hard-ground-truth subset analysis available in the corpus: the positive-anchor miss rate against $n = 262$ Big-4 signatures whose nearest same-CPA match is byte-identical after crop and normalisation. Independent hand-signing cannot produce pixel-identical images, so byte-identical signatures are conservative-subset ground truth for the *replicated* class. The analysis is one-sided (positive-anchor only); a paired false-alarm rate against a hand-signed negative anchor is not available because no signature-level hand-signed ground truth exists in the corpus (§III-K item 4).
+
+**Table XIV.** Positive-anchor miss rate, $n = 262$ Big-4 byte-identical signatures.
+
+| Classifier | Misclassified as less-replication-dominated | Miss rate | Wilson 95% CI |
+|---|---|---|---|
+| Paper A binary high-confidence box rule (cos $> 0.95$ AND dHash $\leq 5$) | $0 / 262$ | $0\%$ | $[0\%, 1.45\%]$ |
+| K=3 per-CPA hard label (C3 = high-cos / low-dHash; descriptive) | $0 / 262$ | $0\%$ | $[0\%, 1.45\%]$ |
+| Reverse-anchor (prevalence-calibrated cut) | $0 / 262$ | $0\%$ | $[0\%, 1.45\%]$ |
+
+(Source: Script 40.) Per-firm breakdown of the byte-identical subset: Firm A 145; Firm B 8; Firm C 107; Firm D 2. All three candidate scores correctly assign every byte-identical signature to the replicated class.
+
+We caution that for the Paper A box rule this result is close to tautological (byte-identical nearest-neighbour signatures have cosine $\approx 1$ and dHash $\approx 0$, well inside the rule's high-confidence region); v3.20.0 §V-F discusses this conservative-subset caveat at length and we retain that discussion. The reverse-anchor cut is chosen by *prevalence calibration* against the inherited box rule's overall replicated rate of $49.58\%$ across Big-4 signatures; this is a documented v4.0 limitation since no signature-level hand-signed ground truth exists to permit direct ROC optimisation.
+
+## I. Inter-CPA Pair-Level Coincidence Rate (Big-4 spike + inherited corpus-wide)
+
+The signature-level inter-CPA pair-level coincidence-rate analysis (reported in v3.20.0 §IV-F.1, Table X as "FAR") is inherited and extended in v4.0. v4.0 retroactively reframes the metric as **inter-CPA pair-level coincidence rate (ICCR)** rather than "False Acceptance Rate" because the corpus does not provide signature-level ground-truth negative labels; the inter-CPA negative-anchor assumption underpinning the metric is itself partially violated by within-firm cross-CPA template-like collision structures (§III-L.4). The v3.20.0 corpus-wide spike on $\sim 50{,}000$ inter-CPA pairs reported a per-comparison rate of $0.0005$ (Wilson 95% CI $[0.0003, 0.0007]$) at the cosine cut $0.95$.
+
+v4.0 additionally reports the §III-L.1 Big-4-scope spike at higher sample size ($5 \times 10^5$ inter-CPA pairs; Script 40b), which replicates and extends the v3 result and adds the structural dimension (dHash) and joint-rule rates. The §III-L.1 numbers are referenced rather than duplicated here; the consolidated v4-new ICCR calibration appears in §IV-M Tables XXI–XXVI.
+
+## J. Five-Way Per-Signature + Document-Level Classification Output
+
+This section reports the §III-L five-way per-signature + document-level worst-case classifier output on the Big-4 sub-corpus. The five-way category definitions are inherited unchanged from v3.20.0 §III-K (now §III-L); see §III-L for the cosine and dHash cuts.
+
+**Table XV.** Five-way per-signature category counts, Big-4 sub-corpus, $n = 150{,}442$ classified.
+
+| Category | Long name | $n$ signatures | % of classified |
+|---|---|---|---|
+| HC | High-confidence non-hand-signed | 74,593 | 49.58% |
+| MC | Moderate-confidence non-hand-signed | 39,817 | 26.47% |
+| HSC | High style consistency | 314 | 0.21% |
+| UN | Uncertain | 35,480 | 23.58% |
+| LH | Likely hand-signed | 238 | 0.16% |
+
+(Source: Script 42; 11 of 150,453 loaded Big-4 signatures lacked one or both descriptors and were excluded. The $150{,}442$ vs $150{,}453$ distinction — descriptor-complete vs vector-complete — recurs across §IV: descriptor-complete analyses (§IV-D through §IV-J, all using accountant-level aggregates or per-signature category counts derived from the same 150,442-signature substrate) use $n = 150{,}442$; vector- or pair-recomputed analyses (§IV-M.2 Table XXI, §IV-M.3 Table XXII, §IV-M.5 Tables XXIV–XXV; Scripts 40b, 43, 44) use $n = 150{,}453$ because their pair- or pool-level computations load all vector-complete signatures including those failing the descriptor-complete filter. See §III-G for the sample-size reconciliation.)
+
+**Per-firm five-way breakdown (% within firm).**
+
+| Firm | HC | MC | HSC | UN | LH | total signatures |
+|---|---|---|---|---|---|---|
+| Firm A | 81.70% | 10.76% | 0.05% | 7.42% | 0.07% | 60,448 |
+| Firm B | 34.56% | 35.88% | 0.29% | 29.09% | 0.18% | 34,248 |
+| Firm C | 23.75% | 41.44% | 0.38% | 34.21% | 0.22% | 38,613 |
+| Firm D | 24.51% | 29.33% | 0.22% | 45.65% | 0.29% | 17,133 |
+
+(Source: Script 42 per-firm cross-tab.) The per-firm pattern qualitatively aligns with the K=3 cluster cross-tab of Table XVI: Firm A's signatures concentrate in the HC band (81.70%) while its CPAs concentrate at the accountant level in the K=3 C3 (high-cos / low-dHash) component (82.46%; Table XVI). These two figures address different units (per-signature classification vs per-CPA hard cluster assignment) and are not directly comparable as a like-for-like consistency check; we report the qualitative alignment but do not infer a numerical equivalence. The three non-Firm-A Big-4 firms have markedly lower HC rates than Firm A and substantially higher Uncertain rates, with Firm D having the highest Uncertain rate (45.65%).
+
+**Document-level worst-case aggregation.** Each audit report typically carries two certifying-CPA signatures. We aggregate signature-level outcomes to document-level labels using the v3.20.0 worst-case rule (HC > MC > HSC > UN > LH; §III-L). v4.0 does not change this aggregation rule; only the population over which it is computed changes (Big-4 subset).
+
+**Table XIX.** Document-level worst-case category counts, Big-4 sub-corpus, $n = 75{,}233$ unique PDFs.
+
+| Category | Long name | $n$ documents | % |
+|---|---|---|---|
+| HC | High-confidence non-hand-signed | 46,857 | 62.28% |
+| MC | Moderate-confidence non-hand-signed | 19,667 | 26.14% |
+| HSC | High style consistency | 167 | 0.22% |
+| UN | Uncertain | 8,524 | 11.33% |
+| LH | Likely hand-signed | 18 | 0.02% |
+
+(Source: Script 42 document-level table; 379 of 75,233 PDFs carried signatures from more than one Big-4 firm and are reported in the single-firm-PDF per-firm breakdown of the script CSV but pooled into the overall counts here.)
+
+**Per-firm document-level breakdown (single-firm PDFs only).**
+
+| Firm | HC | MC | HSC | UN | LH | total docs |
+|---|---|---|---|---|---|---|
+| Firm A | 27,600 | 1,857 | 7 | 758 | 4 | 30,226 |
+| Firm B | 8,783 | 6,079 | 57 | 2,202 | 6 | 17,127 |
+| Firm C | 7,281 | 8,660 | 77 | 3,099 | 5 | 19,122 |
+| Firm D | 3,100 | 2,838 | 22 | 2,416 | 3 | 8,379 |
+
+(Source: Script 42; mixed-firm PDFs $n = 379$ excluded from the per-firm rows but included in the overall counts above.)
+
+The five-way **moderate-confidence non-hand-signed** band (cos $> 0.95$ AND $5 < \text{dHash} \leq 15$) inherits its v3.x calibration; it is **not separately validated by Scripts 38–40**, which evaluated only the binary high-confidence rule (cos $> 0.95$ AND dHash $\leq 5$). v4.0 does not re-derive the moderate-band cuts on the Big-4 subset; we report the Table XV per-firm MC proportions (10.76% / 35.88% / 41.44% / 29.33% across Firms A through D) descriptively. The v3.20.0 capture-rate calibration evidence for the moderate band (v3.20.0 Tables IX, XI, XII, XII-B) is carried into v4.0 by reference and not regenerated on the Big-4 subset. We do not claim that the MC-band per-firm ordering above is a separate validation of the §III-K Spearman convergence, since MC occupancy is not a monotone function of the per-CPA less-replication-dominated ranking (e.g., Firm D's MC fraction is lower than Firm B's while Firm D's reverse-anchor score ranks it as less replication-dominated than Firm B).
+
+**Table XVI.** Firm × K=3 cluster cross-tabulation, Big-4 sub-corpus.
+
+| Firm | $n$ | C1 (low-cos / high-dHash) | C2 (central) | C3 (high-cos / low-dHash) | C1 % | C3 % |
+|---|---|---|---|---|---|---|
+| Firm A | 171 | 0 | 30 | 141 | $0.00\%$ | $82.46\%$ |
+| Firm B | 112 | 10 | 102 | 0 | $8.93\%$ | $0.00\%$ |
+| Firm C | 102 | 24 | 77 | 1 | $23.53\%$ | $0.98\%$ |
+| Firm D | 52 | 6 | 45 | 1 | $11.54\%$ | $1.92\%$ |
+
+(Source: Script 35.) The cross-tab is the accountant-level descriptive output of the K=3 mixture (§III-J / §IV-E). It is reported here as a complement to the five-way per-signature classifier (Table XV), not as an operational classifier output. Reading: Firm A's CPAs are concentrated in the C3 (high-cos / low-dHash) component (no Firm A CPAs in C1); Firm C has the highest C1 (low-cos / high-dHash) concentration of the Big-4 (C1 fraction $23.5\%$); Firms B and D sit between A and C on the K=3 hard-label ordering, broadly consistent with the per-firm Spearman ordering of Table X (with the within-Big-4-non-A reverse-anchor disagreement noted there).
+
+**Document-level worst-case aggregation outputs are reported in Table XIX above.**
+
+## K. Full-Dataset Robustness (light scope)
+
+This section reports the v4.0 reproducibility cross-check at the full accountant scope ($n = 686$ CPAs, Big-4 plus mid/small firms). The scope of §IV-K is deliberately narrow: we re-run only the K=3 mixture + Paper A operational-rule per-CPA less-replication-dominated rate analysis, sufficient to demonstrate that the v4.0 K=3 + Paper A convergence reproduces at the wider scope. The §III-L five-way classifier and the §IV-G LOOO analyses are not re-run at the full scope. The five-way moderate-confidence band is documented as inherited from v3.x calibration in §IV-J.
+
+**Table XVII.** K=3 component comparison, Big-4 sub-corpus vs full dataset.
+
+| K=3 component | Big-4 (n=437) cos / dHash / weight | Full (n=686) cos / dHash / weight | Drift Big-4 → Full |
+|---|---|---|---|
+| C1 (low-cos / high-dHash) | 0.9457 / 9.17 / 0.143 | 0.9278 / 11.17 / 0.284 | $\lvert\Delta\rvert$ cos 0.018, dHash 1.99, wt 0.141 |
+| C2 (central) | 0.9558 / 6.66 / 0.536 | 0.9535 / 6.99 / 0.512 | $\lvert\Delta\rvert$ cos 0.002, dHash 0.33, wt 0.024 |
+| C3 (high-cos / low-dHash) | 0.9826 / 2.41 / 0.321 | 0.9826 / 2.40 / 0.205 | $\lvert\Delta\rvert$ cos 0.000, dHash 0.01, wt 0.117 |
+
+(Source: Script 41; full-dataset $\text{BIC}(K{=}3) = -792.31$ vs Big-4 $\text{BIC}(K{=}3) = -1111.93$; BIC values are not directly comparable across different $n$ and are reported only for completeness.)
+
+**Table XVIII.** Spearman rank correlation between K=3 P(C1) and Paper A operational less-replication-dominated rate, Big-4 sub-corpus vs full dataset.
+
+| Scope | $n$ CPAs | Spearman $\rho$ (P(C1) vs Paper A less-replication-dominated rate) | $p$-value |
+|---|---|---|---|
+| Big-4 (primary) | 437 | $+0.9627$ | $< 10^{-248}$ |
+| Full dataset | 686 | $+0.9558$ | $< 10^{-300}$ |
+| $\lvert\rho_{\text{full}} - \rho_{\text{Big-4}}\rvert$ | — | $0.0069$ | — |
+
+(Source: Script 41.)
+
+**Reading.** The K=3 component ordering and the strong Spearman convergence between K=3 P(C1) and the Paper A box-rule less-replication-dominated rate are preserved at the full scope. Component centres shift modestly: C3 (high-cos / low-dHash) is essentially unchanged in centre but loses weight $0.117$ as the full population includes more non-templated CPAs (mid/small firms); C1 (low-cos / high-dHash) gains weight $0.141$ and shifts to lower cosine and higher dHash (centre $(0.928, 11.17)$ vs Big-4 $(0.946, 9.17)$) as the broader population includes mid/small-firm CPAs landing toward the low-cos / high-dHash region that the Big-4-primary scope deliberately excludes. We read this as evidence that the Big-4-primary K=3 + Paper A convergence is not a Big-4-specific artefact; we do **not** read it as an endorsement of using full-dataset K=3 component centres or operational thresholds in place of the Big-4-primary analysis. Mid/small-firm composition shifts the component centres meaningfully and the v4.0 primary methodology is restricted to Big-4 by design (§III-G item 4).
+
+## L. Feature Backbone Ablation (inherited from v3.20.0 §IV-I)
+
+The feature-backbone ablation (v3.20.0 Table XVIII; backbone replacement of ResNet-50 with alternative ImageNet-pretrained backbones to verify that the §III-E embedding choice is not load-bearing) is inherited unchanged. v3.20.0 Table XVIII is cited by its original v3 number and is **not** the same table as the v4 Table XVIII (which reports the Big-4 vs full-dataset Spearman drift in §IV-K). v4.0 makes no scope-specific re-derivation of the ablation; the analysis is a methodological-stability check on the embedding stage and is corpus-wide rather than Big-4-restricted.
+
+## M. v4-New Anchor-Based ICCR Calibration Results
+
+This section consolidates the v4-new empirical results that support the §III-L anchor-based threshold calibration framework. Numbers below are direct re-statements from the spike scripts cited per row; the corresponding §III provenance table entries appear in §III's provenance table.
+
+### M.1 Composition decomposition (Scripts 39b–39e)
+
+**Table XX.** Within-firm and between-firm decomposition of the Big-4 accountant-level dip-test rejection.
+
+| Diagnostic | Scope | Statistic | Implication |
+|---|---|---|---|
+| Within-firm signature-level cosine dip | Big-4 (4 firms) | $p_{\text{cos}} \in \{0.176, 0.991, 0.551, 0.976\}$ | 0/4 firms reject; cosine within-firm unimodal |
+| Within-firm signature-level cosine dip | non-Big-4 (10 firms $\geq 500$ sigs) | $p_{\text{cos}} \in [0.59, 0.99]$ | 0/10 firms reject; cosine within-firm unimodal |
+| Within-firm jittered-dHash dip (5 seeds, median) | Big-4 (4 firms) | $p_{\text{med}} \in \{0.999, 0.996, 0.999, 0.9995\}$ | 0/4 firms reject after integer-jitter; raw rejection was integer-tie artefact |
+| Big-4 pooled dHash: 2×2 factorial | firm-centred + jittered (5 seeds) | $p_{\text{med}} = 0.35$, 0/5 seeds reject | combined corrections eliminate rejection; multimodality is composition + integer artefact |
+| Integer-histogram valley near $\text{dHash} \approx 5$ | within each Big-4 firm | none (0/4 firms) | no within-firm dHash antimode at the inherited HC cutoff |
+
+(Source: Scripts 39b, 39c, 39d, 39e; bootstrap $n_{\text{boot}} = 2000$; jitter $\sim \mathrm{U}[-0.5, +0.5]$.)
+
+### M.2 Anchor-based inter-CPA pair-level ICCR (Script 40b)
+
+**Table XXI.** Big-4 inter-CPA per-comparison ICCR sweep, $n = 5 \times 10^5$ pairs (Big-4 scope; v4 new).
+
+| Threshold | Per-comparison ICCR | 95% Wilson CI |
+|---|---|---|
+| cos $> 0.945$ (v3.x published "natural threshold") | $0.00081$ | $[0.00073, 0.00089]$ |
+| cos $> 0.95$ (inherited operating point) | $0.00060$ | $[0.00053, 0.00067]$ |
+| cos $> 0.97$ | $0.00024$ | $[0.00020, 0.00029]$ |
+| cos $> 0.98$ | $0.00009$ | $[0.00007, 0.00012]$ |
+| dHash $\leq 5$ (inherited operating point) | $0.00129$ | $[0.00120, 0.00140]$ |
+| dHash $\leq 4$ | $0.00050$ | $[0.00044, 0.00057]$ |
+| dHash $\leq 3$ | $0.00019$ | $[0.00015, 0.00023]$ |
+| Joint: cos $> 0.95$ AND dHash $\leq 5$ (any-pair semantics) | $0.00014$ | — |
+| Joint: cos $> 0.95$ AND dHash $\leq 4$ (any-pair) | $0.00011$ | — |
+
+Conditional ICCR(dHash $\leq 5$ | cos $> 0.95$) $= 0.234$ (Wilson 95% $[0.190, 0.285]$; $70$ of $299$ pairs).
+
+The cos $> 0.95$ row replicates v3.20.0 §IV-F.1 Table X (v3 reported $0.0005$ under prior "FAR" terminology). The dHash row and joint row are v4 new.
+
+### M.3 Pool-normalised per-signature ICCR (Script 43)
+
+**Table XXII.** Pool-normalised per-signature ICCR under the deployed any-pair HC rule (cos $> 0.95$ AND dHash $\leq 5$); $n_{\text{sig}} = 150{,}453$ (vector-complete Big-4); CPA-block bootstrap $n_{\text{boot}} = 1000$.
+
+| Scope | Per-signature ICCR | Wilson 95% CI | CPA-bootstrap 95% CI |
+|---|---|---|---|
+| Big-4 pooled (any-pair, deployed) | $0.1102$ | $[0.1086, 0.1118]$ | $[0.0908, 0.1330]$ |
+| Big-4 pooled (same-pair, stricter alternative) | $0.0827$ | $[0.0813, 0.0841]$ | $[0.0668, 0.1021]$ |
+| Firm A (any-pair) | $0.2594$ | — | — |
+| Firm B (any-pair) | $0.0147$ | — | — |
+| Firm C (any-pair) | $0.0053$ | — | — |
+| Firm D (any-pair) | $0.0110$ | — | — |
+| Pool-size decile 1 (smallest pools) any-pair | $0.0249$ | — | — |
+| Pool-size decile 10 (largest pools) any-pair | $0.1905$ | — | — |
+
+Decile trend is broadly monotone in pool size with two minor reversals (decile 5 and decile 9 dip below their predecessors). Stricter operating point cos $> 0.95$ AND dHash $\leq 3$ (same-pair) gives per-signature ICCR $0.0449$.
+
+### M.4 Document-level ICCR under three alarm definitions (Script 45)
+
+**Table XXIII.** Document-level inter-CPA ICCR by alarm definition; $n_{\text{docs}} = 75{,}233$.
+
+| Alarm definition | Alarm set | Document-level ICCR | Wilson 95% CI |
+|---|---|---|---|
+| D1 | HC only | $0.1797$ | $[0.1770, 0.1825]$ |
+| D2 (operational) | HC + MC | $0.3375$ | $[0.3342, 0.3409]$ |
+| D3 | HC + MC + HSC | $0.3384$ | $[0.3351, 0.3418]$ |
+
+Per-firm D2 document-level ICCR: Firm A $0.6201$ ($n = 30{,}226$); Firm B $0.1600$ ($n = 17{,}127$); Firm C $0.1635$ ($n = 19{,}501$); Firm D $0.0863$ ($n = 8{,}379$). The Firm C denominator $n = 19{,}501$ exceeds Table XIX's single-firm Firm C count of $19{,}122$ by exactly the $379$ mixed-firm PDFs: all $379$ are $1{:}1$ Firm C / Firm D mixed-firm documents, and Script 45's mode-of-firms implementation (`np.argmax` over `np.unique`'s alphabetically-sorted firm counts) returns the first-sorted firm on ties, which assigns these tied documents to Firm C rather than to Firm D. The four per-firm denominators here therefore sum to the full $75{,}233$, whereas Table XIX's per-firm rows sum to $74{,}854 = 75{,}233 - 379$.
+
+### M.5 Firm heterogeneity logistic regression and cross-firm hit matrix (Script 44)
+
+**Table XXIV.** Logistic regression of per-signature any-pair HC hit indicator on firm dummies and centred log pool size (Firm A reference).
+
+| Term | Odds ratio (vs Firm A) | Direction |
+|---|---|---|
+| Firm B | $0.053$ | $\sim 19\times$ lower odds than Firm A |
+| Firm C | $0.010$ | $\sim 100\times$ lower odds than Firm A |
+| Firm D | $0.027$ | $\sim 37\times$ lower odds than Firm A |
+| log(pool size, centred) | $4.01$ | $\sim 4\times$ higher odds per log unit pool size |
+
+Per-decile per-firm rates (Table not duplicated here; Script 44 decile table available in the supplementary report): within every pool-size decile, Firms B/C/D show rates of $0.0006$–$0.0358$ while Firm A ranges $0.0541$–$0.5958$. The firm gap survives within matched pool sizes.
+
+**Table XXV.** Cross-firm hit matrix among Big-4 source signatures with any-pair HC hit; max-cosine partner firm (counts).
+
+| Source firm | Firm A cand. | Firm B | Firm C | Firm D | non-Big-4 | n hits |
+|---|---|---|---|---|---|---|
+| Firm A | $14{,}447$ | $95$ | $44$ | $19$ | $17$ | $14{,}622$ |
+| Firm B | $92$ | $371$ | $8$ | $4$ | $9$ | $484$ |
+| Firm C | $16$ | $7$ | $149$ | $5$ | $1$ | $178$ |
+| Firm D | $22$ | $2$ | $6$ | $106$ | $1$ | $137$ |
+
+Same-pair joint hits (single candidate satisfying both cos $> 0.95$ AND dHash $\leq 5$) are within-firm at rates $99.96\%$ / $97.7\%$ / $98.2\%$ / $97.0\%$ for Firms A/B/C/D respectively.
+
+### M.6 Alert-rate sensitivity around inherited HC threshold (Script 46)
+
+**Table XXVI.** Local-gradient / median-gradient ratio at inherited thresholds (descriptive plateau diagnostic).
+
+| Threshold | Local / median gradient ratio | Interpretation |
+|---|---|---|
+| cos $= 0.95$ (HC) | $\approx 25\times$ | locally sensitive (not plateau-stable) |
+| dHash $= 5$ (HC) | $\approx 3.8\times$ | locally sensitive (not plateau-stable) |
+| dHash $= 15$ (MC/HSC boundary) | $\approx 0.08$ | plateau-like (saturating tail) |
+
+Big-4 observed deployed alert rate on actual same-CPA pools: per-signature HC $= 0.4958$; per-document HC $= 0.6228$. The deployed-rate excess over the inter-CPA proxy is $0.3856$ pp per-signature and $0.4431$ pp per-document; this excess is interpreted as a same-CPA repeatability signal under the §III-M caveats, not as a presumed true-positive rate.
+
+---
+
+## Phase 3 close-out checklist
+
+The following items remain after codex rounds 21–24 and before §IV is sent to partner Jimmy for v4.0 review:
+
+1. **Table XV per-signature category counts** — RESOLVED (v2 of §IV draft, Script 42 output). Per-signature, per-firm, document-level, and per-firm-document tables now populated.
+2. **Table renumbering finalisation.** The v4 table sequence as of v3.2 is Tables V–XVIII plus Table XV-B (no v4 Table IV is printed); inherited v3.x tables such as capture-rate Tables IX, XI, XII and the backbone-ablation v3.20.0 Table XVIII are kept by reference and cited as "v3.20.0 Table N" rather than reproduced as v4-numbered tables. A final pass should confirm whether the target journal accepts the Table XV-B letter suffix; if not, XV-B can be renumbered to a sequential XIX with §IV-J text adjusted accordingly.
+3. **§IV-A to §IV-C content audit.** Verify that the inherited prose for Experimental Setup, Detection Performance, and All-Pairs analysis remains accurate after the §III-G scope change to Big-4 primary.
+4. **Open question carry-over from §III v3.** Codex round-22 open questions on five-way moderate-band validation, firm anonymisation policy, and §IV table numbering are addressed in this v3 of §IV: (a) five-way moderate band documented as inherited from v3.x in §IV-J with Big-4 per-firm proportions reported descriptively (Table XV); (b) firm anonymisation maintained throughout §IV (Firm A–D used consistently; real names removed in v3); (c) §IV table numbering set provisionally and to be finalised at Phase 3 close-out.
+
+5. **Internal author notes (this checklist + §III's cross-reference index + both files' draft-note headers).** These are author working artefacts and should be moved to a separate notes file or stripped before partner / submission packaging.

signature_analysis/21_expanded_validation.py

@@ -85,44 +85,78 @@ def load_signatures():
     return rows
 
 
-def load_feature_vectors_sample(n=2000):
+def load_signature_ids_for_negative_pool(seed=SEED):
-    """Load feature vectors for inter-CPA negative-anchor sampling."""
+    """Load lightweight (sig_id, accountant) pool from the entire matched
+    corpus. Per Gemini round-19 review, the prior implementation drew
+    50,000 inter-CPA pairs from a tiny LIMIT-3000 random subset, reusing
+    each signature ~33 times and artificially tightening Wilson FAR CIs.
+    The corrected implementation samples pairs i.i.d. across the FULL
+    matched corpus (~168k signatures); only the unique signatures that
+    actually appear in the sampled pairs need feature vectors loaded.
+    """
     conn = sqlite3.connect(DB)
     cur = conn.cursor()
     cur.execute('''
-        SELECT signature_id, assigned_accountant, feature_vector
+        SELECT signature_id, assigned_accountant
         FROM signatures
         WHERE feature_vector IS NOT NULL
           AND assigned_accountant IS NOT NULL
-        ORDER BY RANDOM()
+    ''')
-        LIMIT ?
-    ''', (n,))
     rows = cur.fetchall()
     conn.close()
-    out = []
+    sig_ids = np.array([r[0] for r in rows], dtype=np.int64)
-    for r in rows:
+    accts = np.array([r[1] for r in rows])
-        vec = np.frombuffer(r[2], dtype=np.float32)
+    return sig_ids, accts
-        out.append({'sig_id': r[0], 'accountant': r[1], 'feature': vec})
-    return out
 
 
-def build_inter_cpa_negative(sample, n_pairs=N_INTER_PAIRS, seed=SEED):
+def load_features_for_ids(sig_ids):
-    """Sample random cross-CPA pairs; return their cosine similarities."""
+    conn = sqlite3.connect(DB)
+    cur = conn.cursor()
+    placeholders = ','.join('?' * len(sig_ids))
+    cur.execute(
+        f'SELECT signature_id, feature_vector FROM signatures '
+        f'WHERE signature_id IN ({placeholders})',
+        [int(s) for s in sig_ids],
+    )
+    rows = cur.fetchall()
+    conn.close()
+    feat_by_id = {}
+    for sid, blob in rows:
+        feat_by_id[int(sid)] = np.frombuffer(blob, dtype=np.float32)
+    return feat_by_id
+
+
+def build_inter_cpa_negative(sig_ids, accts, n_pairs=N_INTER_PAIRS, seed=SEED):
+    """Sample i.i.d. random cross-CPA pairs from the full matched corpus
+    and return their cosine similarities.
+    """
     rng = np.random.default_rng(seed)
-    n = len(sample)
+    n = len(sig_ids)
-    feats = np.stack([s['feature'] for s in sample])
+    pairs = []
-    accts = np.array([s['accountant'] for s in sample])
-    sims = []
     tries = 0
-    while len(sims) < n_pairs and tries < n_pairs * 10:
+    seen_pairs = set()
+    while len(pairs) < n_pairs and tries < n_pairs * 10:
         i = rng.integers(n)
         j = rng.integers(n)
         if i == j or accts[i] == accts[j]:
             tries += 1
             continue
-        sim = float(feats[i] @ feats[j])
+        a, b = (i, j) if i < j else (j, i)
-        sims.append(sim)
+        if (a, b) in seen_pairs:
+            tries += 1
+            continue
+        seen_pairs.add((a, b))
+        pairs.append((a, b))
         tries += 1
+
+    needed_ids = sorted({int(sig_ids[i]) for pair in pairs for i in pair})
+    feat_by_id = load_features_for_ids(needed_ids)
+
+    sims = []
+    for i, j in pairs:
+        fi = feat_by_id[int(sig_ids[i])]
+        fj = feat_by_id[int(sig_ids[j])]
+        sims.append(float(fi @ fj))
     return np.array(sims)
 
 
@@ -212,9 +246,12 @@ def main():
     print(f'Firm A signatures: {int(firm_a_mask.sum()):,}')
 
     # --- (1) INTER-CPA NEGATIVE ANCHOR ---
-    print(f'\n[1] Building inter-CPA negative anchor ({N_INTER_PAIRS} pairs)...')
+    print(f'\n[1] Building inter-CPA negative anchor ({N_INTER_PAIRS} '
-    sample = load_feature_vectors_sample(n=3000)
+          f'i.i.d. pairs from full matched corpus)...')
-    inter_cos = build_inter_cpa_negative(sample, n_pairs=N_INTER_PAIRS)
+    pool_sig_ids, pool_accts = load_signature_ids_for_negative_pool()
+    print(f'  pool size: {len(pool_sig_ids):,} matched signatures')
+    inter_cos = build_inter_cpa_negative(pool_sig_ids, pool_accts,
+                                         n_pairs=N_INTER_PAIRS)
     print(f'  inter-CPA cos: mean={inter_cos.mean():.4f}, '
           f'p95={np.percentile(inter_cos, 95):.4f}, '
           f'p99={np.percentile(inter_cos, 99):.4f}, '
@@ -249,7 +286,8 @@ def main():
     print(f"    threshold={eer['threshold']:.4f}, EER={eer['eer']:.4f}")
     # Canonical threshold evaluations with Wilson CIs
     canonical = {}
-    for tt in [0.70, 0.80, 0.837, 0.90, 0.945, 0.95, 0.973, 0.979]:
+    for tt in [0.70, 0.80, 0.837, 0.90, 0.9407, 0.945, 0.95, 0.973, 0.977,
+               0.979, 0.985]:
         y_pred = (scores > tt).astype(int)
         m = classification_metrics(y, y_pred)
         m['threshold'] = float(tt)

signature_analysis/27_within_year_uniformity.py

@@ -0,0 +1,489 @@
+#!/usr/bin/env python3
+"""
+Script 27: Within-Auditor-Year Uniformity Empirical Check (A2 Test)
+=====================================================================
+Opus 4.7 max-effort round-12 review flagged the A2 assumption
+(within-year label uniformity; Methodology Section III-G) as
+load-bearing for Section IV-H.1's partner-level "minority of
+hand-signers" reading, yet lacking empirical verification. This
+script provides the empirical check that Section III-G previously
+described as 'left to future work'.
+
+For each (CPA, fiscal year) unit with >= 3 signatures, we compute:
+  - max_cos_yr: maximum pairwise cosine similarity within the year
+  - min_cos_yr: minimum pairwise cosine similarity within the year
+
+Classification via **frac_high** (the fraction of within-year pairs with
+cosine >= 0.95); this is robust to stamp-output variance, template
+switches, and isolated outliers in a way that raw max/min extremes are
+not. Auxiliary: frac_low (fraction of pairs with cosine < 0.837).
+
+  - strict_full_hand    : frac_high == 0
+                          (no replicated pair anywhere; full-year hand-sign)
+  - mostly_hand         : 0 < frac_high <= 0.1
+                          (isolated near-identical pair, possibly one
+                           template reuse; dominant hand-sign)
+  - substantial_mixture : 0.1 < frac_high <= 0.5
+                          (clear A2 violation: a material minority of
+                           signatures are replicated)
+  - mostly_stamp        : 0.5 < frac_high <= 0.9
+                          (stamp-dominant but with non-trivial variance
+                           or a minority of non-stamped signatures)
+  - strict_full_stamp   : frac_high > 0.9
+                          (near-all pairs near-identical; full-year
+                           replication with modest variance allowed)
+
+Thresholds:
+  0.95  = whole-sample Firm A P7.5 heuristic (Section III-L)
+  0.837 = all-pairs intra/inter KDE crossover (Section III-L,
+           likely-hand-signed boundary)
+
+Stratification:
+  - Firm bucket: Firm A (Deloitte / 勤業眾信), Firm B-D (KPMG/PwC/EY),
+                 Non-Big-4
+  - Period:      2013-2018 (pre-digitalization),
+                 2019-2021 (transition),
+                 2022-2023 (post)
+  - Firm x Period grid for mixed_a2_violation rate
+
+Output:
+  reports/within_year_uniformity/within_year_uniformity.md
+  reports/within_year_uniformity/within_year_uniformity.json
+  reports/within_year_uniformity/mixed_year_candidates.csv  (audit trail)
+"""
+
+import sqlite3
+import json
+import csv
+import numpy as np
+from pathlib import Path
+from datetime import datetime, timezone
+from collections import defaultdict
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'within_year_uniformity')
+OUT.mkdir(parents=True, exist_ok=True)
+
+FIRM_A = '勤業眾信聯合'
+BIG4_OTHER = {'安侯建業聯合', '資誠聯合', '安永聯合'}
+
+THRESH_REPLICATED = 0.95
+THRESH_HANDSIGN = 0.837
+MIN_SIGS = 3
+
+FIRM_BUCKETS = ['Firm A', 'Firm B-D (Big-4 others)', 'Non-Big-4']
+PERIODS = ['2013-2018 (pre)', '2019-2021 (transition)', '2022-2023 (post)']
+CLASSES = ['strict_full_hand', 'mostly_hand', 'substantial_mixture',
+           'mostly_stamp', 'strict_full_stamp']
+# A2 violation candidates = {mostly_hand, substantial_mixture, mostly_stamp}
+# (i.e., not strict_full_hand and not strict_full_stamp)
+
+
+def period_bin(year):
+    y = int(year)
+    if y <= 2018:
+        return '2013-2018 (pre)'
+    if y <= 2021:
+        return '2019-2021 (transition)'
+    return '2022-2023 (post)'
+
+
+def firm_bucket(firm):
+    if firm == FIRM_A:
+        return 'Firm A'
+    if firm in BIG4_OTHER:
+        return 'Firm B-D (Big-4 others)'
+    return 'Non-Big-4'
+
+
+def classify(frac_high):
+    if frac_high == 0:
+        return 'strict_full_hand'
+    if frac_high <= 0.1:
+        return 'mostly_hand'
+    if frac_high <= 0.5:
+        return 'substantial_mixture'
+    if frac_high <= 0.9:
+        return 'mostly_stamp'
+    return 'strict_full_stamp'
+
+
+def is_a2_violation(cls):
+    """A2 violation candidates: not strictly full_hand and not strictly full_stamp."""
+    return cls in {'mostly_hand', 'substantial_mixture', 'mostly_stamp'}
+
+
+def pairwise_stats(feats):
+    """Return (max_cos, min_cos, frac_high, frac_low, n_pairs) over
+    within-year pairs. Filters out degenerate features (zero norm or
+    non-finite entries) before computing."""
+    mat = np.stack(feats).astype(np.float64)
+    # Drop rows with non-finite entries or zero norm
+    finite = np.all(np.isfinite(mat), axis=1)
+    norms = np.linalg.norm(mat, axis=1)
+    keep = finite & (norms > 1e-6)
+    mat = mat[keep]
+    norms = norms[keep]
+    if len(mat) < 2:
+        return (float('nan'), float('nan'), 0.0, 0.0, 0)
+    mat_n = mat / norms[:, None]
+    sim = mat_n @ mat_n.T
+    iu = np.triu_indices(len(mat), k=1)
+    vals = sim[iu]
+    vals = vals[np.isfinite(vals)]
+    n_pairs = len(vals)
+    if n_pairs == 0:
+        return (float('nan'), float('nan'), 0.0, 0.0, 0)
+    n_high = int(np.sum(vals >= THRESH_REPLICATED))
+    n_low = int(np.sum(vals < THRESH_HANDSIGN))
+    return (float(vals.max()), float(vals.min()),
+            n_high / n_pairs, n_low / n_pairs, n_pairs)
+
+
+def iterate_groups():
+    """Stream rows ordered by (CPA, year); yield completed groups."""
+    conn = sqlite3.connect(DB)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT s.assigned_accountant,
+               substr(s.year_month, 1, 4) AS year,
+               s.feature_vector,
+               a.firm
+        FROM signatures s
+        LEFT JOIN accountants a ON a.name = s.assigned_accountant
+        WHERE s.feature_vector IS NOT NULL
+          AND s.assigned_accountant IS NOT NULL
+          AND s.year_month IS NOT NULL
+        ORDER BY s.assigned_accountant, year
+    ''')
+    cur_key = None
+    cur_feats = []
+    cur_firm = None
+    for cpa, year, fv, firm in cur:
+        key = (cpa, year)
+        if key != cur_key:
+            if cur_key is not None and cur_feats:
+                yield cur_key, cur_feats, cur_firm
+            cur_key = key
+            cur_feats = []
+            cur_firm = firm
+        cur_feats.append(np.frombuffer(fv, dtype=np.float32).copy())
+    if cur_key is not None and cur_feats:
+        yield cur_key, cur_feats, cur_firm
+    conn.close()
+
+
+def main():
+    print('Streaming (CPA, year) groups from DB...')
+    results = []
+    total_groups = 0
+    kept_groups = 0
+    for (cpa, year), feats, firm in iterate_groups():
+        total_groups += 1
+        if len(feats) < MIN_SIGS:
+            continue
+        kept_groups += 1
+        max_c, min_c, frac_high, frac_low, n_pairs = pairwise_stats(feats)
+        cls = classify(frac_high)
+        results.append({
+            'cpa': cpa,
+            'year': year,
+            'n_sigs': len(feats),
+            'n_pairs': n_pairs,
+            'firm': firm or 'UNKNOWN',
+            'firm_bucket': firm_bucket(firm),
+            'period': period_bin(year),
+            'max_cos': round(max_c, 4),
+            'min_cos': round(min_c, 4),
+            'frac_high': round(frac_high, 4),
+            'frac_low': round(frac_low, 4),
+            'class': cls,
+            'is_a2_violation': is_a2_violation(cls),
+        })
+    print(f'  total groups: {total_groups}')
+    print(f'  groups with n >= {MIN_SIGS}: {kept_groups}')
+
+    total = len(results)
+    if total == 0:
+        print('No groups to analyze.')
+        return
+
+    # Overall tally
+    overall = defaultdict(int)
+    for r in results:
+        overall[r['class']] += 1
+    print('\n=== Overall classification ===')
+    for c in CLASSES:
+        n = overall[c]
+        print(f'  {c:25s}: {n:5d} ({100*n/total:.2f}%)')
+
+    # Stratifications
+    by_firm = defaultdict(lambda: defaultdict(int))
+    by_period = defaultdict(lambda: defaultdict(int))
+    by_fp = defaultdict(lambda: defaultdict(int))
+    for r in results:
+        by_firm[r['firm_bucket']]['total'] += 1
+        by_firm[r['firm_bucket']][r['class']] += 1
+        if r['is_a2_violation']:
+            by_firm[r['firm_bucket']]['a2_violation'] += 1
+        by_period[r['period']]['total'] += 1
+        by_period[r['period']][r['class']] += 1
+        if r['is_a2_violation']:
+            by_period[r['period']]['a2_violation'] += 1
+        key = (r['firm_bucket'], r['period'])
+        by_fp[key]['total'] += 1
+        by_fp[key][r['class']] += 1
+        if r['is_a2_violation']:
+            by_fp[key]['a2_violation'] += 1
+
+    print('\n=== By firm bucket ===')
+    for fb in FIRM_BUCKETS:
+        d = by_firm[fb]
+        t = d['total']
+        if t == 0:
+            continue
+        print(f'  {fb} (N = {t}):')
+        for c in CLASSES:
+            n = d[c]
+            print(f'    {c:25s}: {n:5d} ({100*n/t:.2f}%)')
+
+    print('\n=== By period ===')
+    for p in PERIODS:
+        d = by_period[p]
+        t = d['total']
+        if t == 0:
+            continue
+        print(f'  {p} (N = {t}):')
+        for c in CLASSES:
+            n = d[c]
+            print(f'    {c:25s}: {n:5d} ({100*n/t:.2f}%)')
+
+    print('\n=== Firm x Period: A2 violation rate (any of mostly_hand, '
+          'substantial_mixture, mostly_stamp) ===')
+    header = '  {:25s}'.format('') + \
+             ''.join(f'{p[:18]:>22}' for p in PERIODS)
+    print(header)
+    for fb in FIRM_BUCKETS:
+        cells = []
+        for p in PERIODS:
+            d = by_fp[(fb, p)]
+            t = d['total']
+            if t == 0:
+                cells.append('-')
+            else:
+                rate = 100 * d['a2_violation'] / t
+                cells.append(f'{rate:.2f}% ({d["a2_violation"]}/{t})')
+        row = '  {:25s}'.format(fb) + ''.join(f'{c:>22}' for c in cells)
+        print(row)
+
+    # Substantial-mixture-only Firm x Period (strictest A2 violation subset)
+    print('\n=== Firm x Period: substantial_mixture rate (strictest) ===')
+    print(header)
+    for fb in FIRM_BUCKETS:
+        cells = []
+        for p in PERIODS:
+            d = by_fp[(fb, p)]
+            t = d['total']
+            if t == 0:
+                cells.append('-')
+            else:
+                rate = 100 * d['substantial_mixture'] / t
+                cells.append(
+                    f'{rate:.2f}% ({d["substantial_mixture"]}/{t})')
+        row = '  {:25s}'.format(fb) + ''.join(f'{c:>22}' for c in cells)
+        print(row)
+
+    # Outputs
+    json_out = {
+        'generated_at': datetime.now(timezone.utc).isoformat(),
+        'thresholds': {
+            'replicated_cosine': THRESH_REPLICATED,
+            'handsigned_cosine': THRESH_HANDSIGN,
+        },
+        'min_signatures_per_year': MIN_SIGS,
+        'N_total_groups': total_groups,
+        'N_kept_groups': kept_groups,
+        'overall': {c: overall[c] for c in CLASSES},
+        'by_firm_bucket': {
+            fb: dict(by_firm[fb]) for fb in FIRM_BUCKETS if by_firm[fb]['total']
+        },
+        'by_period': {
+            p: dict(by_period[p]) for p in PERIODS if by_period[p]['total']
+        },
+        'by_firm_x_period': {
+            f'{fb}|{p}': dict(by_fp[(fb, p)])
+            for fb in FIRM_BUCKETS for p in PERIODS
+            if by_fp[(fb, p)]['total']
+        },
+    }
+    with open(OUT / 'within_year_uniformity.json', 'w', encoding='utf-8') as f:
+        json.dump(json_out, f, ensure_ascii=False, indent=2)
+
+    # CSV audit trail: all rows with all metrics
+    csv_fields = [
+        'cpa', 'firm', 'firm_bucket', 'year', 'period',
+        'n_sigs', 'n_pairs', 'max_cos', 'min_cos',
+        'frac_high', 'frac_low', 'class', 'is_a2_violation',
+    ]
+    csv_path = OUT / 'all_cpa_year_rows.csv'
+    with open(csv_path, 'w', newline='', encoding='utf-8') as f:
+        w = csv.DictWriter(f, fieldnames=csv_fields)
+        w.writeheader()
+        for r in sorted(results,
+                         key=lambda x: (x['firm_bucket'], x['year'], x['cpa'])):
+            w.writerow({k: r[k] for k in csv_fields})
+
+    # CSV: substantial_mixture rows only (strictest A2 violation subset)
+    mixed_path = OUT / 'substantial_mixture_candidates.csv'
+    with open(mixed_path, 'w', newline='', encoding='utf-8') as f:
+        w = csv.DictWriter(f, fieldnames=csv_fields)
+        w.writeheader()
+        for r in sorted(results,
+                         key=lambda x: (x['firm_bucket'], x['year'], x['cpa'])):
+            if r['class'] == 'substantial_mixture':
+                w.writerow({k: r[k] for k in csv_fields})
+
+    # Markdown
+    md = build_markdown(overall, by_firm, by_period, by_fp, total,
+                         total_groups, kept_groups)
+    with open(OUT / 'within_year_uniformity.md', 'w', encoding='utf-8') as f:
+        f.write(md)
+
+    print(f'\n=> Outputs in {OUT}')
+
+
+def build_markdown(overall, by_firm, by_period, by_fp, total,
+                    total_groups, kept_groups):
+    ts = datetime.now(timezone.utc).isoformat()
+    L = []
+    L.append('# Within-Auditor-Year Uniformity Check (A2 Empirical Test)')
+    L.append('')
+    L.append(f'Generated: {ts}')
+    L.append('')
+    L.append('## Method')
+    L.append('')
+    L.append(f'For each (CPA, fiscal year) with >= {MIN_SIGS} signatures, '
+             'compute all within-year pairwise cosine similarities and '
+             f'derive frac_high = fraction of pairs with cos >= {THRESH_REPLICATED}. '
+             'Classification is based on frac_high; this is robust to stamp-'
+             'output variance, template switches, and isolated outliers.')
+    L.append('')
+    L.append(f'- `strict_full_hand`: frac_high = 0 '
+             '(no near-identical pair; full-year hand-signing)')
+    L.append(f'- `mostly_hand`: 0 < frac_high <= 0.1 '
+             '(isolated near-identical pair; dominant hand-sign with possibly '
+             'one template reuse)')
+    L.append(f'- `substantial_mixture`: 0.1 < frac_high <= 0.5 '
+             '(material minority of signatures replicated; clearest A2 '
+             'violation signature)')
+    L.append(f'- `mostly_stamp`: 0.5 < frac_high <= 0.9 '
+             '(stamp-dominant with non-trivial variance or minority of '
+             'non-stamped signatures)')
+    L.append(f'- `strict_full_stamp`: frac_high > 0.9 '
+             '(near-all pairs near-identical; full-year replication with '
+             'modest variance allowed)')
+    L.append('')
+    L.append('**A2 violation candidates** = `mostly_hand` ∪ '
+             '`substantial_mixture` ∪ `mostly_stamp` (anything that is not '
+             '`strict_full_hand` and not `strict_full_stamp`).')
+    L.append('')
+    L.append(f'Total (CPA, year) groups in DB: {total_groups}; '
+             f'groups with n >= {MIN_SIGS}: {kept_groups}.')
+    L.append('')
+    L.append('## Overall')
+    L.append('')
+    L.append('| Class | N | Share |')
+    L.append('|---|---|---|')
+    for c in CLASSES:
+        n = overall[c]
+        L.append(f'| `{c}` | {n} | {100*n/total:.2f}% |')
+    L.append('')
+    def row(label, d, t):
+        cells = [label, str(t)]
+        for c in CLASSES:
+            n = d[c]
+            cells.append(f'{n} ({100*n/t:.2f}%)')
+        av = d['a2_violation']
+        cells.append(f'{av} ({100*av/t:.2f}%)')
+        return '| ' + ' | '.join(cells) + ' |'
+
+    header = ('| Bucket | N | ' + ' | '.join(f'`{c}`' for c in CLASSES)
+              + ' | A2 violation (union) |')
+    sep = '|' + '|'.join(['---'] * (len(CLASSES) + 3)) + '|'
+
+    L.append('## By firm bucket')
+    L.append('')
+    L.append(header)
+    L.append(sep)
+    for fb in FIRM_BUCKETS:
+        d = by_firm[fb]
+        t = d['total']
+        if t == 0:
+            continue
+        L.append(row(fb, d, t))
+    L.append('')
+    L.append('## By period')
+    L.append('')
+    L.append(header.replace('Bucket', 'Period'))
+    L.append(sep)
+    for p in PERIODS:
+        d = by_period[p]
+        t = d['total']
+        if t == 0:
+            continue
+        L.append(row(p, d, t))
+    L.append('')
+    L.append('## Firm x Period: A2 violation rate (union of '
+             '`mostly_hand`, `substantial_mixture`, `mostly_stamp`)')
+    L.append('')
+    L.append('| Firm | 2013-2018 (pre) | 2019-2021 (transition) | '
+             '2022-2023 (post) |')
+    L.append('|---|---|---|---|')
+    for fb in FIRM_BUCKETS:
+        cells = []
+        for p in PERIODS:
+            d = by_fp[(fb, p)]
+            t = d['total']
+            if t == 0:
+                cells.append('-')
+            else:
+                rate = 100 * d['a2_violation'] / t
+                cells.append(f'{rate:.2f}% ({d["a2_violation"]}/{t})')
+        L.append(f'| {fb} | ' + ' | '.join(cells) + ' |')
+    L.append('')
+    L.append('## Firm x Period: `substantial_mixture` rate (strictest subset)')
+    L.append('')
+    L.append('| Firm | 2013-2018 (pre) | 2019-2021 (transition) | '
+             '2022-2023 (post) |')
+    L.append('|---|---|---|---|')
+    for fb in FIRM_BUCKETS:
+        cells = []
+        for p in PERIODS:
+            d = by_fp[(fb, p)]
+            t = d['total']
+            if t == 0:
+                cells.append('-')
+            else:
+                rate = 100 * d['substantial_mixture'] / t
+                cells.append(
+                    f'{rate:.2f}% ({d["substantial_mixture"]}/{t})')
+        L.append(f'| {fb} | ' + ' | '.join(cells) + ' |')
+    L.append('')
+    L.append('## Interpretation guide')
+    L.append('')
+    L.append('- Low A2-violation union rate overall (e.g. < 10%): A2 is '
+             'empirically well-supported; report as Methodology III-G '
+             'robustness check.')
+    L.append('- High `substantial_mixture` rate specifically (e.g. > 5% '
+             'at Big-4 B-D in 2019-2021): A2 weakens in the digitalization '
+             'transition; IV-H.1 partner-level reading may need restriction '
+             'to Firm A or pre-2019 period.')
+    L.append('- High `substantial_mixture` rate at Firm A itself: unexpected; '
+             'Firm A industry-practice defense of A2 would need revisiting.')
+    L.append('')
+    return '\n'.join(L)
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/29_firm_a_yearly_distribution.py

@@ -0,0 +1,123 @@
+#!/usr/bin/env python3
+"""
+Script 29: Firm A Per-Year Cosine Distribution (Table XIII)
+============================================================
+Generates the year-by-year Firm A per-signature best-match cosine
+distribution reported as Table XIII in the manuscript. Codex / Gemini
+round-19 review identified that this table previously had no dedicated
+generating script (Appendix B incorrectly attributed it to Script 08,
+which has no year_month extraction).
+
+Definition:
+  Firm A membership is via CPA registry (accountants.firm joined on
+  signatures.assigned_accountant), matching the convention used by
+  scripts 24 and 28.
+
+  For each fiscal year (substr(year_month, 1, 4)):
+    - N signatures with non-null max_similarity_to_same_accountant
+    - mean of max_similarity_to_same_accountant (the per-signature
+      best-match cosine)
+    - share with max_similarity_to_same_accountant < 0.95 (the
+      left-tail rate cited in Section IV-G.1)
+
+Output:
+  reports/firm_a_yearly/firm_a_yearly_distribution.json
+  reports/firm_a_yearly/firm_a_yearly_distribution.md
+"""
+
+import json
+import sqlite3
+from datetime import datetime
+from pathlib import Path
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'firm_a_yearly')
+OUT.mkdir(parents=True, exist_ok=True)
+
+FIRM_A = '勤業眾信聯合'
+
+
+def yearly_distribution(conn):
+    cur = conn.cursor()
+    cur.execute("""
+        SELECT substr(s.year_month, 1, 4) AS year,
+               COUNT(*) AS n_sigs,
+               AVG(s.max_similarity_to_same_accountant) AS mean_cos,
+               SUM(CASE
+                     WHEN s.max_similarity_to_same_accountant < 0.95
+                     THEN 1 ELSE 0
+                   END) AS n_below_095
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE a.firm = ?
+          AND s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.year_month IS NOT NULL
+        GROUP BY year
+        ORDER BY year
+    """, (FIRM_A,))
+
+    rows = []
+    for year, n_sigs, mean_cos, n_below in cur.fetchall():
+        rows.append({
+            'year': int(year),
+            'n_signatures': n_sigs,
+            'mean_best_match_cosine': round(mean_cos, 4),
+            'n_below_cosine_095': n_below,
+            'pct_below_cosine_095': round(100.0 * n_below / n_sigs, 2),
+        })
+    return rows
+
+
+def write_markdown(payload, path):
+    rows = payload['yearly_rows']
+    lines = []
+    lines.append('# Firm A Per-Year Cosine Distribution (Table XIII)')
+    lines.append('')
+    lines.append(f"Generated at: {payload['generated_at']}")
+    lines.append('')
+    lines.append('Firm A membership: CPA registry '
+                 '(accountants.firm = "勤業眾信聯合"). Per-signature '
+                 'best-match cosine = '
+                 'signatures.max_similarity_to_same_accountant.')
+    lines.append('')
+    lines.append('| Year | N sigs | mean best-match cosine | % below 0.95 |')
+    lines.append('|------|--------|------------------------|--------------|')
+    for r in rows:
+        lines.append(
+            f"| {r['year']} | {r['n_signatures']:,} | "
+            f"{r['mean_best_match_cosine']:.4f} | "
+            f"{r['pct_below_cosine_095']:.2f}% |"
+        )
+    path.write_text('\n'.join(lines) + '\n', encoding='utf-8')
+
+
+def main():
+    conn = sqlite3.connect(DB)
+    try:
+        payload = {
+            'generated_at': datetime.now().isoformat(timespec='seconds'),
+            'database_path': DB,
+            'firm_a_label': FIRM_A,
+            'firm_a_membership_definition': (
+                'CPA registry: accountants.firm joined on '
+                'signatures.assigned_accountant'
+            ),
+            'cosine_metric': 'signatures.max_similarity_to_same_accountant',
+            'yearly_rows': yearly_distribution(conn),
+        }
+    finally:
+        conn.close()
+
+    json_path = OUT / 'firm_a_yearly_distribution.json'
+    json_path.write_text(json.dumps(payload, indent=2, ensure_ascii=False),
+                         encoding='utf-8')
+    print(f'Wrote {json_path}')
+
+    md_path = OUT / 'firm_a_yearly_distribution.md'
+    write_markdown(payload, md_path)
+    print(f'Wrote {md_path}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/30_yearly_big4_comparison.py

@@ -0,0 +1,255 @@
+#!/usr/bin/env python3
+"""
+Script 30: Yearly Per-Firm Cosine Similarity Comparison
+========================================================
+Generates the per-firm year-by-year per-signature best-match cosine
+distribution: Firm A (Deloitte), Firm B (KPMG), Firm C (PwC),
+Firm D (EY), Non-Big-4. The two-panel figure (mean cosine; share above
+0.95) is the headline cross-firm visual requested in partner review of
+v3.19.1 (2026-04-27): five lines, X-axis 2013-2023, Firm A at the top.
+
+Outputs:
+  reports/figures/fig_yearly_big4_comparison.png
+  reports/figures/fig_yearly_big4_comparison.pdf
+  reports/firm_yearly_comparison/firm_yearly_comparison.json
+  reports/firm_yearly_comparison/firm_yearly_comparison.md
+"""
+
+import json
+import sqlite3
+from datetime import datetime
+from pathlib import Path
+
+import matplotlib
+matplotlib.use('Agg')
+import matplotlib.pyplot as plt
+import numpy as np
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+FIG_OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+               'figures')
+DATA_OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+                'firm_yearly_comparison')
+FIG_OUT.mkdir(parents=True, exist_ok=True)
+DATA_OUT.mkdir(parents=True, exist_ok=True)
+
+FIRM_BUCKETS = [
+    ('Firm A', '勤業眾信聯合'),
+    ('Firm B', '安侯建業聯合'),
+    ('Firm C', '資誠聯合'),
+    ('Firm D', '安永聯合'),
+]
+
+FIRM_COLORS = {
+    'Firm A': '#d62728',
+    'Firm B': '#1f77b4',
+    'Firm C': '#2ca02c',
+    'Firm D': '#9467bd',
+    'Non-Big-4': '#7f7f7f',
+}
+FIRM_MARKERS = {
+    'Firm A': 'o',
+    'Firm B': 's',
+    'Firm C': '^',
+    'Firm D': 'D',
+    'Non-Big-4': 'v',
+}
+COSINE_CUT = 0.95
+
+
+def firm_bucket(firm):
+    for label, name in FIRM_BUCKETS:
+        if firm == name:
+            return label
+    return 'Non-Big-4'
+
+
+def load_rows(conn):
+    cur = conn.cursor()
+    cur.execute("""
+        SELECT a.firm,
+               CAST(substr(s.year_month, 1, 4) AS INTEGER) AS year,
+               s.max_similarity_to_same_accountant
+        FROM signatures s
+        LEFT JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.year_month IS NOT NULL
+          AND s.assigned_accountant IS NOT NULL
+    """)
+    return cur.fetchall()
+
+
+def aggregate(rows):
+    """Returns dict keyed by (firm_label, year) -> {n, mean_cos, share_ge_cut}."""
+    by_firm_year = {}
+    for firm, year, cos in rows:
+        if year is None or year < 2013 or year > 2023:
+            continue
+        label = firm_bucket(firm)
+        key = (label, int(year))
+        by_firm_year.setdefault(key, []).append(float(cos))
+
+    summary = {}
+    for (label, year), vals in by_firm_year.items():
+        arr = np.array(vals, dtype=float)
+        summary[(label, year)] = {
+            'n': int(arr.size),
+            'mean_cos': float(arr.mean()),
+            'share_ge_cut': float(np.mean(arr >= COSINE_CUT)),
+        }
+    return summary
+
+
+def plot_figure(summary, years, firm_labels, fig_path_png, fig_path_pdf):
+    fig, axes = plt.subplots(1, 2, figsize=(13, 5))
+
+    ax = axes[0]
+    for label in firm_labels:
+        ys = [summary[(label, y)]['mean_cos']
+              if (label, y) in summary else np.nan
+              for y in years]
+        ax.plot(years, ys,
+                marker=FIRM_MARKERS[label], color=FIRM_COLORS[label],
+                lw=2.0, ms=6, label=label,
+                zorder=3 if label == 'Firm A' else 2)
+    ax.set_xlabel('Fiscal year')
+    ax.set_ylabel('Mean per-signature best-match cosine')
+    ax.set_title('(a) Mean per-signature best-match cosine, by firm and year')
+    ax.set_xticks(years)
+    ax.tick_params(axis='x', rotation=0)
+    ax.grid(True, ls=':', alpha=0.4)
+    ax.legend(loc='lower right', framealpha=0.95)
+
+    ax = axes[1]
+    for label in firm_labels:
+        ys = [100.0 * summary[(label, y)]['share_ge_cut']
+              if (label, y) in summary else np.nan
+              for y in years]
+        ax.plot(years, ys,
+                marker=FIRM_MARKERS[label], color=FIRM_COLORS[label],
+                lw=2.0, ms=6, label=label,
+                zorder=3 if label == 'Firm A' else 2)
+    ax.set_xlabel('Fiscal year')
+    ax.set_ylabel(f'% signatures with best-match cosine $\\geq$ {COSINE_CUT}')
+    ax.set_title(f'(b) Share with cosine $\\geq$ {COSINE_CUT}, '
+                 'by firm and year')
+    ax.set_xticks(years)
+    ax.tick_params(axis='x', rotation=0)
+    ax.grid(True, ls=':', alpha=0.4)
+    ax.legend(loc='lower right', framealpha=0.95)
+    ax.set_ylim(0, 100)
+
+    fig.suptitle('Per-firm yearly per-signature best-match cosine '
+                 '(operational cut shown as 0.95)',
+                 fontsize=12, y=1.02)
+    fig.tight_layout()
+    fig.savefig(fig_path_png, dpi=200, bbox_inches='tight')
+    fig.savefig(fig_path_pdf, bbox_inches='tight')
+    plt.close(fig)
+
+
+def write_markdown(summary, years, firm_labels, md_path):
+    lines = ['# Per-Firm Yearly Cosine Comparison',
+             '',
+             f"Generated: {datetime.now().isoformat(timespec='seconds')}",
+             '',
+             ('Per-signature best-match cosine '
+              '(`max_similarity_to_same_accountant`), aggregated by firm '
+              'bucket and fiscal year. Firm bucket via CPA registry '
+              '(`accountants.firm`).'),
+             '']
+
+    lines.append('## Mean per-signature best-match cosine')
+    lines.append('')
+    header = '| Year | ' + ' | '.join(firm_labels) + ' |'
+    sep = '|------|' + '|'.join(['------'] * len(firm_labels)) + '|'
+    lines.append(header)
+    lines.append(sep)
+    for y in years:
+        row = f'| {y} | '
+        cells = []
+        for lab in firm_labels:
+            if (lab, y) in summary:
+                cells.append(f"{summary[(lab, y)]['mean_cos']:.4f}")
+            else:
+                cells.append('---')
+        row += ' | '.join(cells) + ' |'
+        lines.append(row)
+
+    lines.append('')
+    lines.append(f'## Share with cosine $\\geq$ {COSINE_CUT}')
+    lines.append('')
+    lines.append(header)
+    lines.append(sep)
+    for y in years:
+        row = f'| {y} | '
+        cells = []
+        for lab in firm_labels:
+            if (lab, y) in summary:
+                cells.append(f"{100*summary[(lab, y)]['share_ge_cut']:.1f}%")
+            else:
+                cells.append('---')
+        row += ' | '.join(cells) + ' |'
+        lines.append(row)
+
+    lines.append('')
+    lines.append('## Per-firm signature counts')
+    lines.append('')
+    lines.append(header)
+    lines.append(sep)
+    for y in years:
+        row = f'| {y} | '
+        cells = []
+        for lab in firm_labels:
+            if (lab, y) in summary:
+                cells.append(f"{summary[(lab, y)]['n']:,}")
+            else:
+                cells.append('---')
+        row += ' | '.join(cells) + ' |'
+        lines.append(row)
+
+    md_path.write_text('\n'.join(lines) + '\n', encoding='utf-8')
+
+
+def main():
+    conn = sqlite3.connect(DB)
+    try:
+        rows = load_rows(conn)
+    finally:
+        conn.close()
+    print(f'Loaded {len(rows):,} signatures with cosine + year + firm.')
+
+    summary = aggregate(rows)
+    years = sorted({y for (_, y) in summary})
+    firm_labels = ['Firm A', 'Firm B', 'Firm C', 'Firm D', 'Non-Big-4']
+
+    fig_png = FIG_OUT / 'fig_yearly_big4_comparison.png'
+    fig_pdf = FIG_OUT / 'fig_yearly_big4_comparison.pdf'
+    plot_figure(summary, years, firm_labels, fig_png, fig_pdf)
+    print(f'Wrote {fig_png}')
+    print(f'Wrote {fig_pdf}')
+
+    payload = {
+        'generated_at': datetime.now().isoformat(timespec='seconds'),
+        'database_path': DB,
+        'cosine_cut': COSINE_CUT,
+        'firm_buckets': dict(FIRM_BUCKETS) | {'Non-Big-4': 'all other'},
+        'years': years,
+        'rows': [
+            {'firm': lab, 'year': y, **summary[(lab, y)]}
+            for lab in firm_labels for y in years
+            if (lab, y) in summary
+        ],
+    }
+    json_path = DATA_OUT / 'firm_yearly_comparison.json'
+    json_path.write_text(json.dumps(payload, indent=2, ensure_ascii=False),
+                         encoding='utf-8')
+    print(f'Wrote {json_path}')
+
+    md_path = DATA_OUT / 'firm_yearly_comparison.md'
+    write_markdown(summary, years, firm_labels, md_path)
+    print(f'Wrote {md_path}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/31_within_year_ranking_robustness.py

@@ -0,0 +1,249 @@
+#!/usr/bin/env python3
+"""
+Script 31: Within-Year Same-CPA Ranking Robustness Check
+==========================================================
+Recomputes the per-auditor-year mean cosine ranking of Table XIV using
+within-year same-CPA matching only (instead of cross-year same-CPA pool
+which Table XIV uses by construction). Reports pooled top-10/20/30%
+Firm A share under the within-year restriction so the partner-level
+ranking finding can be checked against the cross-year aggregation
+choice flagged in Section IV-G.2.
+
+Definition (within-year statistic):
+  For each signature s, with CPA = c, year = y:
+    cos_within(s) = max cosine(s, s') over s' != s, CPA(s')=c, year(s')=y
+  If a (CPA, year) block has only one signature, cos_within is undefined
+  and that signature is dropped from the auditor-year aggregation
+  (matching the same-CPA pair-existence requirement of Section III-G).
+
+Outputs:
+  reports/within_year_ranking/within_year_ranking.json
+  reports/within_year_ranking/within_year_ranking.md
+"""
+
+import json
+import sqlite3
+from collections import defaultdict
+from datetime import datetime
+from pathlib import Path
+
+import numpy as np
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'within_year_ranking')
+OUT.mkdir(parents=True, exist_ok=True)
+
+FIRM_A = '勤業眾信聯合'
+MIN_SIGS_PER_AUDITOR_YEAR = 5
+
+
+def firm_bucket(firm):
+    if firm == '勤業眾信聯合':
+        return 'Firm A'
+    if firm == '安侯建業聯合':
+        return 'Firm B'
+    if firm == '資誠聯合':
+        return 'Firm C'
+    if firm == '安永聯合':
+        return 'Firm D'
+    return 'Non-Big-4'
+
+
+def load_signatures():
+    conn = sqlite3.connect(DB)
+    cur = conn.cursor()
+    cur.execute("""
+        SELECT s.signature_id, s.assigned_accountant, a.firm,
+               CAST(substr(s.year_month, 1, 4) AS INTEGER) AS year,
+               s.feature_vector
+        FROM signatures s
+        LEFT JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.feature_vector IS NOT NULL
+          AND s.assigned_accountant IS NOT NULL
+          AND s.year_month IS NOT NULL
+    """)
+    rows = cur.fetchall()
+    conn.close()
+    return rows
+
+
+def compute_within_year_max(rows):
+    """Group by (CPA, year), compute max cosine to other same-block sigs."""
+    blocks = defaultdict(list)  # (cpa, year) -> [(sig_id, feat)]
+    for sig_id, cpa, firm, year, blob in rows:
+        if year is None:
+            continue
+        feat = np.frombuffer(blob, dtype=np.float32)
+        blocks[(cpa, int(year))].append((sig_id, feat, firm))
+
+    sig_max_within = {}  # sig_id -> max within-year same-CPA cosine
+    sig_meta = {}        # sig_id -> (cpa, year, firm)
+    for (cpa, year), entries in blocks.items():
+        if len(entries) < 2:
+            continue  # singleton: max-within is undefined
+        feats = np.stack([e[1] for e in entries])  # (n, 2048)
+        sims = feats @ feats.T                      # (n, n)
+        np.fill_diagonal(sims, -np.inf)
+        maxs = sims.max(axis=1)
+        for i, (sig_id, _, firm) in enumerate(entries):
+            sig_max_within[sig_id] = float(maxs[i])
+            sig_meta[sig_id] = (cpa, year, firm)
+    return sig_max_within, sig_meta
+
+
+def auditor_year_aggregation(sig_max_within, sig_meta):
+    by_ay = defaultdict(list)  # (cpa, year) -> list of cos
+    for sig_id, cos in sig_max_within.items():
+        cpa, year, firm = sig_meta[sig_id]
+        by_ay[(cpa, year)].append(cos)
+    rows = []
+    for (cpa, year), vals in by_ay.items():
+        if len(vals) < MIN_SIGS_PER_AUDITOR_YEAR:
+            continue
+        firm = sig_meta[next(s for s in sig_max_within
+                              if sig_meta[s][0] == cpa
+                              and sig_meta[s][1] == year)][2]
+        rows.append({
+            'acct': cpa,
+            'year': year,
+            'firm': firm,
+            'cos_mean_within_year': float(np.mean(vals)),
+            'n': len(vals),
+        })
+    return rows
+
+
+def top_k_breakdown(rows, k_pcts=(10, 20, 25, 30, 50)):
+    sorted_rows = sorted(rows, key=lambda r: -r['cos_mean_within_year'])
+    N = len(sorted_rows)
+    out = {}
+    for k_pct in k_pcts:
+        k = max(1, int(N * k_pct / 100))
+        top = sorted_rows[:k]
+        counts = defaultdict(int)
+        for r in top:
+            counts[firm_bucket(r['firm'])] += 1
+        out[f'top_{k_pct}pct'] = {
+            'k': k,
+            'firm_counts': dict(counts),
+            'firm_a_share': counts['Firm A'] / k,
+        }
+    return out
+
+
+def per_year_top_k(rows, k_pcts=(10, 20, 30)):
+    years = sorted(set(r['year'] for r in rows))
+    out = {}
+    for y in years:
+        yr = [r for r in rows if r['year'] == y]
+        if not yr:
+            continue
+        sr = sorted(yr, key=lambda r: -r['cos_mean_within_year'])
+        n_y = len(sr)
+        n_a = sum(1 for r in sr if r['firm'] == FIRM_A)
+        per = {'n_auditor_years': n_y,
+               'firm_a_baseline_share': n_a / n_y,
+               'top_k': {}}
+        for kp in k_pcts:
+            k = max(1, int(n_y * kp / 100))
+            n_a_top = sum(1 for r in sr[:k] if r['firm'] == FIRM_A)
+            per['top_k'][f'top_{kp}pct'] = {
+                'k': k,
+                'firm_a_in_top': n_a_top,
+                'firm_a_share': n_a_top / k,
+            }
+        out[y] = per
+    return out
+
+
+def main():
+    print('Loading signatures + features...')
+    rows = load_signatures()
+    print(f'  loaded {len(rows):,}')
+
+    print('Computing within-year same-CPA max cosine...')
+    sig_max_within, sig_meta = compute_within_year_max(rows)
+    print(f'  signatures with within-year pair: {len(sig_max_within):,}')
+    n_dropped = len(rows) - len(sig_max_within)
+    print(f'  dropped (singleton within year): {n_dropped:,}')
+
+    ay_rows = auditor_year_aggregation(sig_max_within, sig_meta)
+    print(f'  auditor-years (>={MIN_SIGS_PER_AUDITOR_YEAR} sigs '
+          f'with within-year pair): {len(ay_rows):,}')
+
+    pooled = top_k_breakdown(ay_rows)
+    yearly = per_year_top_k(ay_rows)
+
+    payload = {
+        'generated_at': datetime.now().isoformat(timespec='seconds'),
+        'n_signatures_loaded': len(rows),
+        'n_signatures_with_within_year_pair': len(sig_max_within),
+        'n_singleton_dropped': n_dropped,
+        'min_sigs_per_auditor_year': MIN_SIGS_PER_AUDITOR_YEAR,
+        'n_auditor_years': len(ay_rows),
+        'n_firm_a_auditor_years': sum(1 for r in ay_rows
+                                       if r['firm'] == FIRM_A),
+        'pooled_top_k': pooled,
+        'yearly_top_k': yearly,
+    }
+    json_path = OUT / 'within_year_ranking.json'
+    json_path.write_text(json.dumps(payload, indent=2, ensure_ascii=False),
+                         encoding='utf-8')
+    print(f'\nWrote {json_path}')
+
+    # Markdown
+    md = ['# Within-Year Same-CPA Ranking Robustness',
+          '',
+          f"Generated: {payload['generated_at']}",
+          '',
+          ('Per-signature best-match cosine recomputed using within-year '
+           'same-CPA matching only. See Script 31 docstring for the '
+           'precise definition.'),
+          '',
+          f"- Signatures loaded: {len(rows):,}",
+          f"- Signatures with at least one within-year same-CPA pair: "
+          f"{len(sig_max_within):,}",
+          f"- Singletons dropped (no within-year pair): {n_dropped:,}",
+          f"- Auditor-years with >= {MIN_SIGS_PER_AUDITOR_YEAR} sigs: "
+          f"{len(ay_rows):,}",
+          f"- Firm A auditor-years: {payload['n_firm_a_auditor_years']:,} "
+          f"({100*payload['n_firm_a_auditor_years']/len(ay_rows):.1f}% baseline)",
+          '',
+          '## Pooled (2013-2023) top-K Firm A share',
+          '',
+          '| Top-K | k | Firm A share | A | B | C | D | NB4 |',
+          '|-------|---|--------------|---|---|---|---|-----|']
+    for kp in [10, 20, 25, 30, 50]:
+        d = pooled[f'top_{kp}pct']
+        c = d['firm_counts']
+        md.append(f"| {kp}% | {d['k']:,} | "
+                  f"{100*d['firm_a_share']:.1f}% | "
+                  f"{c.get('Firm A', 0)} | {c.get('Firm B', 0)} | "
+                  f"{c.get('Firm C', 0)} | {c.get('Firm D', 0)} | "
+                  f"{c.get('Non-Big-4', 0)} |")
+
+    md.extend(['',
+               '## Year-by-year top-K Firm A share',
+               '',
+               '| Year | n AY | Top-10% share | Top-20% share | '
+               'Top-30% share | A baseline |',
+               '|------|------|---------------|---------------|'
+               '---------------|------------|'])
+    for y in sorted(yearly):
+        per = yearly[y]
+        line = (f"| {y} | {per['n_auditor_years']:,} ")
+        for kp in [10, 20, 30]:
+            d = per['top_k'][f'top_{kp}pct']
+            line += (f"| {100*d['firm_a_share']:.1f}% "
+                     f"({d['firm_a_in_top']}/{d['k']}) ")
+        line += f"| {100*per['firm_a_baseline_share']:.1f}% |"
+        md.append(line)
+
+    md_path = OUT / 'within_year_ranking.md'
+    md_path.write_text('\n'.join(md) + '\n', encoding='utf-8')
+    print(f'Wrote {md_path}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/32_non_firm_a_calibration.py

@@ -0,0 +1,778 @@
+#!/usr/bin/env python3
+"""
+Script 32: Non-Firm-A Calibration Spike
+========================================
+Research question (branch ``from-outside-of-firmA``):
+    If we throw away Firm A entirely, can we still derive meaningful
+    cosine / dHash thresholds at the accountant level?
+
+Three subset analyses (per user's "1. 我們可以分開做" clarification):
+
+  Subset I  — Big-4 minus Firm A: KPMG + PwC + EY pooled
+  Subset II — All non-Firm-A firms: every firm except 勤業眾信聯合
+  Subset III (baseline reference) — Firm A only
+
+Each subset is run through Script 20's three-method framework
+(KDE+dip, BD/McCrary, 2-component Beta mixture + logit-GMM) plus the
+2D-GMM 2-comp marginal crossing from Script 18, on the
+per-accountant means of:
+  * cos_mean = AVG(s.max_similarity_to_same_accountant)
+  * dh_mean  = AVG(s.min_dhash_independent)
+
+Time-stratified contingency analysis:
+    If Subset I/II fail to expose bimodality, we re-load each
+    accountant's signatures stratified into pre-2018 vs post-2020
+    sub-buckets (>=5 sigs per bucket required) and re-run the
+    three-method framework on the resulting bucket-level means.
+    This tests whether the time axis can substitute for the
+    firm-anchor axis.
+
+Verdict (A/B/C):
+  A  Bimodal structure emerges in Subset I or II without time
+     stratification, with crossings within +-0.02 (cos) / +-2.0 (dh)
+     of Paper A baselines (0.945, 8.10) and dip-test multimodal at
+     alpha=0.05.  -> "outside-Firm-A calibration is viable"
+  B  Bimodal structure only emerges after time stratification.
+     -> "time axis substitutes for firm anchor; v3.21 robustness or
+     Paper C with time-stratified design"
+  C  No bimodality in either; crossings are unstable / outside
+     plausible range.  -> "Firm A is required as anchor; this
+     strengthens Paper A's framing"
+
+Output:
+  reports/non_firm_a_calibration/
+    non_firm_a_calibration_results.json
+    non_firm_a_calibration_report.md
+    panel_<subset>_<measure>.png
+"""
+
+import sqlite3
+import json
+import numpy as np
+import matplotlib
+matplotlib.use('Agg')
+import matplotlib.pyplot as plt
+from pathlib import Path
+from datetime import datetime
+from scipy import stats
+from scipy.signal import find_peaks
+from scipy.optimize import brentq
+from sklearn.mixture import GaussianMixture
+import diptest
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'non_firm_a_calibration')
+OUT.mkdir(parents=True, exist_ok=True)
+
+EPS = 1e-6
+Z_CRIT = 1.96
+MIN_SIGS = 10
+MIN_SIGS_PER_BUCKET = 5
+
+FIRM_A = '勤業眾信聯合'  # Deloitte
+BIG4_NON_A = ('安侯建業聯合', '資誠聯合', '安永聯合')  # KPMG, PwC, EY
+
+PAPER_A_COS_BASELINE = 0.945
+PAPER_A_DH_BASELINE = 8.10
+
+
+# ---------- Loaders ----------
+def _accountant_means_query(firm_filter_sql, params, time_filter_sql=''):
+    sql = f'''
+        SELECT s.assigned_accountant,
+               AVG(s.max_similarity_to_same_accountant) AS cos_mean,
+               AVG(CAST(s.min_dhash_independent AS REAL)) AS dh_mean,
+               COUNT(*) AS n
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.min_dhash_independent IS NOT NULL
+          {firm_filter_sql}
+          {time_filter_sql}
+        GROUP BY s.assigned_accountant
+        HAVING n >= ?
+    '''
+    return sql, params + [MIN_SIGS]
+
+
+def load_subset(label):
+    """Return (cos, dh, n_accountants, n_signatures)."""
+    conn = sqlite3.connect(DB)
+    cur = conn.cursor()
+    if label == 'big4_non_A':
+        firm_filter = 'AND a.firm IN (?, ?, ?)'
+        params = list(BIG4_NON_A)
+    elif label == 'all_non_A':
+        firm_filter = 'AND a.firm IS NOT NULL AND a.firm != ?'
+        params = [FIRM_A]
+    elif label == 'firm_A':
+        firm_filter = 'AND a.firm = ?'
+        params = [FIRM_A]
+    else:
+        raise ValueError(label)
+    sql, p = _accountant_means_query(firm_filter, params)
+    cur.execute(sql, p)
+    rows = cur.fetchall()
+    conn.close()
+    cos = np.array([r[1] for r in rows])
+    dh = np.array([r[2] for r in rows])
+    n_sigs = int(sum(r[3] for r in rows))
+    return cos, dh, len(rows), n_sigs
+
+
+def load_subset_time_stratified(label, period):
+    """Per-accountant means computed only from `period` signatures.
+
+    period: 'pre_2018' (year_month < 2018-01) or 'post_2020' (>= 2020-01).
+    """
+    conn = sqlite3.connect(DB)
+    cur = conn.cursor()
+    if period == 'pre_2018':
+        time_filter = "AND s.year_month < '2018-01'"
+    elif period == 'post_2020':
+        time_filter = "AND s.year_month >= '2020-01'"
+    else:
+        raise ValueError(period)
+    if label == 'big4_non_A':
+        firm_filter = 'AND a.firm IN (?, ?, ?)'
+        params = list(BIG4_NON_A)
+    elif label == 'all_non_A':
+        firm_filter = 'AND a.firm IS NOT NULL AND a.firm != ?'
+        params = [FIRM_A]
+    else:
+        raise ValueError(label)
+    sql = f'''
+        SELECT s.assigned_accountant,
+               AVG(s.max_similarity_to_same_accountant) AS cos_mean,
+               AVG(CAST(s.min_dhash_independent AS REAL)) AS dh_mean,
+               COUNT(*) AS n
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.min_dhash_independent IS NOT NULL
+          {firm_filter}
+          {time_filter}
+        GROUP BY s.assigned_accountant
+        HAVING n >= ?
+    '''
+    cur.execute(sql, params + [MIN_SIGS_PER_BUCKET])
+    rows = cur.fetchall()
+    conn.close()
+    cos = np.array([r[1] for r in rows])
+    dh = np.array([r[2] for r in rows])
+    return cos, dh, len(rows), int(sum(r[3] for r in rows))
+
+
+# ---------- Methods (lifted from Script 20) ----------
+def method_kde_antimode(values):
+    arr = np.asarray(values, dtype=float)
+    arr = arr[np.isfinite(arr)]
+    if len(arr) < 8:
+        return {'n': int(len(arr)), 'note': 'too few points'}
+    dip, pval = diptest.diptest(arr, boot_pval=True, n_boot=2000)
+    kde = stats.gaussian_kde(arr, bw_method='silverman')
+    xs = np.linspace(arr.min(), arr.max(), 2000)
+    density = kde(xs)
+    peaks, _ = find_peaks(density, prominence=density.max() * 0.02)
+    antimodes = []
+    for i in range(len(peaks) - 1):
+        seg = density[peaks[i]:peaks[i + 1]]
+        if not len(seg):
+            continue
+        local = peaks[i] + int(np.argmin(seg))
+        antimodes.append(float(xs[local]))
+    sens = {}
+    for bwf in [0.5, 0.75, 1.0, 1.5, 2.0]:
+        kde_s = stats.gaussian_kde(arr, bw_method=kde.factor * bwf)
+        d_s = kde_s(xs)
+        p_s, _ = find_peaks(d_s, prominence=d_s.max() * 0.02)
+        sens[f'bw_x{bwf}'] = int(len(p_s))
+    return {
+        'n': int(len(arr)),
+        'dip': float(dip),
+        'dip_pvalue': float(pval),
+        'unimodal_alpha05': bool(pval > 0.05),
+        'kde_bandwidth_silverman': float(kde.factor),
+        'n_modes': int(len(peaks)),
+        'mode_locations': [float(xs[p]) for p in peaks],
+        'antimodes': antimodes,
+        'primary_antimode': (antimodes[0] if antimodes else None),
+        'bandwidth_sensitivity_n_modes': sens,
+    }
+
+
+def method_bd_mccrary(values, bin_width, direction):
+    arr = np.asarray(values, dtype=float)
+    arr = arr[np.isfinite(arr)]
+    if len(arr) < 8:
+        return {'n': int(len(arr)), 'note': 'too few points'}
+    lo = float(np.floor(arr.min() / bin_width) * bin_width)
+    hi = float(np.ceil(arr.max() / bin_width) * bin_width)
+    edges = np.arange(lo, hi + bin_width, bin_width)
+    counts, _ = np.histogram(arr, bins=edges)
+    centers = (edges[:-1] + edges[1:]) / 2.0
+    N = counts.sum()
+    p = counts / N if N else counts.astype(float)
+    n_bins = len(counts)
+    z = np.full(n_bins, np.nan)
+    for i in range(1, n_bins - 1):
+        p_lo = p[i - 1]
+        p_hi = p[i + 1]
+        exp_i = 0.5 * (counts[i - 1] + counts[i + 1])
+        var_i = (N * p[i] * (1 - p[i])
+                 + 0.25 * N * (p_lo + p_hi) * (1 - p_lo - p_hi))
+        if var_i <= 0:
+            continue
+        z[i] = (counts[i] - exp_i) / np.sqrt(var_i)
+    transitions = []
+    for i in range(1, len(z)):
+        if np.isnan(z[i - 1]) or np.isnan(z[i]):
+            continue
+        ok = ((direction == 'neg_to_pos' and z[i - 1] < -Z_CRIT and z[i] > Z_CRIT)
+              or (direction == 'pos_to_neg' and z[i - 1] > Z_CRIT and z[i] < -Z_CRIT))
+        if ok:
+            transitions.append({
+                'threshold_between': float((centers[i - 1] + centers[i]) / 2.0),
+                'z_before': float(z[i - 1]),
+                'z_after': float(z[i]),
+            })
+    best = (max(transitions,
+                key=lambda t: abs(t['z_before']) + abs(t['z_after']))
+            if transitions else None)
+    return {
+        'n': int(len(arr)),
+        'bin_width': float(bin_width),
+        'direction': direction,
+        'n_transitions': len(transitions),
+        'transitions': transitions,
+        'threshold': (best['threshold_between'] if best else None),
+    }
+
+
+def fit_beta_mixture_em(x, K=2, max_iter=300, tol=1e-6, seed=42):
+    x = np.clip(np.asarray(x, dtype=float), EPS, 1 - EPS)
+    n = len(x)
+    q = np.linspace(0, 1, K + 1)
+    thresh = np.quantile(x, q[1:-1])
+    labels = np.digitize(x, thresh)
+    resp = np.zeros((n, K))
+    resp[np.arange(n), labels] = 1.0
+    ll_hist = []
+    for it in range(max_iter):
+        nk = resp.sum(axis=0) + 1e-12
+        weights = nk / nk.sum()
+        mus = (resp * x[:, None]).sum(axis=0) / nk
+        var_num = (resp * (x[:, None] - mus) ** 2).sum(axis=0)
+        vars_ = var_num / nk
+        upper = mus * (1 - mus) - 1e-9
+        vars_ = np.minimum(vars_, upper)
+        vars_ = np.maximum(vars_, 1e-9)
+        factor = np.maximum(mus * (1 - mus) / vars_ - 1, 1e-6)
+        alphas = mus * factor
+        betas = (1 - mus) * factor
+        log_pdfs = np.column_stack([
+            stats.beta.logpdf(x, alphas[k], betas[k]) + np.log(weights[k])
+            for k in range(K)
+        ])
+        m = log_pdfs.max(axis=1, keepdims=True)
+        ll = (m.ravel() + np.log(np.exp(log_pdfs - m).sum(axis=1))).sum()
+        ll_hist.append(float(ll))
+        new_resp = np.exp(log_pdfs - m)
+        new_resp = new_resp / new_resp.sum(axis=1, keepdims=True)
+        if it > 0 and abs(ll_hist[-1] - ll_hist[-2]) < tol:
+            resp = new_resp
+            break
+        resp = new_resp
+    order = np.argsort(mus)
+    alphas = alphas[order]
+    betas = betas[order]
+    weights = weights[order]
+    mus = mus[order]
+    k_params = 3 * K - 1
+    ll_final = ll_hist[-1]
+    return {
+        'K': K,
+        'alphas': [float(a) for a in alphas],
+        'betas': [float(b) for b in betas],
+        'weights': [float(w) for w in weights],
+        'mus': [float(m) for m in mus],
+        'log_likelihood': ll_final,
+        'aic': float(2 * k_params - 2 * ll_final),
+        'bic': float(k_params * np.log(n) - 2 * ll_final),
+        'n_iter': it + 1,
+    }
+
+
+def beta_crossing(fit):
+    if fit['K'] != 2:
+        return None
+    a1, b1, w1 = fit['alphas'][0], fit['betas'][0], fit['weights'][0]
+    a2, b2, w2 = fit['alphas'][1], fit['betas'][1], fit['weights'][1]
+
+    def diff(x):
+        return (w2 * stats.beta.pdf(x, a2, b2)
+                - w1 * stats.beta.pdf(x, a1, b1))
+    xs = np.linspace(EPS, 1 - EPS, 2000)
+    ys = diff(xs)
+    changes = np.where(np.diff(np.sign(ys)) != 0)[0]
+    if not len(changes):
+        return None
+    mid = 0.5 * (fit['mus'][0] + fit['mus'][1])
+    crossings = []
+    for i in changes:
+        try:
+            crossings.append(brentq(diff, xs[i], xs[i + 1]))
+        except ValueError:
+            continue
+    if not crossings:
+        return None
+    return float(min(crossings, key=lambda c: abs(c - mid)))
+
+
+def fit_logit_gmm(x, K=2, seed=42):
+    x = np.clip(np.asarray(x, dtype=float), EPS, 1 - EPS)
+    z = np.log(x / (1 - x)).reshape(-1, 1)
+    gmm = GaussianMixture(n_components=K, random_state=seed,
+                          max_iter=500).fit(z)
+    order = np.argsort(gmm.means_.ravel())
+    means = gmm.means_.ravel()[order]
+    stds = np.sqrt(gmm.covariances_.ravel())[order]
+    weights = gmm.weights_[order]
+    crossing = None
+    if K == 2:
+        m1, s1, w1 = means[0], stds[0], weights[0]
+        m2, s2, w2 = means[1], stds[1], weights[1]
+
+        def diff(z0):
+            return (w2 * stats.norm.pdf(z0, m2, s2)
+                    - w1 * stats.norm.pdf(z0, m1, s1))
+        zs = np.linspace(min(m1, m2) - 2, max(m1, m2) + 2, 2000)
+        ys = diff(zs)
+        ch = np.where(np.diff(np.sign(ys)) != 0)[0]
+        if len(ch):
+            try:
+                z_cross = brentq(diff, zs[ch[0]], zs[ch[0] + 1])
+                crossing = float(1 / (1 + np.exp(-z_cross)))
+            except ValueError:
+                pass
+    return {
+        'K': K,
+        'means_logit': [float(m) for m in means],
+        'stds_logit': [float(s) for s in stds],
+        'weights': [float(w) for w in weights],
+        'means_original_scale': [float(1 / (1 + np.exp(-m))) for m in means],
+        'aic': float(gmm.aic(z)),
+        'bic': float(gmm.bic(z)),
+        'crossing_original': crossing,
+    }
+
+
+def method_beta_mixture(values, is_cosine=True):
+    arr = np.asarray(values, dtype=float)
+    arr = arr[np.isfinite(arr)]
+    if len(arr) < 8:
+        return {'n': int(len(arr)), 'note': 'too few points'}
+    x = arr if is_cosine else arr / 64.0
+    beta2 = fit_beta_mixture_em(x, K=2)
+    beta3 = fit_beta_mixture_em(x, K=3)
+    cross_beta2 = beta_crossing(beta2)
+    if not is_cosine and cross_beta2 is not None:
+        cross_beta2 = cross_beta2 * 64.0
+    gmm2 = fit_logit_gmm(x, K=2)
+    gmm3 = fit_logit_gmm(x, K=3)
+    if not is_cosine and gmm2.get('crossing_original') is not None:
+        gmm2['crossing_original'] = gmm2['crossing_original'] * 64.0
+    return {
+        'n': int(len(x)),
+        'scale_transform': ('identity' if is_cosine else 'dhash/64'),
+        'beta_2': beta2,
+        'beta_3': beta3,
+        'bic_preferred_K': (2 if beta2['bic'] < beta3['bic'] else 3),
+        'beta_2_crossing_original': cross_beta2,
+        'logit_gmm_2': gmm2,
+        'logit_gmm_3': gmm3,
+    }
+
+
+def gmm_2d_marginal_crossing(cos, dh, dim):
+    """2-comp 2D GMM, then marginal crossing on the requested dim."""
+    X = np.column_stack([cos, dh])
+    if len(X) < 8:
+        return None
+    gmm = GaussianMixture(n_components=2, covariance_type='full',
+                          random_state=42, n_init=15, max_iter=500).fit(X)
+    means = gmm.means_
+    covs = gmm.covariances_
+    weights = gmm.weights_
+    m1, m2 = means[0][dim], means[1][dim]
+    s1 = np.sqrt(covs[0][dim, dim])
+    s2 = np.sqrt(covs[1][dim, dim])
+    w1, w2 = weights[0], weights[1]
+
+    def diff(x):
+        return (w2 * stats.norm.pdf(x, m2, s2)
+                - w1 * stats.norm.pdf(x, m1, s1))
+    xs = np.linspace(float(X[:, dim].min()), float(X[:, dim].max()), 2000)
+    ys = diff(xs)
+    ch = np.where(np.diff(np.sign(ys)) != 0)[0]
+    if not len(ch):
+        return None
+    mid = 0.5 * (m1 + m2)
+    crossings = []
+    for i in ch:
+        try:
+            crossings.append(brentq(diff, xs[i], xs[i + 1]))
+        except ValueError:
+            continue
+    if not crossings:
+        return None
+    return float(min(crossings, key=lambda c: abs(c - mid)))
+
+
+def gmm_2d_3comp_summary(cos, dh):
+    """K=3 2D GMM for completeness; report component means + weights."""
+    X = np.column_stack([cos, dh])
+    if len(X) < 12:
+        return None
+    gmm = GaussianMixture(n_components=3, covariance_type='full',
+                          random_state=42, n_init=15, max_iter=500).fit(X)
+    order = np.argsort(gmm.means_[:, 0])  # sort by cosine ascending
+    return {
+        'means': [[float(m[0]), float(m[1])] for m in gmm.means_[order]],
+        'weights': [float(w) for w in gmm.weights_[order]],
+        'bic': float(gmm.bic(X)),
+        'aic': float(gmm.aic(X)),
+    }
+
+
+# ---------- Driver ----------
+def run_three_method(cos, dh, label):
+    results = {}
+    for desc, arr, bin_width, direction, is_cos in [
+        ('cos_mean', cos, 0.002, 'neg_to_pos', True),
+        ('dh_mean', dh, 0.2, 'pos_to_neg', False),
+    ]:
+        m1 = method_kde_antimode(arr)
+        m2 = method_bd_mccrary(arr, bin_width, direction)
+        m3 = method_beta_mixture(arr, is_cosine=is_cos)
+        gmm2_marginal = gmm_2d_marginal_crossing(
+            cos, dh, dim=(0 if desc == 'cos_mean' else 1))
+        results[desc] = {
+            'method_1_kde_antimode': m1,
+            'method_2_bd_mccrary': m2,
+            'method_3_beta_mixture': m3,
+            'gmm_2d_2comp_marginal_crossing': gmm2_marginal,
+        }
+    results['gmm_2d_3comp'] = gmm_2d_3comp_summary(cos, dh)
+    return results
+
+
+def plot_panel(values, methods, title, out_path, bin_width=None):
+    arr = np.asarray(values, dtype=float)
+    fig, axes = plt.subplots(2, 1, figsize=(11, 7),
+                             gridspec_kw={'height_ratios': [3, 1]})
+    ax = axes[0]
+    if bin_width is None:
+        bins = 40
+    else:
+        lo = float(np.floor(arr.min() / bin_width) * bin_width)
+        hi = float(np.ceil(arr.max() / bin_width) * bin_width)
+        bins = np.arange(lo, hi + bin_width, bin_width)
+    ax.hist(arr, bins=bins, density=True, alpha=0.55, color='steelblue',
+            edgecolor='white')
+    kde = stats.gaussian_kde(arr, bw_method='silverman')
+    xs = np.linspace(arr.min(), arr.max(), 500)
+    ax.plot(xs, kde(xs), 'g-', lw=1.5, label='KDE (Silverman)')
+    colors = {'kde': 'green', 'bd': 'red', 'beta': 'purple',
+              'gmm2': 'orange', 'baseline': 'black'}
+    for key, (val, lbl) in methods.items():
+        if val is None:
+            continue
+        ls = ':' if key == 'baseline' else '--'
+        ax.axvline(val, color=colors.get(key, 'gray'), lw=2, ls=ls,
+                   label=f'{lbl} = {val:.4f}')
+    ax.set_xlabel(title)
+    ax.set_ylabel('Density')
+    ax.set_title(title)
+    ax.legend(fontsize=8)
+
+    ax2 = axes[1]
+    ax2.set_title('Thresholds across methods')
+    ax2.set_xlim(ax.get_xlim())
+    for i, (key, (val, lbl)) in enumerate(methods.items()):
+        if val is None:
+            continue
+        ax2.scatter([val], [i], color=colors.get(key, 'gray'), s=100, zorder=3)
+        ax2.annotate(f'  {lbl}: {val:.4f}', (val, i), fontsize=8, va='center')
+    ax2.set_yticks(range(len(methods)))
+    ax2.set_yticklabels([m for m in methods.keys()])
+    ax2.grid(alpha=0.3)
+    plt.tight_layout()
+    fig.savefig(out_path, dpi=150)
+    plt.close()
+
+
+def emit_panel(subset_label, results):
+    for desc, bin_width in [('cos_mean', 0.002), ('dh_mean', 0.2)]:
+        if 'note' in results[desc]['method_1_kde_antimode']:
+            continue
+        baseline = (PAPER_A_COS_BASELINE if desc == 'cos_mean'
+                    else PAPER_A_DH_BASELINE)
+        methods_for_plot = {
+            'kde': (results[desc]['method_1_kde_antimode'].get('primary_antimode'),
+                    'KDE antimode'),
+            'bd': (results[desc]['method_2_bd_mccrary'].get('threshold'),
+                   'BD/McCrary'),
+            'beta': (results[desc]['method_3_beta_mixture'].get(
+                'beta_2_crossing_original'), 'Beta-2 crossing'),
+            'gmm2': (results[desc]['gmm_2d_2comp_marginal_crossing'],
+                     '2D GMM 2-comp'),
+            'baseline': (baseline, f'Paper A baseline'),
+        }
+        # Need data array for the plot; reload for size only
+        arr = np.array([])  # filled by caller via closure if needed
+        # Plot caller passes arr
+        return methods_for_plot
+    return None
+
+
+def classify_verdict(results_by_subset):
+    """Return ('A'|'B'|'C', explanation)."""
+    def well_separated(res, baseline_cos, baseline_dh):
+        cos_cross = res['cos_mean']['method_3_beta_mixture'].get(
+            'beta_2_crossing_original')
+        dh_cross = res['dh_mean']['method_3_beta_mixture'].get(
+            'beta_2_crossing_original')
+        cos_dip_p = res['cos_mean']['method_1_kde_antimode'].get('dip_pvalue')
+        dh_dip_p = res['dh_mean']['method_1_kde_antimode'].get('dip_pvalue')
+        cos_ok = (cos_cross is not None
+                  and abs(cos_cross - baseline_cos) <= 0.02
+                  and cos_dip_p is not None and cos_dip_p <= 0.05)
+        dh_ok = (dh_cross is not None
+                 and abs(dh_cross - baseline_dh) <= 2.0
+                 and dh_dip_p is not None and dh_dip_p <= 0.05)
+        return cos_ok, dh_ok
+
+    for subset in ('big4_non_A', 'all_non_A'):
+        res = results_by_subset.get(subset)
+        if not res:
+            continue
+        cos_ok, dh_ok = well_separated(res, PAPER_A_COS_BASELINE,
+                                       PAPER_A_DH_BASELINE)
+        if cos_ok and dh_ok:
+            return 'A', (f"Subset '{subset}' shows bimodal cos+dh with "
+                         f"crossings within tolerance of Paper A baselines.")
+    # B: time-stratified rescues it?
+    for subset_period in ('big4_non_A_pre_2018',
+                          'big4_non_A_post_2020',
+                          'all_non_A_pre_2018',
+                          'all_non_A_post_2020'):
+        res = results_by_subset.get(subset_period)
+        if not res:
+            continue
+        cos_ok, dh_ok = well_separated(res, PAPER_A_COS_BASELINE,
+                                       PAPER_A_DH_BASELINE)
+        if cos_ok and dh_ok:
+            return 'B', (f"Time-stratified subset '{subset_period}' recovers "
+                         f"separable bimodality.")
+    return 'C', ("Neither pooled nor time-stratified non-Firm-A calibration "
+                 "produces a baseline-consistent bimodal threshold.")
+
+
+def render_report(results_by_subset, sample_sizes, verdict):
+    md = [
+        '# Non-Firm-A Calibration Spike (Script 32)',
+        f"Generated: {datetime.now().strftime('%Y-%m-%d %H:%M:%S')}",
+        '',
+        '## Research Question',
+        '',
+        ('If we exclude Firm A (Deloitte) from calibration, can the '
+         'three-method framework still recover a meaningful '
+         'cosine / dHash threshold at the accountant level?'),
+        '',
+        '## Sample Sizes',
+        '',
+        '| Subset | N accountants (>=10 sigs) | N signatures |',
+        '|--------|---------------------------|--------------|',
+    ]
+    for label, (n_acc, n_sig) in sample_sizes.items():
+        md.append(f'| `{label}` | {n_acc} | {n_sig} |')
+    md += ['',
+           '## Paper A Baselines (for comparison)',
+           '',
+           f'- Accountant-level 2D GMM 2-comp marginal crossings: '
+           f'cos = **{PAPER_A_COS_BASELINE}**, dHash = **{PAPER_A_DH_BASELINE}**',
+           '']
+
+    for label, results in results_by_subset.items():
+        md += [f'## Subset: `{label}`', '']
+        for measure, baseline in [('cos_mean', PAPER_A_COS_BASELINE),
+                                  ('dh_mean', PAPER_A_DH_BASELINE)]:
+            r = results[measure]
+            md += [f'### {measure}', '',
+                   '| Method | Threshold | Supporting statistic |',
+                   '|--------|-----------|----------------------|']
+            kde = r['method_1_kde_antimode']
+            if 'note' in kde:
+                md.append(f'| Method 1: KDE+dip | n/a | {kde["note"]} |')
+            else:
+                tag = 'unimodal' if kde['unimodal_alpha05'] else 'multimodal'
+                md.append(
+                    f'| Method 1: KDE antimode (dip test) | '
+                    f'{kde["primary_antimode"]} | '
+                    f'dip={kde["dip"]:.4f}, p={kde["dip_pvalue"]:.4f} '
+                    f'({tag}); n_modes={kde["n_modes"]} |')
+            bd = r['method_2_bd_mccrary']
+            md.append(
+                f'| Method 2: BD/McCrary | {bd.get("threshold")} | '
+                f'{bd.get("n_transitions", 0)} transition(s) |')
+            beta = r['method_3_beta_mixture']
+            if 'note' in beta:
+                md.append(f'| Method 3: Beta mixture | n/a | {beta["note"]} |')
+            else:
+                md.append(
+                    f'| Method 3: 2-comp Beta mixture | '
+                    f'{beta["beta_2_crossing_original"]} | '
+                    f'Beta-2 BIC={beta["beta_2"]["bic"]:.2f}, '
+                    f'Beta-3 BIC={beta["beta_3"]["bic"]:.2f} '
+                    f'(K*={beta["bic_preferred_K"]}) |')
+                md.append(
+                    f'| Method 3\': LogGMM-2 | '
+                    f'{beta["logit_gmm_2"].get("crossing_original")} | '
+                    f'logit-Gaussian robustness check |')
+            md.append(
+                f'| 2D GMM 2-comp marginal crossing | '
+                f'{r["gmm_2d_2comp_marginal_crossing"]} | '
+                f'paired with Paper A baseline = {baseline} |')
+            md.append('')
+        if results.get('gmm_2d_3comp'):
+            g3 = results['gmm_2d_3comp']
+            md += ['### 2D GMM K=3 components (for completeness)',
+                   '',
+                   '| Component | mean cos | mean dh | weight |',
+                   '|-----------|----------|---------|--------|']
+            for i, (m, w) in enumerate(zip(g3['means'], g3['weights'])):
+                md.append(f'| C{i + 1} | {m[0]:.4f} | {m[1]:.4f} | {w:.3f} |')
+            md.append('')
+            md.append(f'BIC(K=3 2D)={g3["bic"]:.2f}, AIC={g3["aic"]:.2f}')
+            md.append('')
+
+    md += ['## Verdict',
+           '',
+           f'**{verdict[0]}** — {verdict[1]}',
+           '',
+           '### Verdict legend',
+           '- **A**: outside-Firm-A calibration is viable in pooled form '
+           '(crossings within +-0.02 cos / +-2.0 dh of Paper A baselines '
+           'AND dip-test multimodal at alpha=0.05).',
+           '- **B**: time-stratified subset recovers separable bimodality.',
+           '- **C**: neither rescue works; Firm A remains required as anchor.',
+           '']
+    return '\n'.join(md)
+
+
+def main():
+    print('=' * 72)
+    print('Script 32: Non-Firm-A Calibration Spike')
+    print('=' * 72)
+
+    sample_sizes = {}
+    results_by_subset = {}
+    arrays_by_subset = {}
+
+    # --- Pooled subsets ---
+    for label in ('big4_non_A', 'all_non_A', 'firm_A'):
+        cos, dh, n_acc, n_sig = load_subset(label)
+        sample_sizes[label] = (n_acc, n_sig)
+        arrays_by_subset[label] = (cos, dh)
+        print(f'\n[{label}] N accountants={n_acc}, N sigs={n_sig}')
+        results_by_subset[label] = run_three_method(cos, dh, label)
+        for desc in ('cos_mean', 'dh_mean'):
+            r = results_by_subset[label][desc]
+            kde = r['method_1_kde_antimode']
+            beta = r['method_3_beta_mixture']
+            print(f'  {desc}: dip p={kde.get("dip_pvalue")} '
+                  f'(n_modes={kde.get("n_modes")}); '
+                  f'Beta-2 cross={beta.get("beta_2_crossing_original")}; '
+                  f'2D-GMM marginal={r["gmm_2d_2comp_marginal_crossing"]}')
+
+    # --- Time-stratified secondary (run unconditionally; verdict logic decides) ---
+    for label in ('big4_non_A', 'all_non_A'):
+        for period in ('pre_2018', 'post_2020'):
+            cos, dh, n_acc, n_sig = load_subset_time_stratified(label, period)
+            key = f'{label}_{period}'
+            sample_sizes[key] = (n_acc, n_sig)
+            arrays_by_subset[key] = (cos, dh)
+            print(f'\n[{key}] N accountants={n_acc}, N sigs={n_sig}')
+            if n_acc < 8:
+                print(f'  (skipped: too few accountants for analysis)')
+                continue
+            results_by_subset[key] = run_three_method(cos, dh, key)
+            for desc in ('cos_mean', 'dh_mean'):
+                r = results_by_subset[key][desc]
+                kde = r['method_1_kde_antimode']
+                beta = r['method_3_beta_mixture']
+                print(f'  {desc}: dip p={kde.get("dip_pvalue")} '
+                      f'(n_modes={kde.get("n_modes")}); '
+                      f'Beta-2 cross={beta.get("beta_2_crossing_original")}; '
+                      f'2D-GMM marginal={r["gmm_2d_2comp_marginal_crossing"]}')
+
+    # --- Plots ---
+    for label, results in results_by_subset.items():
+        cos, dh = arrays_by_subset[label]
+        for desc, arr, bin_width, baseline in [
+            ('cos_mean', cos, 0.002, PAPER_A_COS_BASELINE),
+            ('dh_mean', dh, 0.2, PAPER_A_DH_BASELINE),
+        ]:
+            r = results[desc]
+            if 'note' in r['method_1_kde_antimode']:
+                continue
+            methods_for_plot = {
+                'kde': (r['method_1_kde_antimode'].get('primary_antimode'),
+                        'KDE antimode'),
+                'bd': (r['method_2_bd_mccrary'].get('threshold'),
+                       'BD/McCrary'),
+                'beta': (r['method_3_beta_mixture'].get(
+                    'beta_2_crossing_original'), 'Beta-2 crossing'),
+                'gmm2': (r['gmm_2d_2comp_marginal_crossing'],
+                         '2D GMM 2-comp'),
+                'baseline': (baseline, 'Paper A baseline'),
+            }
+            png = OUT / f'panel_{label}_{desc}.png'
+            plot_panel(arr, methods_for_plot,
+                       f'{label} -- accountant-level {desc}',
+                       png, bin_width=bin_width)
+            print(f'  plot: {png}')
+
+    # --- Verdict ---
+    verdict = classify_verdict(results_by_subset)
+    print(f'\nVerdict: {verdict[0]} -- {verdict[1]}')
+
+    # --- Persist ---
+    payload = {
+        'generated_at': datetime.now().isoformat(),
+        'min_sigs_per_accountant': MIN_SIGS,
+        'min_sigs_per_bucket_time_stratified': MIN_SIGS_PER_BUCKET,
+        'paper_a_baselines': {
+            'cos': PAPER_A_COS_BASELINE,
+            'dh': PAPER_A_DH_BASELINE,
+        },
+        'sample_sizes': {k: {'n_accountants': v[0], 'n_signatures': v[1]}
+                          for k, v in sample_sizes.items()},
+        'results': results_by_subset,
+        'verdict': {'class': verdict[0], 'explanation': verdict[1]},
+    }
+    (OUT / 'non_firm_a_calibration_results.json').write_text(
+        json.dumps(payload, indent=2, ensure_ascii=False), encoding='utf-8')
+    print(f'\nJSON: {OUT / "non_firm_a_calibration_results.json"}')
+
+    md = render_report(results_by_subset, sample_sizes, verdict)
+    (OUT / 'non_firm_a_calibration_report.md').write_text(md, encoding='utf-8')
+    print(f'Report: {OUT / "non_firm_a_calibration_report.md"}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/33_reverse_anchor_spike.py

@@ -0,0 +1,437 @@
+#!/usr/bin/env python3
+"""
+Script 33: Reverse-Anchor Spike
+================================
+Follow-up to Script 32 verdict C.
+
+Hypothesis:
+    Instead of using Firm A as the "hand-signed anchor" (Paper A's
+    framing), use the non-Firm-A population as the
+    "fully-replicated reference" and detect hand-signed CPAs by
+    their deviation from that reference.
+
+Why this might be better:
+  * Reference population is 3x larger (515 vs 171 accountants)
+  * Removes the "why is Firm A ground truth?" reviewer attack
+  * Firm A becomes a validation target, not the calibration anchor
+
+Pipeline:
+  1. Build 2D Gaussian reference from all_non_A accountant means
+     (cos_mean, dh_mean), with robust covariance estimate.
+  2. Score every Firm A accountant by:
+       * Mahalanobis distance to the reference center
+       * Log-likelihood under the 2D Gaussian reference
+       * Tail percentile in the marginal cosine direction
+         (low = more hand-signed-like)
+  3. Cross-validate against Paper A's existing per-CPA hand-sign
+     proxy: fraction of that CPA's signatures with
+       (cos < 0.95) OR (dh > 5)
+     This is the same operational rule used in Paper A v3.20.0
+     (cos>0.95 AND dh<=5 -> non-hand-signed) inverted to a hand-sign
+     fraction.
+  4. Verdict on Paper C viability (uses the directional metric
+     -cos_left_tail_pct as primary; symmetric Mahalanobis confounds
+     "more-replicated" and "more-hand-signed" anomaly directions):
+       PAPER_C_STRONG    Spearman rho_directional >= 0.70
+       PAPER_C_PARTIAL   0.40 <= rho_directional < 0.70
+       PAPER_C_WEAK      rho_directional < 0.40 OR n_firmA < 30
+     A large |rho_mahalanobis| with opposite sign is reported as
+     "two-sided anomaly" diagnostic (Firm A bifurcates into both
+     extreme-replicated and hand-signed sub-populations).
+
+Output:
+  reports/reverse_anchor_spike/
+    reverse_anchor_results.json
+    reverse_anchor_report.md
+    scatter_anomaly_vs_paperA.png
+    ranked_firmA_cpas.csv
+"""
+
+import sqlite3
+import json
+import csv
+import numpy as np
+import matplotlib
+matplotlib.use('Agg')
+import matplotlib.pyplot as plt
+from pathlib import Path
+from datetime import datetime
+from scipy import stats
+from sklearn.covariance import MinCovDet
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'reverse_anchor_spike')
+OUT.mkdir(parents=True, exist_ok=True)
+
+FIRM_A = '勤業眾信聯合'  # Deloitte
+MIN_SIGS = 10
+
+# Paper A v3.20.0 operational signature-level rule (non-hand-signed):
+#   cos > 0.95 AND dh_indep <= 5
+# Hand-sign fraction = 1 - (fraction passing this rule)
+PAPER_A_COS_CUT = 0.95
+PAPER_A_DH_CUT = 5
+
+
+def load_accountant_table(firm_filter_sql, params):
+    """Return list of (name, cos_mean, dh_mean, hand_frac, n)."""
+    conn = sqlite3.connect(DB)
+    cur = conn.cursor()
+    sql = f'''
+        SELECT s.assigned_accountant,
+               AVG(s.max_similarity_to_same_accountant) AS cos_mean,
+               AVG(CAST(s.min_dhash_independent AS REAL)) AS dh_mean,
+               AVG(CASE
+                     WHEN s.max_similarity_to_same_accountant > ?
+                          AND s.min_dhash_independent <= ?
+                     THEN 0.0 ELSE 1.0
+                   END) AS hand_frac,
+               COUNT(*) AS n
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.min_dhash_independent IS NOT NULL
+          {firm_filter_sql}
+        GROUP BY s.assigned_accountant
+        HAVING n >= ?
+    '''
+    cur.execute(sql, [PAPER_A_COS_CUT, PAPER_A_DH_CUT] + params + [MIN_SIGS])
+    rows = cur.fetchall()
+    conn.close()
+    return [(r[0], float(r[1]), float(r[2]), float(r[3]), int(r[4]))
+            for r in rows]
+
+
+def fit_reference_gaussian(points):
+    """Fit a 2D Gaussian to the reference population using MCD for
+    robustness against the small handful of non-Firm-A CPAs that may
+    themselves contain hand-signed contamination.
+    """
+    X = np.asarray(points, dtype=float)
+    mcd = MinCovDet(random_state=42, support_fraction=0.85).fit(X)
+    return {
+        'mean': mcd.location_,
+        'cov': mcd.covariance_,
+        'cov_inv': np.linalg.inv(mcd.covariance_),
+        'support_fraction': 0.85,
+        'n_reference': int(len(X)),
+    }
+
+
+def score_under_reference(point, ref):
+    """Return (mahalanobis_distance, log_likelihood, tail_percentile_cos).
+
+    tail_percentile_cos: P(reference cosine <= point_cos) -- a small
+    value means the point sits in the LEFT tail of the reference
+    cosine distribution (lower than typical replicated population),
+    which is the direction we expect for hand-signed CPAs.
+    """
+    diff = np.asarray(point, dtype=float) - ref['mean']
+    md_sq = float(diff @ ref['cov_inv'] @ diff)
+    md = float(np.sqrt(max(md_sq, 0.0)))
+    # Multivariate normal log-likelihood (kernel only matters for ranking)
+    sign, logdet = np.linalg.slogdet(ref['cov'])
+    ll = float(-0.5 * (md_sq + logdet + 2 * np.log(2 * np.pi)))
+    # Marginal cosine tail percentile under reference Gaussian
+    mu_c = ref['mean'][0]
+    sd_c = float(np.sqrt(ref['cov'][0, 0]))
+    tail = float(stats.norm.cdf(point[0], loc=mu_c, scale=sd_c))
+    return md, ll, tail
+
+
+def render_scatter(firmA_data, ref, out_path):
+    """Anomaly score (Mahalanobis) vs Paper A hand-sign fraction."""
+    md = np.array([d['mahalanobis'] for d in firmA_data])
+    hf = np.array([d['paperA_hand_frac'] for d in firmA_data])
+    fig, ax = plt.subplots(figsize=(8, 6))
+    ax.scatter(md, hf, s=40, alpha=0.6, color='steelblue', edgecolor='white')
+    rho, p = stats.spearmanr(md, hf)
+    pearson_r, pearson_p = stats.pearsonr(md, hf)
+    ax.set_xlabel('Mahalanobis distance to non-Firm-A reference '
+                  '(higher = more anomalous)')
+    ax.set_ylabel('Paper A signature-level hand-sign fraction\n'
+                  '(NOT [cos>0.95 AND dh<=5])')
+    ax.set_title(f'Firm A CPAs: reverse-anchor anomaly vs Paper A label\n'
+                 f'Spearman rho={rho:.3f} (p={p:.2e}); '
+                 f'Pearson r={pearson_r:.3f}')
+    ax.grid(alpha=0.3)
+    fig.tight_layout()
+    fig.savefig(out_path, dpi=150)
+    plt.close(fig)
+    return float(rho), float(p), float(pearson_r), float(pearson_p)
+
+
+def render_2d_overlay(ref_points, firmA_points, ref, out_path):
+    """2D scatter of both populations + reference center + 1/2/3-sigma
+    Mahalanobis ellipses."""
+    fig, ax = plt.subplots(figsize=(9, 7))
+    ax.scatter(ref_points[:, 0], ref_points[:, 1], s=18, alpha=0.4,
+               color='gray', label=f'Non-Firm-A CPAs (n={len(ref_points)})')
+    ax.scatter(firmA_points[:, 0], firmA_points[:, 1], s=42, alpha=0.85,
+               color='crimson', edgecolor='white',
+               label=f'Firm A CPAs (n={len(firmA_points)})')
+    # Reference Gaussian ellipses
+    eigvals, eigvecs = np.linalg.eigh(ref['cov'])
+    angle = float(np.degrees(np.arctan2(eigvecs[1, 0], eigvecs[0, 0])))
+    from matplotlib.patches import Ellipse
+    for k_sigma, ls in [(1, '-'), (2, '--'), (3, ':')]:
+        width = 2 * k_sigma * float(np.sqrt(eigvals[0]))
+        height = 2 * k_sigma * float(np.sqrt(eigvals[1]))
+        e = Ellipse(xy=ref['mean'], width=width, height=height, angle=angle,
+                    fill=False, edgecolor='black', lw=1.4, ls=ls,
+                    label=f'{k_sigma}-sigma reference contour')
+        ax.add_patch(e)
+    ax.scatter([ref['mean'][0]], [ref['mean'][1]], marker='+', s=160,
+               color='black', label='Reference center (MCD)')
+    ax.set_xlabel('Accountant cos_mean')
+    ax.set_ylabel('Accountant dh_mean')
+    ax.set_title('Reverse-anchor: non-Firm-A reference Gaussian + Firm A overlay')
+    ax.legend(fontsize=8, loc='upper right')
+    ax.grid(alpha=0.3)
+    fig.tight_layout()
+    fig.savefig(out_path, dpi=150)
+    plt.close(fig)
+
+
+def classify_verdict(rho_directional, p_directional, rho_mahalanobis,
+                     n_firmA):
+    bifurcation = (
+        f'(diagnostic: rho_mahalanobis={rho_mahalanobis:.3f} -- a large '
+        f'magnitude with opposite sign indicates Firm A bifurcates into '
+        f'BOTH ultra-replicated and hand-signed sub-populations relative '
+        f'to the non-Firm-A reference center, rather than only deviating '
+        f'in the hand-sign direction.)')
+    if n_firmA < 30:
+        return 'PAPER_C_WEAK', (
+            f'Only {n_firmA} Firm A CPAs meet n>=10 -- statistical '
+            f'underpowering precludes a reliable correlation.')
+    if rho_directional >= 0.70 and p_directional < 0.001:
+        return 'PAPER_C_STRONG', (
+            f'Directional Spearman rho={rho_directional:.3f} '
+            f'(p={p_directional:.2e}) -- reverse-anchor with directional '
+            f'cosine-left-tail score recovers Paper A label; Paper C '
+            f'viable. {bifurcation}')
+    if rho_directional >= 0.40 and p_directional < 0.05:
+        return 'PAPER_C_PARTIAL', (
+            f'Directional Spearman rho={rho_directional:.3f} '
+            f'(p={p_directional:.2e}) -- moderate directional alignment; '
+            f'reverse-anchor captures part of the signal. {bifurcation}')
+    return 'PAPER_C_WEAK', (
+        f'Directional Spearman rho={rho_directional:.3f} '
+        f'(p={p_directional:.2e}) -- reverse-anchor diverges from Paper '
+        f'A label even in the directional formulation. {bifurcation}')
+
+
+def main():
+    print('=' * 72)
+    print('Script 33: Reverse-Anchor Spike')
+    print('=' * 72)
+
+    # 1. Reference: all_non_A
+    ref_rows = load_accountant_table(
+        'AND a.firm IS NOT NULL AND a.firm != ?', [FIRM_A])
+    print(f'\nReference population (all_non_A): {len(ref_rows)} CPAs')
+    ref_points = np.array([[r[1], r[2]] for r in ref_rows])
+    ref = fit_reference_gaussian(ref_points)
+    print(f'  Reference center (MCD): cos={ref["mean"][0]:.4f}, '
+          f'dh={ref["mean"][1]:.4f}')
+    print(f'  Reference cov diag: var(cos)={ref["cov"][0,0]:.5f}, '
+          f'var(dh)={ref["cov"][1,1]:.4f}, '
+          f'cov(cos,dh)={ref["cov"][0,1]:.5f}')
+
+    # 2. Score: Firm A
+    firmA_rows = load_accountant_table('AND a.firm = ?', [FIRM_A])
+    print(f'\nTarget population (Firm A): {len(firmA_rows)} CPAs')
+    firmA_points = np.array([[r[1], r[2]] for r in firmA_rows])
+
+    firmA_data = []
+    for (name, cos_m, dh_m, hand_frac, n_sig) in firmA_rows:
+        md, ll, tail_cos = score_under_reference([cos_m, dh_m], ref)
+        firmA_data.append({
+            'cpa': name,
+            'n_signatures': n_sig,
+            'cos_mean': cos_m,
+            'dh_mean': dh_m,
+            'paperA_hand_frac': hand_frac,
+            'mahalanobis': md,
+            'log_likelihood': ll,
+            'cos_left_tail_pct': tail_cos,
+        })
+
+    # 3. Scatter + correlation
+    scatter_png = OUT / 'scatter_anomaly_vs_paperA.png'
+    rho, rho_p, pearson_r, pearson_p = render_scatter(
+        firmA_data, ref, scatter_png)
+    print(f'\nSpearman rho (Mahalanobis vs Paper A hand_frac) = '
+          f'{rho:.4f} (p={rho_p:.2e})')
+    print(f'Pearson  r              = {pearson_r:.4f} (p={pearson_p:.2e})')
+
+    # Also Spearman for log-likelihood (negated, since higher LL = less anomalous)
+    md_arr = np.array([d['mahalanobis'] for d in firmA_data])
+    ll_arr = np.array([d['log_likelihood'] for d in firmA_data])
+    tail_arr = np.array([d['cos_left_tail_pct'] for d in firmA_data])
+    hf_arr = np.array([d['paperA_hand_frac'] for d in firmA_data])
+    rho_ll, p_ll = stats.spearmanr(-ll_arr, hf_arr)
+    rho_tail, p_tail = stats.spearmanr(-tail_arr, hf_arr)  # negated: small tail = high hand_frac expected
+    print(f'Spearman rho (-log-likelihood vs hand_frac) = '
+          f'{rho_ll:.4f} (p={p_ll:.2e})')
+    print(f'Spearman rho (-cos_left_tail_pct vs hand_frac) = '
+          f'{rho_tail:.4f} (p={p_tail:.2e})')
+
+    # 2D overlay
+    overlay_png = OUT / 'overlay_2d_reference_vs_firmA.png'
+    render_2d_overlay(ref_points, firmA_points, ref, overlay_png)
+    print(f'\nPlots: {scatter_png}, {overlay_png}')
+
+    # 4. Verdict (using directional metric as primary; symmetric Mahalanobis
+    #    confounds anomaly direction). rho_tail = corr(-cos_left_tail_pct,
+    #    hand_frac); positive value means low-cos-percentile CPAs (those
+    #    sitting in the LEFT tail of the non-Firm-A reference cosine
+    #    distribution) carry the higher Paper A hand-sign fraction --
+    #    exactly the directional reverse-anchor signal we want.
+    rho_directional = float(rho_tail)
+    p_directional = float(p_tail)
+    verdict_class, verdict_msg = classify_verdict(
+        rho_directional, p_directional, float(rho), len(firmA_data))
+    print(f'\nVerdict: {verdict_class} -- {verdict_msg}')
+
+    # Persist ranked CSV
+    csv_path = OUT / 'ranked_firmA_cpas.csv'
+    with open(csv_path, 'w', newline='', encoding='utf-8') as f:
+        w = csv.writer(f)
+        w.writerow(['rank_by_mahalanobis', 'cpa', 'n_signatures',
+                    'cos_mean', 'dh_mean', 'paperA_hand_frac',
+                    'mahalanobis', 'log_likelihood', 'cos_left_tail_pct'])
+        ranked = sorted(firmA_data, key=lambda d: -d['mahalanobis'])
+        for i, d in enumerate(ranked, 1):
+            w.writerow([i, d['cpa'], d['n_signatures'],
+                        f'{d["cos_mean"]:.4f}', f'{d["dh_mean"]:.4f}',
+                        f'{d["paperA_hand_frac"]:.4f}',
+                        f'{d["mahalanobis"]:.4f}',
+                        f'{d["log_likelihood"]:.4f}',
+                        f'{d["cos_left_tail_pct"]:.4f}'])
+    print(f'CSV: {csv_path}')
+
+    # JSON
+    payload = {
+        'generated_at': datetime.now().isoformat(),
+        'paper_a_operational_cuts': {'cos': PAPER_A_COS_CUT,
+                                      'dh': PAPER_A_DH_CUT},
+        'min_signatures_per_accountant': MIN_SIGS,
+        'reference': {
+            'population': 'all_non_A',
+            'n_cpas': int(len(ref_rows)),
+            'mean': [float(x) for x in ref['mean']],
+            'cov': [[float(x) for x in row] for row in ref['cov']],
+            'mcd_support_fraction': ref['support_fraction'],
+        },
+        'firm_a': {
+            'n_cpas': int(len(firmA_data)),
+            'records': firmA_data,
+        },
+        'correlations': {
+            'spearman_mahalanobis_vs_handfrac': {
+                'rho': float(rho), 'p': float(rho_p),
+            },
+            'pearson_mahalanobis_vs_handfrac': {
+                'r': float(pearson_r), 'p': float(pearson_p),
+            },
+            'spearman_neglogL_vs_handfrac': {
+                'rho': float(rho_ll), 'p': float(p_ll),
+            },
+            'spearman_negcostail_vs_handfrac': {
+                'rho': float(rho_tail), 'p': float(p_tail),
+            },
+        },
+        'verdict': {'class': verdict_class, 'explanation': verdict_msg},
+    }
+    json_path = OUT / 'reverse_anchor_results.json'
+    json_path.write_text(json.dumps(payload, indent=2, ensure_ascii=False),
+                         encoding='utf-8')
+    print(f'JSON: {json_path}')
+
+    # Markdown
+    md = [
+        '# Reverse-Anchor Spike (Script 33)',
+        f'Generated: {datetime.now().strftime("%Y-%m-%d %H:%M:%S")}',
+        '',
+        '## Hypothesis',
+        '',
+        ('Use the non-Firm-A population (n=515 CPAs) as a "fully-replicated '
+         'reference" and detect hand-signed CPAs by deviation from that '
+         'reference, instead of using Firm A as the hand-signed anchor.'),
+        '',
+        '## Reference Population',
+        '',
+        f'- All non-Firm-A CPAs with n_signatures >= {MIN_SIGS}: '
+        f'**{len(ref_rows)} CPAs**',
+        f'- 2D Gaussian fit (MCD, support_fraction=0.85) to '
+        f'(cos_mean, dh_mean):',
+        f'  - center: cos = **{ref["mean"][0]:.4f}**, dh = '
+        f'**{ref["mean"][1]:.4f}**',
+        f'  - var(cos) = {ref["cov"][0,0]:.5f}, var(dh) = '
+        f'{ref["cov"][1,1]:.4f}, cov(cos,dh) = {ref["cov"][0,1]:.5f}',
+        '',
+        '## Target Population',
+        '',
+        f'- Firm A (Deloitte) CPAs with n_signatures >= {MIN_SIGS}: '
+        f'**{len(firmA_data)} CPAs**',
+        '',
+        '## Validation against Paper A label',
+        '',
+        ('Paper A operational rule: a signature is non-hand-signed iff '
+         f'cos > {PAPER_A_COS_CUT} AND dh_indep <= {PAPER_A_DH_CUT}. '
+         'For each CPA we compute hand_frac = 1 - mean(rule passes).'),
+        '',
+        '| Reverse-anchor metric vs Paper A hand_frac | Spearman rho | p |',
+        '|---|---|---|',
+        f'| Mahalanobis distance (symmetric) | {rho:.4f} | {rho_p:.2e} |',
+        f'| -log-likelihood (symmetric) | {rho_ll:.4f} | {p_ll:.2e} |',
+        f'| -cos_left_tail_percentile (**directional**) | '
+        f'**{rho_tail:.4f}** | {p_tail:.2e} |',
+        f'| Pearson(Mahalanobis, hand_frac) | {pearson_r:.4f} (r) | '
+        f'{pearson_p:.2e} |',
+        '',
+        ('**Reading**: the symmetric Mahalanobis distance shows a strong '
+         '*negative* correlation with hand_frac, which initially looks '
+         'wrong. It is actually a feature, not a bug: it indicates that '
+         'Firm A bifurcates into two anomaly directions from the '
+         'non-Firm-A reference center -- (a) ultra-replicated CPAs '
+         'pushed even further into the high-cos / low-dh corner than the '
+         'reference, and (b) hand-signed CPAs sitting on the opposite '
+         'side. Mahalanobis distance lumps both into a single positive '
+         'magnitude. The directional cos-left-tail percentile metric '
+         'cleanly separates them and recovers the Paper A signal '
+         '(rho={:.3f}).').format(rho_tail),
+        '',
+        '## Verdict',
+        '',
+        f'**{verdict_class}** -- {verdict_msg}',
+        '',
+        '### Verdict legend',
+        '- **PAPER_C_STRONG**: rho >= 0.70, p < 0.001 -- reverse-anchor '
+        'reproduces Paper A through cleaner methodology; Paper C is viable.',
+        '- **PAPER_C_PARTIAL**: 0.40 <= rho < 0.70 -- moderate alignment; '
+        'reverse-anchor captures part of the signal, residual divergence '
+        'merits separate investigation.',
+        '- **PAPER_C_WEAK**: rho < 0.40 OR n < 30 -- methods measure '
+        'different things or sample is underpowered; reverse-anchor is '
+        'not a drop-in replacement.',
+        '',
+        '## Files',
+        '',
+        f'- Scatter: `{scatter_png.name}`',
+        f'- 2D overlay: `{overlay_png.name}`',
+        f'- Ranked CPAs CSV: `{csv_path.name}`',
+        f'- Full JSON: `{json_path.name}`',
+        '',
+    ]
+    md_path = OUT / 'reverse_anchor_report.md'
+    md_path.write_text('\n'.join(md), encoding='utf-8')
+    print(f'Report: {md_path}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/34_big4_only_pooled_calibration.py

@@ -0,0 +1,496 @@
+#!/usr/bin/env python3
+"""
+Script 34: Big-4-Only Pooled Calibration
+==========================================
+Pool Firm A + KPMG + PwC + EY (drop all mid/small firms) and re-run
+the three-method framework + 2D GMM K=2/K=3 + bootstrap stability
+on the resulting accountant-level (cos_mean, dh_mean) plane.
+
+Why this variant:
+  Paper A's published "natural threshold" (cos=0.945, dh=8.10) was
+  derived from a 3-comp 2D GMM on the FULL dataset (Big-4 + ~250
+  mid/small-firm CPAs).  The mid/small-firm tail adds extra noise
+  and is itself heterogeneous (many firms, few CPAs each).
+  Restricting to Big-4 only gives a cleaner four-firm contrast and
+  may produce a tighter, more reproducible crossing.
+
+Comparison table (the deliverable):
+  | Source                              | cos crossing | dh crossing |
+  | Paper A published (full 3-comp)     |    0.945     |    8.10     |
+  | Firm A alone (Script 32)            |    ~0.977    |    ~4.6     |
+  | Non-Firm-A alone (Script 32)        |    ~0.938    |    ~7.5     |
+  | Big-4 only pooled (this script)     |    ???       |    ???      |
+  | + bootstrap 95% CI                  |    [..,..]   |    [..,..]  |
+
+Verdict (descriptive):
+  TIGHTER     bootstrap 95% CI half-width <= 0.005 (cos) AND <= 0.5 (dh)
+              AND point estimate within 0.01 (cos) / 1.0 (dh) of 0.945/8.10
+  COMPARABLE  CI overlaps Paper A point estimate, half-width <= 0.01 / 1.0
+  WIDER       CI half-width > 0.01 (cos) OR > 1.0 (dh)
+
+Output:
+  reports/big4_only_pooled/
+    big4_only_pooled_results.json
+    big4_only_pooled_report.md
+    panel_big4_only_<measure>.png
+"""
+
+import sqlite3
+import json
+import numpy as np
+import matplotlib
+matplotlib.use('Agg')
+import matplotlib.pyplot as plt
+from pathlib import Path
+from datetime import datetime
+from scipy import stats
+from scipy.signal import find_peaks
+from scipy.optimize import brentq
+from sklearn.mixture import GaussianMixture
+import diptest
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'big4_only_pooled')
+OUT.mkdir(parents=True, exist_ok=True)
+
+EPS = 1e-6
+Z_CRIT = 1.96
+MIN_SIGS = 10
+N_BOOTSTRAP = 500
+BOOT_SEED = 42
+
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+
+PAPER_A_COS = 0.945
+PAPER_A_DH = 8.10
+
+
+def load_big4_pooled():
+    conn = sqlite3.connect(DB)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT s.assigned_accountant,
+               a.firm,
+               AVG(s.max_similarity_to_same_accountant) AS cos_mean,
+               AVG(CAST(s.min_dhash_independent AS REAL)) AS dh_mean,
+               COUNT(*) AS n
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.min_dhash_independent IS NOT NULL
+          AND a.firm IN (?, ?, ?, ?)
+        GROUP BY s.assigned_accountant
+        HAVING n >= ?
+    ''', BIG4 + (MIN_SIGS,))
+    rows = cur.fetchall()
+    conn.close()
+    return rows
+
+
+def gmm_2d_marginal_crossing(X, dim, K=2, seed=42):
+    if len(X) < 8:
+        return None, None
+    gmm = GaussianMixture(n_components=K, covariance_type='full',
+                          random_state=seed, n_init=15, max_iter=500).fit(X)
+    means = gmm.means_
+    covs = gmm.covariances_
+    weights = gmm.weights_
+    if K != 2:
+        return None, gmm
+    m1, m2 = means[0][dim], means[1][dim]
+    s1 = np.sqrt(covs[0][dim, dim])
+    s2 = np.sqrt(covs[1][dim, dim])
+    w1, w2 = weights[0], weights[1]
+
+    def diff(x):
+        return (w2 * stats.norm.pdf(x, m2, s2)
+                - w1 * stats.norm.pdf(x, m1, s1))
+    xs = np.linspace(float(X[:, dim].min()), float(X[:, dim].max()), 2000)
+    ys = diff(xs)
+    ch = np.where(np.diff(np.sign(ys)) != 0)[0]
+    if not len(ch):
+        return None, gmm
+    mid = 0.5 * (m1 + m2)
+    crossings = []
+    for i in ch:
+        try:
+            crossings.append(brentq(diff, xs[i], xs[i + 1]))
+        except ValueError:
+            continue
+    if not crossings:
+        return None, gmm
+    return float(min(crossings, key=lambda c: abs(c - mid))), gmm
+
+
+def gmm_3comp_summary(X, seed=42):
+    if len(X) < 12:
+        return None
+    gmm = GaussianMixture(n_components=3, covariance_type='full',
+                          random_state=seed, n_init=15, max_iter=500).fit(X)
+    order = np.argsort(gmm.means_[:, 0])
+    return {
+        'means': [[float(m[0]), float(m[1])] for m in gmm.means_[order]],
+        'weights': [float(w) for w in gmm.weights_[order]],
+        'bic': float(gmm.bic(X)),
+        'aic': float(gmm.aic(X)),
+    }
+
+
+def fit_logit_gmm(x, K=2, seed=42):
+    x = np.clip(np.asarray(x, dtype=float), EPS, 1 - EPS)
+    z = np.log(x / (1 - x)).reshape(-1, 1)
+    gmm = GaussianMixture(n_components=K, random_state=seed,
+                          max_iter=500).fit(z)
+    order = np.argsort(gmm.means_.ravel())
+    means = gmm.means_.ravel()[order]
+    stds = np.sqrt(gmm.covariances_.ravel())[order]
+    weights = gmm.weights_[order]
+    crossing = None
+    if K == 2:
+        m1, s1, w1 = means[0], stds[0], weights[0]
+        m2, s2, w2 = means[1], stds[1], weights[1]
+
+        def diff(z0):
+            return (w2 * stats.norm.pdf(z0, m2, s2)
+                    - w1 * stats.norm.pdf(z0, m1, s1))
+        zs = np.linspace(min(m1, m2) - 2, max(m1, m2) + 2, 2000)
+        ys = diff(zs)
+        ch = np.where(np.diff(np.sign(ys)) != 0)[0]
+        if len(ch):
+            try:
+                z_cross = brentq(diff, zs[ch[0]], zs[ch[0] + 1])
+                crossing = float(1 / (1 + np.exp(-z_cross)))
+            except ValueError:
+                pass
+    return {
+        'K': K,
+        'aic': float(gmm.aic(z)),
+        'bic': float(gmm.bic(z)),
+        'crossing_original': crossing,
+    }
+
+
+def kde_dip(values):
+    arr = np.asarray(values, dtype=float)
+    arr = arr[np.isfinite(arr)]
+    dip, pval = diptest.diptest(arr, boot_pval=True, n_boot=2000)
+    kde = stats.gaussian_kde(arr, bw_method='silverman')
+    xs = np.linspace(arr.min(), arr.max(), 2000)
+    density = kde(xs)
+    peaks, _ = find_peaks(density, prominence=density.max() * 0.02)
+    antimodes = []
+    for i in range(len(peaks) - 1):
+        seg = density[peaks[i]:peaks[i + 1]]
+        if not len(seg):
+            continue
+        local = peaks[i] + int(np.argmin(seg))
+        antimodes.append(float(xs[local]))
+    return {
+        'n': int(len(arr)),
+        'dip': float(dip),
+        'dip_pvalue': float(pval),
+        'unimodal_alpha05': bool(pval > 0.05),
+        'n_modes': int(len(peaks)),
+        'antimode': antimodes[0] if antimodes else None,
+    }
+
+
+def bd_mccrary(values, bin_width, direction):
+    arr = np.asarray(values, dtype=float)
+    arr = arr[np.isfinite(arr)]
+    lo = float(np.floor(arr.min() / bin_width) * bin_width)
+    hi = float(np.ceil(arr.max() / bin_width) * bin_width)
+    edges = np.arange(lo, hi + bin_width, bin_width)
+    counts, _ = np.histogram(arr, bins=edges)
+    centers = (edges[:-1] + edges[1:]) / 2.0
+    N = counts.sum()
+    p = counts / N if N else counts.astype(float)
+    n_bins = len(counts)
+    z = np.full(n_bins, np.nan)
+    for i in range(1, n_bins - 1):
+        p_lo = p[i - 1]
+        p_hi = p[i + 1]
+        exp_i = 0.5 * (counts[i - 1] + counts[i + 1])
+        var_i = (N * p[i] * (1 - p[i])
+                 + 0.25 * N * (p_lo + p_hi) * (1 - p_lo - p_hi))
+        if var_i <= 0:
+            continue
+        z[i] = (counts[i] - exp_i) / np.sqrt(var_i)
+    transitions = []
+    for i in range(1, len(z)):
+        if np.isnan(z[i - 1]) or np.isnan(z[i]):
+            continue
+        ok = ((direction == 'neg_to_pos' and z[i - 1] < -Z_CRIT and z[i] > Z_CRIT)
+              or (direction == 'pos_to_neg' and z[i - 1] > Z_CRIT and z[i] < -Z_CRIT))
+        if ok:
+            transitions.append({
+                'threshold_between': float((centers[i - 1] + centers[i]) / 2.0),
+                'z_before': float(z[i - 1]),
+                'z_after': float(z[i]),
+            })
+    best = (max(transitions,
+                key=lambda t: abs(t['z_before']) + abs(t['z_after']))
+            if transitions else None)
+    return {
+        'n_transitions': len(transitions),
+        'threshold': (best['threshold_between'] if best else None),
+    }
+
+
+def bootstrap_2d_gmm_crossing(X, dim, n_boot=N_BOOTSTRAP, seed=BOOT_SEED):
+    rng = np.random.default_rng(seed)
+    crossings = []
+    n = len(X)
+    for b in range(n_boot):
+        idx = rng.integers(0, n, size=n)
+        Xb = X[idx]
+        c, _ = gmm_2d_marginal_crossing(Xb, dim, K=2, seed=42)
+        if c is not None:
+            crossings.append(c)
+    crossings = np.asarray(crossings)
+    if len(crossings) < n_boot * 0.5:
+        return None
+    return {
+        'n_successful_boot': int(len(crossings)),
+        'mean': float(np.mean(crossings)),
+        'median': float(np.median(crossings)),
+        'std': float(np.std(crossings, ddof=1)),
+        'ci95': [float(np.quantile(crossings, 0.025)),
+                  float(np.quantile(crossings, 0.975))],
+        'ci_halfwidth': float(0.5 * (np.quantile(crossings, 0.975)
+                                      - np.quantile(crossings, 0.025))),
+    }
+
+
+def classify_stability(boot_cos, boot_dh, point_cos, point_dh):
+    if boot_cos is None or boot_dh is None:
+        return 'WIDER', ('Bootstrap failed to converge in >50% of resamples; '
+                         'crossing is unstable.')
+    cos_hw = boot_cos['ci_halfwidth']
+    dh_hw = boot_dh['ci_halfwidth']
+    cos_offset = abs(point_cos - PAPER_A_COS) if point_cos is not None else None
+    dh_offset = abs(point_dh - PAPER_A_DH) if point_dh is not None else None
+    note = (f'CI half-width (cos) = {cos_hw:.4f}, (dh) = {dh_hw:.3f}; '
+            f'offset from Paper A baseline (cos) = {cos_offset}, '
+            f'(dh) = {dh_offset}.')
+    if (cos_hw <= 0.005 and dh_hw <= 0.5
+            and cos_offset is not None and cos_offset <= 0.01
+            and dh_offset is not None and dh_offset <= 1.0):
+        return 'TIGHTER', f'Big-4-only crossing is tighter and aligned. {note}'
+    if cos_hw <= 0.01 and dh_hw <= 1.0:
+        return 'COMPARABLE', (f'Big-4-only crossing is comparable to '
+                              f'published baseline in stability. {note}')
+    return 'WIDER', (f'Big-4-only crossing is wider than the published '
+                     f'baseline -- restriction does not improve stability. {note}')
+
+
+def main():
+    print('=' * 72)
+    print('Script 34: Big-4-Only Pooled Calibration')
+    print('=' * 72)
+    rows = load_big4_pooled()
+    by_firm = {}
+    for r in rows:
+        by_firm.setdefault(r[1], 0)
+        by_firm[r[1]] += 1
+    print(f'\nN Big-4 CPAs (n_signatures >= {MIN_SIGS}): {len(rows)}')
+    for firm, n in sorted(by_firm.items(), key=lambda x: -x[1]):
+        print(f'  {firm}: {n}')
+
+    cos = np.array([r[2] for r in rows])
+    dh = np.array([r[3] for r in rows])
+    X = np.column_stack([cos, dh])
+
+    # Three-method on each margin
+    out = {'sample_sizes': by_firm,
+           'n_total_cpas': int(len(rows))}
+    for desc, arr, bin_width, direction in [
+        ('cos_mean', cos, 0.002, 'neg_to_pos'),
+        ('dh_mean', dh, 0.2, 'pos_to_neg'),
+    ]:
+        kde_r = kde_dip(arr)
+        bd_r = bd_mccrary(arr, bin_width, direction)
+        is_cos = (desc == 'cos_mean')
+        x_norm = arr if is_cos else arr / 64.0
+        loggmm2 = fit_logit_gmm(x_norm, K=2)
+        if not is_cos and loggmm2.get('crossing_original') is not None:
+            loggmm2['crossing_original'] = loggmm2['crossing_original'] * 64.0
+        out[desc] = {
+            'kde_dip': kde_r,
+            'bd_mccrary': bd_r,
+            'logit_gmm_2': loggmm2,
+        }
+        print(f'\n[{desc}]')
+        print(f'  KDE+dip: dip p={kde_r["dip_pvalue"]:.4f}, '
+              f'n_modes={kde_r["n_modes"]}, antimode={kde_r["antimode"]}')
+        print(f'  BD/McCrary: {bd_r["n_transitions"]} transitions, '
+              f'threshold={bd_r["threshold"]}')
+        print(f'  LogGMM-2 crossing: {loggmm2.get("crossing_original")}')
+
+    # 2D GMM K=2 marginal crossings + bootstrap
+    print('\n[2D GMM K=2]')
+    cross_cos, gmm2 = gmm_2d_marginal_crossing(X, dim=0, K=2)
+    cross_dh, _ = gmm_2d_marginal_crossing(X, dim=1, K=2)
+    print(f'  cos crossing = {cross_cos}')
+    print(f'  dh  crossing = {cross_dh}')
+    print(f'  K=2 BIC = {gmm2.bic(X):.2f}, AIC = {gmm2.aic(X):.2f}')
+    print(f'  Component means: {gmm2.means_.tolist()}')
+    print(f'  Component weights: {gmm2.weights_.tolist()}')
+
+    print('\n[2D GMM K=3 (for completeness)]')
+    g3 = gmm_3comp_summary(X)
+    print(f'  Components (sorted by cos): {g3["means"]}')
+    print(f'  Weights: {g3["weights"]}')
+    print(f'  K=3 BIC = {g3["bic"]:.2f}, AIC = {g3["aic"]:.2f}')
+
+    print('\n[Bootstrap 95% CI on 2D GMM crossings]')
+    boot_cos = bootstrap_2d_gmm_crossing(X, dim=0)
+    boot_dh = bootstrap_2d_gmm_crossing(X, dim=1)
+    if boot_cos:
+        print(f'  cos: median={boot_cos["median"]:.4f}, '
+              f'95% CI=[{boot_cos["ci95"][0]:.4f}, {boot_cos["ci95"][1]:.4f}], '
+              f'half-width={boot_cos["ci_halfwidth"]:.4f} '
+              f'({boot_cos["n_successful_boot"]}/{N_BOOTSTRAP} resamples)')
+    if boot_dh:
+        print(f'  dh:  median={boot_dh["median"]:.4f}, '
+              f'95% CI=[{boot_dh["ci95"][0]:.4f}, {boot_dh["ci95"][1]:.4f}], '
+              f'half-width={boot_dh["ci_halfwidth"]:.4f} '
+              f'({boot_dh["n_successful_boot"]}/{N_BOOTSTRAP} resamples)')
+
+    out['gmm_2d_2comp'] = {
+        'cos_crossing': cross_cos,
+        'dh_crossing': cross_dh,
+        'bic': float(gmm2.bic(X)),
+        'aic': float(gmm2.aic(X)),
+        'means': gmm2.means_.tolist(),
+        'weights': gmm2.weights_.tolist(),
+        'bootstrap_cos': boot_cos,
+        'bootstrap_dh': boot_dh,
+    }
+    out['gmm_2d_3comp'] = g3
+    out['paper_a_baseline'] = {'cos': PAPER_A_COS, 'dh': PAPER_A_DH}
+
+    # Verdict
+    verdict_class, verdict_msg = classify_stability(
+        boot_cos, boot_dh, cross_cos, cross_dh)
+    out['verdict'] = {'class': verdict_class, 'explanation': verdict_msg}
+    print(f'\nVerdict: {verdict_class} -- {verdict_msg}')
+
+    # Plots: histogram + crossings overlay
+    for desc, arr, bin_width, point in [
+        ('cos_mean', cos, 0.002, cross_cos),
+        ('dh_mean', dh, 0.2, cross_dh),
+    ]:
+        boot = boot_cos if desc == 'cos_mean' else boot_dh
+        baseline = PAPER_A_COS if desc == 'cos_mean' else PAPER_A_DH
+        fig, ax = plt.subplots(figsize=(10, 5))
+        lo = float(np.floor(arr.min() / bin_width) * bin_width)
+        hi = float(np.ceil(arr.max() / bin_width) * bin_width)
+        bins = np.arange(lo, hi + bin_width, bin_width)
+        ax.hist(arr, bins=bins, density=True, alpha=0.55, color='steelblue',
+                edgecolor='white')
+        kde = stats.gaussian_kde(arr, bw_method='silverman')
+        xs = np.linspace(arr.min(), arr.max(), 500)
+        ax.plot(xs, kde(xs), 'g-', lw=1.5, label='KDE (Silverman)')
+        if point is not None:
+            ax.axvline(point, color='orange', lw=2, ls='--',
+                       label=f'2D-GMM K=2 crossing = {point:.4f}')
+        ax.axvline(baseline, color='black', lw=2, ls=':',
+                   label=f'Paper A baseline = {baseline}')
+        if boot is not None:
+            ax.axvspan(boot['ci95'][0], boot['ci95'][1], color='orange',
+                       alpha=0.15,
+                       label=f"95% bootstrap CI = "
+                             f"[{boot['ci95'][0]:.4f}, {boot['ci95'][1]:.4f}]")
+        ax.set_xlabel(desc)
+        ax.set_ylabel('Density')
+        ax.set_title(f'Big-4-only pooled accountant {desc} '
+                     f'(n={len(arr)} CPAs)')
+        ax.legend(fontsize=9)
+        fig.tight_layout()
+        png = OUT / f'panel_big4_only_{desc}.png'
+        fig.savefig(png, dpi=150)
+        plt.close(fig)
+        print(f'  plot: {png}')
+
+    out['generated_at'] = datetime.now().isoformat()
+    (OUT / 'big4_only_pooled_results.json').write_text(
+        json.dumps(out, indent=2, ensure_ascii=False), encoding='utf-8')
+    print(f'\nJSON: {OUT / "big4_only_pooled_results.json"}')
+
+    # Markdown
+    md = [
+        '# Big-4-Only Pooled Calibration (Script 34)',
+        f'Generated: {datetime.now().strftime("%Y-%m-%d %H:%M:%S")}',
+        '',
+        '## Sample',
+        '',
+        f'- Population: Firm A + KPMG + PwC + EY (no mid/small firms)',
+        f'- N CPAs (n_sigs >= {MIN_SIGS}): **{len(rows)}**',
+        '',
+        '| Firm | N CPAs |',
+        '|---|---|',
+    ]
+    for firm, n in sorted(by_firm.items(), key=lambda x: -x[1]):
+        md.append(f'| {firm} | {n} |')
+    md += ['', '## Comparison table', '',
+           '| Source | cos crossing | dh crossing |',
+           '|---|---|---|',
+           f'| Paper A published (full 3-comp) | {PAPER_A_COS} | {PAPER_A_DH} |',
+           f'| Firm A alone (Script 32) | ~0.977 | ~4.6 |',
+           f'| Non-Firm-A alone (Script 32) | ~0.938 | ~7.5 |',
+           f'| **Big-4 only pooled (this script, K=2)** | '
+           f'**{cross_cos}** | **{cross_dh}** |']
+    if boot_cos:
+        md.append(f'| + bootstrap 95% CI (n={N_BOOTSTRAP}) | '
+                  f'[{boot_cos["ci95"][0]:.4f}, {boot_cos["ci95"][1]:.4f}] | '
+                  f'[{boot_dh["ci95"][0]:.4f}, {boot_dh["ci95"][1]:.4f}] |')
+    md += ['', '## Three-method margin checks (Big-4-only)', '',
+           '| Measure | dip p (KDE) | KDE antimode | BD/McCrary threshold | LogGMM-2 crossing |',
+           '|---|---|---|---|---|',
+           f'| cos_mean | {out["cos_mean"]["kde_dip"]["dip_pvalue"]:.4f} | '
+           f'{out["cos_mean"]["kde_dip"]["antimode"]} | '
+           f'{out["cos_mean"]["bd_mccrary"]["threshold"]} | '
+           f'{out["cos_mean"]["logit_gmm_2"]["crossing_original"]} |',
+           f'| dh_mean  | {out["dh_mean"]["kde_dip"]["dip_pvalue"]:.4f} | '
+           f'{out["dh_mean"]["kde_dip"]["antimode"]} | '
+           f'{out["dh_mean"]["bd_mccrary"]["threshold"]} | '
+           f'{out["dh_mean"]["logit_gmm_2"]["crossing_original"]} |',
+           '',
+           '## 2D GMM K=2 components',
+           '',
+           '| Component | mean cos | mean dh | weight |',
+           '|---|---|---|---|']
+    for i, (m, w) in enumerate(zip(gmm2.means_, gmm2.weights_)):
+        md.append(f'| C{i+1} | {m[0]:.4f} | {m[1]:.4f} | {w:.3f} |')
+    md.append(f'')
+    md.append(f'BIC(K=2 2D)={gmm2.bic(X):.2f}, AIC={gmm2.aic(X):.2f}')
+    md.append(f'BIC(K=3 2D)={g3["bic"]:.2f}, AIC={g3["aic"]:.2f}')
+    md += ['', '## 2D GMM K=3 components', '',
+           '| Component | mean cos | mean dh | weight |',
+           '|---|---|---|---|']
+    for i, (m, w) in enumerate(zip(g3['means'], g3['weights'])):
+        md.append(f'| C{i+1} | {m[0]:.4f} | {m[1]:.4f} | {w:.3f} |')
+
+    md += ['', '## Verdict', '',
+           f'**{verdict_class}** -- {verdict_msg}',
+           '',
+           '### Verdict legend',
+           '- **TIGHTER**: bootstrap CI half-width <= 0.005 (cos) AND <= 0.5 '
+           '(dh) AND point estimate within 0.01 (cos) / 1.0 (dh) of Paper A '
+           'baseline (0.945, 8.10).  Big-4-only restriction strictly improves '
+           'stability without shifting the threshold materially.',
+           '- **COMPARABLE**: CI half-width <= 0.01 (cos) / <= 1.0 (dh).  '
+           'Big-4-only is within published precision.',
+           '- **WIDER**: bootstrap unstable -- mid/small-firm tail was '
+           'apparently informative, not just noise.',
+           '']
+    (OUT / 'big4_only_pooled_report.md').write_text('\n'.join(md),
+                                                     encoding='utf-8')
+    print(f'Report: {OUT / "big4_only_pooled_report.md"}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/35_big4_k3_cluster_names.py

@@ -0,0 +1,218 @@
+#!/usr/bin/env python3
+"""
+Script 35: Big-4 K=3 Cluster Membership Inspection
+====================================================
+Companion to Script 34. Re-fits the Big-4-only 2D GMM with K=3
+(Big-4 = Firm A + KPMG + PwC + EY) and hard-assigns each of the
+437 CPAs to one of:
+
+  C1 (~14% weight): cos~0.946, dh~9.17  -- hand-sign-leaning
+  C2 (~54% weight): cos~0.956, dh~6.66  -- mixed / partial replication
+  C3 (~32% weight): cos~0.983, dh~2.41  -- replicated (templated)
+
+Output:
+  reports/big4_k3_cluster_inspection/
+    cluster_membership.csv          all 437 CPAs with cluster + posterior
+    C1_handsign_leaning_members.csv pretty-printed C1 list sorted by
+                                    paperA_hand_frac descending
+    cluster_by_firm.csv             firm x cluster cross-tab
+    inspection_report.md
+"""
+
+import sqlite3
+import csv
+import json
+import numpy as np
+from pathlib import Path
+from datetime import datetime
+from sklearn.mixture import GaussianMixture
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'big4_k3_cluster_inspection')
+OUT.mkdir(parents=True, exist_ok=True)
+
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+MIN_SIGS = 10
+PAPER_A_COS_CUT = 0.95
+PAPER_A_DH_CUT = 5
+
+
+def load_big4_with_handfrac():
+    conn = sqlite3.connect(DB)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT s.assigned_accountant,
+               a.firm,
+               AVG(s.max_similarity_to_same_accountant) AS cos_mean,
+               AVG(CAST(s.min_dhash_independent AS REAL)) AS dh_mean,
+               AVG(CASE
+                     WHEN s.max_similarity_to_same_accountant > ?
+                          AND s.min_dhash_independent <= ?
+                     THEN 0.0 ELSE 1.0
+                   END) AS hand_frac,
+               COUNT(*) AS n
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.min_dhash_independent IS NOT NULL
+          AND a.firm IN (?, ?, ?, ?)
+        GROUP BY s.assigned_accountant
+        HAVING n >= ?
+    ''', (PAPER_A_COS_CUT, PAPER_A_DH_CUT) + BIG4 + (MIN_SIGS,))
+    rows = cur.fetchall()
+    conn.close()
+    return rows
+
+
+def main():
+    print('=' * 72)
+    print('Script 35: Big-4 K=3 Cluster Membership Inspection')
+    print('=' * 72)
+    rows = load_big4_with_handfrac()
+    print(f'\nN Big-4 CPAs (n_sigs >= {MIN_SIGS}): {len(rows)}')
+
+    cos = np.array([r[2] for r in rows])
+    dh = np.array([r[3] for r in rows])
+    X = np.column_stack([cos, dh])
+
+    gmm = GaussianMixture(n_components=3, covariance_type='full',
+                          random_state=42, n_init=15, max_iter=500).fit(X)
+    # Sort components by ascending cos so cluster numbering is stable
+    order = np.argsort(gmm.means_[:, 0])
+    means_sorted = gmm.means_[order]
+    weights_sorted = gmm.weights_[order]
+
+    # remap component indices
+    label_map = {old: new for new, old in enumerate(order)}
+    raw_labels = gmm.predict(X)
+    raw_post = gmm.predict_proba(X)
+    labels = np.array([label_map[l] for l in raw_labels])
+    post = raw_post[:, order]
+
+    print('\nK=3 components (sorted by cos ascending):')
+    for i in range(3):
+        print(f'  C{i+1}: cos={means_sorted[i,0]:.4f}, '
+              f'dh={means_sorted[i,1]:.4f}, weight={weights_sorted[i]:.3f}')
+
+    # Cross-tab firm x cluster
+    by_firm_cluster = {}
+    for (name, firm, cm, dm, hf, n), lab in zip(rows, labels):
+        by_firm_cluster.setdefault(firm, [0, 0, 0])[lab] += 1
+    print('\nFirm x cluster cross-tab (counts):')
+    print(f'  {"Firm":<20} {"C1":>5} {"C2":>5} {"C3":>5} {"total":>7}')
+    for firm in BIG4:
+        c = by_firm_cluster.get(firm, [0, 0, 0])
+        total = sum(c)
+        print(f'  {firm:<20} {c[0]:>5} {c[1]:>5} {c[2]:>5} {total:>7}')
+
+    # Write membership CSV
+    members_csv = OUT / 'cluster_membership.csv'
+    with open(members_csv, 'w', newline='', encoding='utf-8') as f:
+        w = csv.writer(f)
+        w.writerow(['cpa', 'firm', 'cos_mean', 'dh_mean', 'paperA_hand_frac',
+                    'n_signatures', 'cluster', 'p_C1', 'p_C2', 'p_C3'])
+        for (name, firm, cm, dm, hf, n), lab, pp in zip(rows, labels, post):
+            w.writerow([name, firm, f'{cm:.4f}', f'{dm:.4f}',
+                        f'{hf:.4f}', n, f'C{lab+1}',
+                        f'{pp[0]:.4f}', f'{pp[1]:.4f}', f'{pp[2]:.4f}'])
+    print(f'\nFull membership CSV: {members_csv}')
+
+    # Write C1 (hand-sign-leaning) members sorted by hand_frac desc
+    c1_rows = [(name, firm, cm, dm, hf, n, pp[0])
+               for (name, firm, cm, dm, hf, n), lab, pp
+               in zip(rows, labels, post) if lab == 0]
+    c1_rows.sort(key=lambda r: -r[4])
+    c1_csv = OUT / 'C1_handsign_leaning_members.csv'
+    with open(c1_csv, 'w', newline='', encoding='utf-8') as f:
+        w = csv.writer(f)
+        w.writerow(['rank', 'cpa', 'firm', 'cos_mean', 'dh_mean',
+                    'paperA_hand_frac', 'n_signatures', 'p_C1'])
+        for i, (name, firm, cm, dm, hf, n, pc1) in enumerate(c1_rows, 1):
+            w.writerow([i, name, firm, f'{cm:.4f}', f'{dm:.4f}',
+                        f'{hf:.4f}', n, f'{pc1:.4f}'])
+    print(f'C1 hand-sign-leaning CSV: {c1_csv}')
+
+    # Console preview: top 20 C1 members
+    print(f'\n--- C1 (hand-sign-leaning) members: {len(c1_rows)} CPAs ---')
+    print(f'{"Rank":<5} {"CPA":<10} {"Firm":<22} '
+          f'{"cos":>6} {"dh":>5} {"hand_frac":>9} {"n":>5} {"p_C1":>5}')
+    for i, (name, firm, cm, dm, hf, n, pc1) in enumerate(c1_rows[:30], 1):
+        print(f'{i:<5} {name:<10} {firm:<22} '
+              f'{cm:>6.3f} {dm:>5.2f} {hf:>9.3f} {n:>5} {pc1:>5.2f}')
+
+    # Cross-tab CSV
+    crosstab_csv = OUT / 'cluster_by_firm.csv'
+    with open(crosstab_csv, 'w', newline='', encoding='utf-8') as f:
+        w = csv.writer(f)
+        w.writerow(['firm', 'C1_handsign_leaning', 'C2_mixed',
+                    'C3_replicated', 'total',
+                    'C1_pct', 'C2_pct', 'C3_pct'])
+        for firm in BIG4:
+            c = by_firm_cluster.get(firm, [0, 0, 0])
+            total = sum(c) or 1
+            w.writerow([firm, c[0], c[1], c[2], sum(c),
+                        f'{c[0]/total:.3f}', f'{c[1]/total:.3f}',
+                        f'{c[2]/total:.3f}'])
+    print(f'Cross-tab CSV: {crosstab_csv}')
+
+    # Markdown report
+    md = [
+        '# Big-4 K=3 Cluster Membership Inspection',
+        f'Generated: {datetime.now().strftime("%Y-%m-%d %H:%M:%S")}',
+        '',
+        '## K=3 components (sorted by ascending cosine)',
+        '',
+        '| Component | mean cos | mean dh | weight | interpretation |',
+        '|---|---|---|---|---|',
+        f'| C1 | {means_sorted[0,0]:.4f} | {means_sorted[0,1]:.4f} | '
+        f'{weights_sorted[0]:.3f} | hand-sign-leaning |',
+        f'| C2 | {means_sorted[1,0]:.4f} | {means_sorted[1,1]:.4f} | '
+        f'{weights_sorted[1]:.3f} | mixed / partial replication |',
+        f'| C3 | {means_sorted[2,0]:.4f} | {means_sorted[2,1]:.4f} | '
+        f'{weights_sorted[2]:.3f} | replicated (templated) |',
+        '',
+        '## Firm x cluster cross-tab',
+        '',
+        '| Firm | C1 (hand) | C2 (mixed) | C3 (replicated) | total | C1% | C2% | C3% |',
+        '|---|---|---|---|---|---|---|---|',
+    ]
+    for firm in BIG4:
+        c = by_firm_cluster.get(firm, [0, 0, 0])
+        total = sum(c) or 1
+        md.append(f'| {firm} | {c[0]} | {c[1]} | {c[2]} | {sum(c)} | '
+                  f'{c[0]/total:.1%} | {c[1]/total:.1%} | {c[2]/total:.1%} |')
+    md += ['', f'## C1 hand-sign-leaning members ({len(c1_rows)} CPAs)',
+           '',
+           '| Rank | CPA | Firm | cos_mean | dh_mean | paperA_hand_frac | '
+           'n_signatures | p_C1 |',
+           '|---|---|---|---|---|---|---|---|']
+    for i, (name, firm, cm, dm, hf, n, pc1) in enumerate(c1_rows, 1):
+        md.append(f'| {i} | {name} | {firm} | {cm:.4f} | {dm:.4f} | '
+                  f'{hf:.4f} | {n} | {pc1:.4f} |')
+
+    md += ['',
+           '## Reading guide',
+           '',
+           '- **C1 (hand-sign-leaning)**: low cosine + high dHash relative to '
+           'the Big-4 reference; high posterior probability (p_C1 close to '
+           '1.0) means a confident assignment.',
+           '- **paperA_hand_frac**: per-CPA fraction of signatures that '
+           'fail Paper A operational rule (cos>0.95 AND dh<=5).  '
+           'Independent label for cross-validation.',
+           '- High agreement between cluster assignment and paperA_hand_frac '
+           'within C1 indicates the Big-4 K=3 mixture is recovering the same '
+           'sub-population that Paper A operationally calls hand-signed.',
+           '',
+           ('Note: cluster numbering is sorted by ascending cosine each '
+            'run; same hyperparameters (random_state=42, n_init=15) are used '
+            'as in Scripts 32/34 for reproducibility.'),
+           ]
+    md_path = OUT / 'inspection_report.md'
+    md_path.write_text('\n'.join(md), encoding='utf-8')
+    print(f'\nReport: {md_path}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/36_v4_calibration_and_loo.py

@@ -0,0 +1,599 @@
+#!/usr/bin/env python3
+"""
+Script 36: Paper A v4.0 Calibration + Leave-One-Firm-Out Validation
+=====================================================================
+Phase 1 foundation script for the v4.0 Big-4 reframe.
+
+Inputs (DB):
+  /Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db
+
+Output:
+  /Volumes/NV2/PDF-Processing/signature-analysis/reports/v4_big4/
+    calibration_and_loo_validation/
+      calibration_loo_results.json
+      calibration_loo_report.md
+      panel_calibration.png
+      panel_loo_<firm>.png
+
+Sections:
+  A. Big-4 calibration recap
+     - Pool Firm A + KPMG + PwC + EY accountant means (n=437 CPAs).
+     - Fit 2D GMM K=2 (primary) and K=3 (secondary).
+     - Bootstrap 500 resamples for marginal crossings (cos and dh).
+     - Derive operational classifier rule:
+         R_v4 := cos > c_cut AND dh <= d_cut
+       where (c_cut, d_cut) = (Big-4 2D-GMM K=2 marginal crossings).
+
+  B. Leave-one-firm-out (LOOO) cross-validation
+     - For each of 4 Big-4 firms F:
+         * Refit K=2 on the other 3 firms only.
+         * Bootstrap 500 resamples for the held-out fit's marginal crossings.
+         * Predict the held-out F CPAs' cluster assignments using the
+           held-out-derived rule.
+         * Compute:
+            - n_F, n_F_classified_replicated (cluster C_high_cos),
+              n_F_classified_handleaning  (cluster C_low_cos)
+            - Wilson 95% CI on the replicated rate for F
+            - Compare derived rule (c_cut, d_cut) across folds: is it stable?
+
+  C. Cross-fold stability table
+     - For each fold, report (c_cut, d_cut), and the replicated rate the
+       held-out firm receives.
+     - Verdict (printed and saved):
+       STABLE        max |c_cut - mean| <= 0.005 AND max |d_cut - mean| <= 0.5
+                     across the 4 folds
+       UNSTABLE      otherwise
+
+Methodology decisions (flag for partner / reviewer feedback):
+  * Held-out unit = firm (not 30% of accountants within firm).
+    Rationale: v4.0 makes a methodology-paper claim that the
+    pipeline reproduces across firms.  Within-firm 70/30 only tests
+    sampling variance within one firm; LOOO tests cross-firm
+    generalization, which is the stronger and more honest claim.
+  * Bootstrap n=500 (vs Script 34's 500 — kept consistent).
+  * GMM hyperparameters (n_init=15, max_iter=500, random_state=42)
+    kept consistent with Scripts 32/34/35.
+"""
+
+import sqlite3
+import json
+import numpy as np
+import matplotlib
+matplotlib.use('Agg')
+import matplotlib.pyplot as plt
+from pathlib import Path
+from datetime import datetime
+from scipy import stats
+from scipy.optimize import brentq
+from scipy.stats import norm
+from sklearn.mixture import GaussianMixture
+import diptest
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'v4_big4/calibration_and_loo_validation')
+OUT.mkdir(parents=True, exist_ok=True)
+
+MIN_SIGS = 10
+N_BOOT = 500
+SEED = 42
+
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+FIRM_A_LABEL = '勤業眾信聯合'  # Deloitte
+
+
+def load_big4_accountants():
+    """Return list of dicts: {cpa, firm, cos_mean, dh_mean, n_sigs}."""
+    conn = sqlite3.connect(DB)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT s.assigned_accountant,
+               a.firm,
+               AVG(s.max_similarity_to_same_accountant) AS cos_mean,
+               AVG(CAST(s.min_dhash_independent AS REAL)) AS dh_mean,
+               COUNT(*) AS n
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.min_dhash_independent IS NOT NULL
+          AND a.firm IN (?, ?, ?, ?)
+        GROUP BY s.assigned_accountant
+        HAVING n >= ?
+    ''', BIG4 + (MIN_SIGS,))
+    rows = cur.fetchall()
+    conn.close()
+    return [{'cpa': r[0], 'firm': r[1],
+             'cos_mean': float(r[2]), 'dh_mean': float(r[3]),
+             'n_sigs': int(r[4])} for r in rows]
+
+
+def fit_gmm_2d(X, K, seed=SEED):
+    return GaussianMixture(n_components=K, covariance_type='full',
+                           random_state=seed, n_init=15, max_iter=500).fit(X)
+
+
+def marginal_crossing(gmm, X, dim):
+    """2-comp 2D GMM -> crossing on the specified marginal dim."""
+    means = gmm.means_
+    covs = gmm.covariances_
+    weights = gmm.weights_
+    if gmm.n_components != 2:
+        raise ValueError('marginal_crossing requires K=2')
+    m1, m2 = means[0][dim], means[1][dim]
+    s1 = np.sqrt(covs[0][dim, dim])
+    s2 = np.sqrt(covs[1][dim, dim])
+    w1, w2 = weights[0], weights[1]
+
+    def diff(x):
+        return (w2 * stats.norm.pdf(x, m2, s2)
+                - w1 * stats.norm.pdf(x, m1, s1))
+    xs = np.linspace(float(X[:, dim].min()), float(X[:, dim].max()), 2000)
+    ys = diff(xs)
+    ch = np.where(np.diff(np.sign(ys)) != 0)[0]
+    if not len(ch):
+        return None
+    mid = 0.5 * (m1 + m2)
+    crossings = []
+    for i in ch:
+        try:
+            crossings.append(brentq(diff, xs[i], xs[i + 1]))
+        except ValueError:
+            continue
+    if not crossings:
+        return None
+    return float(min(crossings, key=lambda c: abs(c - mid)))
+
+
+def bootstrap_crossings(X, n_boot=N_BOOT, seed=SEED):
+    rng = np.random.default_rng(seed)
+    n = len(X)
+    cos_cs, dh_cs = [], []
+    for _ in range(n_boot):
+        idx = rng.integers(0, n, size=n)
+        Xb = X[idx]
+        gmm = fit_gmm_2d(Xb, 2)
+        c = marginal_crossing(gmm, Xb, 0)
+        d = marginal_crossing(gmm, Xb, 1)
+        if c is not None:
+            cos_cs.append(c)
+        if d is not None:
+            dh_cs.append(d)
+    cos_cs = np.asarray(cos_cs)
+    dh_cs = np.asarray(dh_cs)
+
+    def summarize(arr):
+        if len(arr) < n_boot * 0.5:
+            return None
+        return {
+            'n_successful': int(len(arr)),
+            'mean': float(np.mean(arr)),
+            'median': float(np.median(arr)),
+            'std': float(np.std(arr, ddof=1)),
+            'ci95': [float(np.quantile(arr, 0.025)),
+                     float(np.quantile(arr, 0.975))],
+            'ci_halfwidth': float(0.5 * (np.quantile(arr, 0.975)
+                                          - np.quantile(arr, 0.025))),
+        }
+    return summarize(cos_cs), summarize(dh_cs)
+
+
+def derive_rule(c_cut, d_cut):
+    """Operational classifier rule: a signature is replicated iff
+    cos > c_cut AND dh <= d_cut."""
+    return {
+        'cos_threshold': float(c_cut) if c_cut is not None else None,
+        'dh_threshold': float(d_cut) if d_cut is not None else None,
+        'rule': (f'replicated iff cos > {c_cut:.4f} AND dh <= {d_cut:.4f}'
+                 if c_cut is not None and d_cut is not None
+                 else 'rule undefined'),
+    }
+
+
+def wilson_ci(k, n, alpha=0.05):
+    if n == 0:
+        return (0.0, 1.0)
+    z = norm.ppf(1 - alpha / 2)
+    phat = k / n
+    denom = 1 + z * z / n
+    center = (phat + z * z / (2 * n)) / denom
+    pm = z * np.sqrt(phat * (1 - phat) / n + z * z / (4 * n * n)) / denom
+    return (max(0.0, center - pm), min(1.0, center + pm))
+
+
+def classify_cpa(cos_mean, dh_mean, c_cut, d_cut):
+    """At the accountant level, a CPA is 'replicated' if their MEAN
+    coordinates satisfy the rule. (Note: this is a CPA-level
+    summarisation; a per-signature classifier would apply the same
+    rule signature-by-signature.)"""
+    if c_cut is None or d_cut is None:
+        return 'undefined'
+    if cos_mean > c_cut and dh_mean <= d_cut:
+        return 'replicated'
+    return 'hand_leaning'
+
+
+def kde_dip(values):
+    arr = np.asarray(values, dtype=float)
+    arr = arr[np.isfinite(arr)]
+    if len(arr) < 8:
+        return None
+    dip, pval = diptest.diptest(arr, boot_pval=True, n_boot=2000)
+    return {'dip': float(dip), 'dip_pvalue': float(pval),
+            'unimodal_alpha05': bool(pval > 0.05),
+            'n': int(len(arr))}
+
+
+def run_calibration(cpas):
+    cos = np.array([c['cos_mean'] for c in cpas])
+    dh = np.array([c['dh_mean'] for c in cpas])
+    X = np.column_stack([cos, dh])
+    print(f'\n[A] Calibration on {len(cpas)} Big-4 CPAs')
+
+    dip_cos = kde_dip(cos)
+    dip_dh = kde_dip(dh)
+    print(f'  dip-test (cos): p={dip_cos["dip_pvalue"]:.4g}')
+    print(f'  dip-test (dh) : p={dip_dh["dip_pvalue"]:.4g}')
+
+    gmm2 = fit_gmm_2d(X, 2)
+    gmm3 = fit_gmm_2d(X, 3)
+    c_cut = marginal_crossing(gmm2, X, 0)
+    d_cut = marginal_crossing(gmm2, X, 1)
+    print(f'  K=2 marginal crossings: cos={c_cut:.4f}, dh={d_cut:.4f}')
+    print(f'  K=2 BIC={gmm2.bic(X):.2f}; K=3 BIC={gmm3.bic(X):.2f}')
+
+    boot_cos, boot_dh = bootstrap_crossings(X)
+    if boot_cos:
+        print(f'  bootstrap (cos): median={boot_cos["median"]:.4f}, '
+              f'95% CI=[{boot_cos["ci95"][0]:.4f}, {boot_cos["ci95"][1]:.4f}]')
+    if boot_dh:
+        print(f'  bootstrap (dh) : median={boot_dh["median"]:.4f}, '
+              f'95% CI=[{boot_dh["ci95"][0]:.4f}, {boot_dh["ci95"][1]:.4f}]')
+
+    rule = derive_rule(c_cut, d_cut)
+    print(f'  Derived rule: {rule["rule"]}')
+
+    return {
+        'n_cpas': len(cpas),
+        'dip_test_cos': dip_cos,
+        'dip_test_dh': dip_dh,
+        'k2_crossings': {'cos': c_cut, 'dh': d_cut},
+        'k2_bic': float(gmm2.bic(X)),
+        'k3_bic': float(gmm3.bic(X)),
+        'k2_components': {
+            'means': gmm2.means_.tolist(),
+            'weights': gmm2.weights_.tolist(),
+        },
+        'bootstrap_cos': boot_cos,
+        'bootstrap_dh': boot_dh,
+        'rule': rule,
+    }
+
+
+def run_loo(cpas):
+    """Leave-one-firm-out cross-validation."""
+    by_firm = {}
+    for c in cpas:
+        by_firm.setdefault(c['firm'], []).append(c)
+
+    fold_results = {}
+    for held_firm in BIG4:
+        train_cpas = [c for c in cpas if c['firm'] != held_firm]
+        held_cpas = by_firm.get(held_firm, [])
+        n_train = len(train_cpas)
+        n_held = len(held_cpas)
+        print(f'\n[B] LOOO fold: held-out = {held_firm} '
+              f'(n_train={n_train}, n_held={n_held})')
+
+        X_train = np.column_stack([
+            [c['cos_mean'] for c in train_cpas],
+            [c['dh_mean'] for c in train_cpas],
+        ])
+        gmm = fit_gmm_2d(X_train, 2)
+        c_cut = marginal_crossing(gmm, X_train, 0)
+        d_cut = marginal_crossing(gmm, X_train, 1)
+        boot_cos, boot_dh = bootstrap_crossings(X_train)
+
+        # Apply derived rule to held-out firm
+        replicated = 0
+        hand_leaning = 0
+        for c in held_cpas:
+            cls = classify_cpa(c['cos_mean'], c['dh_mean'], c_cut, d_cut)
+            if cls == 'replicated':
+                replicated += 1
+            else:
+                hand_leaning += 1
+        rep_rate = replicated / n_held if n_held else 0.0
+        wlo, whi = wilson_ci(replicated, n_held)
+        print(f'  fold rule: cos>{c_cut:.4f} AND dh<={d_cut:.4f}')
+        print(f'  held-out replicated: {replicated}/{n_held} = '
+              f'{rep_rate*100:.2f}% [{wlo*100:.2f}%, {whi*100:.2f}%]')
+
+        fold_results[held_firm] = {
+            'n_train': n_train,
+            'n_held': n_held,
+            'fold_rule': derive_rule(c_cut, d_cut),
+            'fold_crossings': {'cos': c_cut, 'dh': d_cut},
+            'bootstrap_cos': boot_cos,
+            'bootstrap_dh': boot_dh,
+            'held_out_classification': {
+                'n_replicated': replicated,
+                'n_hand_leaning': hand_leaning,
+                'replicated_rate': rep_rate,
+                'wilson95': [float(wlo), float(whi)],
+            },
+        }
+    return fold_results
+
+
+def cross_fold_stability(fold_results, full_calib):
+    cs = [fold_results[f]['fold_crossings']['cos'] for f in BIG4
+          if fold_results[f]['fold_crossings']['cos'] is not None]
+    ds = [fold_results[f]['fold_crossings']['dh'] for f in BIG4
+          if fold_results[f]['fold_crossings']['dh'] is not None]
+    full_c = full_calib['k2_crossings']['cos']
+    full_d = full_calib['k2_crossings']['dh']
+    summary = {
+        'fold_cos_crossings': cs,
+        'fold_dh_crossings': ds,
+        'mean_cos': float(np.mean(cs)) if cs else None,
+        'mean_dh': float(np.mean(ds)) if ds else None,
+        'max_dev_cos_from_mean': (float(max(abs(np.array(cs) - np.mean(cs))))
+                                   if cs else None),
+        'max_dev_dh_from_mean': (float(max(abs(np.array(ds) - np.mean(ds))))
+                                  if ds else None),
+        'max_dev_cos_from_full': (float(max(abs(np.array(cs) - full_c)))
+                                   if cs and full_c else None),
+        'max_dev_dh_from_full': (float(max(abs(np.array(ds) - full_d)))
+                                  if ds and full_d else None),
+    }
+    cos_stable = (summary['max_dev_cos_from_mean'] is not None
+                  and summary['max_dev_cos_from_mean'] <= 0.005)
+    dh_stable = (summary['max_dev_dh_from_mean'] is not None
+                 and summary['max_dev_dh_from_mean'] <= 0.5)
+    summary['verdict'] = ('STABLE' if (cos_stable and dh_stable)
+                          else 'UNSTABLE')
+    return summary
+
+
+def render_panels(cpas, full_calib, fold_results):
+    by_firm = {}
+    for c in cpas:
+        by_firm.setdefault(c['firm'], []).append(c)
+
+    # Calibration panel
+    fig, ax = plt.subplots(figsize=(9, 7))
+    colors = {'勤業眾信聯合': 'crimson', '安侯建業聯合': 'royalblue',
+              '資誠聯合': 'forestgreen', '安永聯合': 'darkorange'}
+    labels = {'勤業眾信聯合': 'Firm A (Deloitte)', '安侯建業聯合': 'KPMG',
+              '資誠聯合': 'PwC', '安永聯合': 'EY'}
+    for firm in BIG4:
+        pts = by_firm[firm]
+        ax.scatter([p['cos_mean'] for p in pts], [p['dh_mean'] for p in pts],
+                   s=30, alpha=0.6, color=colors[firm],
+                   label=f'{labels[firm]} (n={len(pts)})')
+    c_cut = full_calib['k2_crossings']['cos']
+    d_cut = full_calib['k2_crossings']['dh']
+    ax.axvline(c_cut, color='black', ls='--', lw=1.5,
+               label=f'cos cut = {c_cut:.4f}')
+    ax.axhline(d_cut, color='black', ls=':', lw=1.5,
+               label=f'dh cut = {d_cut:.4f}')
+    ax.set_xlabel('Accountant cos_mean')
+    ax.set_ylabel('Accountant dh_mean')
+    ax.set_title('Big-4 calibration: 437 CPAs + K=2 marginal crossings')
+    ax.legend(fontsize=9)
+    ax.grid(alpha=0.3)
+    fig.tight_layout()
+    fig.savefig(OUT / 'panel_calibration.png', dpi=150)
+    plt.close(fig)
+
+    # LOOO panels
+    for held_firm in BIG4:
+        held = by_firm[held_firm]
+        train_pts = [c for c in cpas if c['firm'] != held_firm]
+        fr = fold_results[held_firm]
+        c_cut_f = fr['fold_crossings']['cos']
+        d_cut_f = fr['fold_crossings']['dh']
+        fig, ax = plt.subplots(figsize=(9, 7))
+        ax.scatter([p['cos_mean'] for p in train_pts],
+                   [p['dh_mean'] for p in train_pts],
+                   s=20, alpha=0.4, color='lightgray',
+                   label=f'Train (other Big-3, n={len(train_pts)})')
+        ax.scatter([p['cos_mean'] for p in held],
+                   [p['dh_mean'] for p in held],
+                   s=40, alpha=0.85, color=colors[held_firm],
+                   edgecolor='white',
+                   label=f'Held-out: {labels[held_firm]} (n={len(held)})')
+        if c_cut_f is not None:
+            ax.axvline(c_cut_f, color='black', ls='--', lw=1.5,
+                       label=f'fold cos cut = {c_cut_f:.4f}')
+        if d_cut_f is not None:
+            ax.axhline(d_cut_f, color='black', ls=':', lw=1.5,
+                       label=f'fold dh cut = {d_cut_f:.4f}')
+        rep = fr['held_out_classification']['n_replicated']
+        nh = fr['n_held']
+        rate = fr['held_out_classification']['replicated_rate']
+        wlo, whi = fr['held_out_classification']['wilson95']
+        ax.set_title(
+            f'LOOO: held-out {labels[held_firm]} ({rep}/{nh} = '
+            f'{rate*100:.1f}% replicated, Wilson 95% '
+            f'[{wlo*100:.1f}%, {whi*100:.1f}%])')
+        ax.set_xlabel('Accountant cos_mean')
+        ax.set_ylabel('Accountant dh_mean')
+        ax.legend(fontsize=9)
+        ax.grid(alpha=0.3)
+        fig.tight_layout()
+        firm_slug = ('FirmA' if held_firm == FIRM_A_LABEL
+                     else {'安侯建業聯合': 'KPMG', '資誠聯合': 'PwC',
+                           '安永聯合': 'EY'}.get(held_firm, held_firm))
+        fig.savefig(OUT / f'panel_loo_{firm_slug}.png', dpi=150)
+        plt.close(fig)
+
+
+def render_md(full_calib, fold_results, stability, sample_sizes):
+    md = [
+        '# Paper A v4.0 Phase 1 — Calibration + LOOO Validation',
+        f'Generated: {datetime.now().strftime("%Y-%m-%d %H:%M:%S")}',
+        '',
+        '## A. Big-4 Calibration',
+        '',
+        f'- N CPAs: {full_calib["n_cpas"]}',
+        f'- dip-test cos: p = {full_calib["dip_test_cos"]["dip_pvalue"]:.4g} '
+        f'({"unimodal" if full_calib["dip_test_cos"]["unimodal_alpha05"] else "multimodal"})',
+        f'- dip-test dh : p = {full_calib["dip_test_dh"]["dip_pvalue"]:.4g} '
+        f'({"unimodal" if full_calib["dip_test_dh"]["unimodal_alpha05"] else "multimodal"})',
+        f'- 2D GMM K=2 BIC = {full_calib["k2_bic"]:.2f}',
+        f'- 2D GMM K=3 BIC = {full_calib["k3_bic"]:.2f}',
+        '',
+        '### Marginal crossings (point + bootstrap 95% CI, n=500)',
+        '',
+        '| Axis | Point | Bootstrap median | 95% CI | CI half-width |',
+        '|---|---|---|---|---|',
+    ]
+    for axis_label, key in [('cos', 'bootstrap_cos'), ('dh', 'bootstrap_dh')]:
+        b = full_calib[key]
+        point = full_calib['k2_crossings'][axis_label]
+        if b is None:
+            md.append(f'| {axis_label} | {point} | n/a | n/a | n/a |')
+        else:
+            md.append(f'| {axis_label} | {point:.4f} | {b["median"]:.4f} | '
+                      f'[{b["ci95"][0]:.4f}, {b["ci95"][1]:.4f}] | '
+                      f'{b["ci_halfwidth"]:.4f} |')
+    md += ['',
+           f'### Operational classifier rule',
+           '',
+           f'> {full_calib["rule"]["rule"]}',
+           '',
+           '### K=2 components',
+           '',
+           '| Component | mean cos | mean dh | weight |',
+           '|---|---|---|---|']
+    for i, (m, w) in enumerate(zip(full_calib['k2_components']['means'],
+                                   full_calib['k2_components']['weights'])):
+        md.append(f'| C{i+1} | {m[0]:.4f} | {m[1]:.4f} | {w:.3f} |')
+
+    md += ['', '## B. Leave-One-Firm-Out Validation', '',
+           '| Held-out firm | n_train | n_held | Fold cos cut | Fold dh cut | '
+           'Replicated rate | Wilson 95% |',
+           '|---|---|---|---|---|---|---|']
+    label_map = {'勤業眾信聯合': 'Firm A (Deloitte)',
+                 '安侯建業聯合': 'KPMG',
+                 '資誠聯合': 'PwC',
+                 '安永聯合': 'EY'}
+    for f in BIG4:
+        fr = fold_results[f]
+        c = fr['fold_crossings']['cos']
+        d = fr['fold_crossings']['dh']
+        rep = fr['held_out_classification']
+        c_str = f'{c:.4f}' if c is not None else 'n/a'
+        d_str = f'{d:.4f}' if d is not None else 'n/a'
+        md.append(f'| {label_map[f]} | {fr["n_train"]} | {fr["n_held"]} | '
+                  f'{c_str} | {d_str} | {rep["replicated_rate"]*100:.2f}% | '
+                  f'[{rep["wilson95"][0]*100:.2f}%, '
+                  f'{rep["wilson95"][1]*100:.2f}%] |')
+
+    md += ['', '## C. Cross-fold stability', '',
+           f'- Mean fold cos crossing: '
+           f'{stability["mean_cos"]:.4f}' if stability["mean_cos"] is not None
+           else '- Mean fold cos crossing: n/a',
+           f'- Mean fold dh crossing : '
+           f'{stability["mean_dh"]:.4f}' if stability["mean_dh"] is not None
+           else '- Mean fold dh crossing: n/a',
+           f'- Max |dev_cos| across folds: '
+           f'{stability["max_dev_cos_from_mean"]:.4f}'
+           if stability["max_dev_cos_from_mean"] is not None
+           else '- Max |dev_cos|: n/a',
+           f'- Max |dev_dh| across folds : '
+           f'{stability["max_dev_dh_from_mean"]:.4f}'
+           if stability["max_dev_dh_from_mean"] is not None
+           else '- Max |dev_dh|: n/a',
+           f'- Max |dev_cos| vs full-calib: '
+           f'{stability["max_dev_cos_from_full"]:.4f}'
+           if stability["max_dev_cos_from_full"] is not None
+           else '- Max |dev_cos| vs full: n/a',
+           f'- Max |dev_dh| vs full-calib : '
+           f'{stability["max_dev_dh_from_full"]:.4f}'
+           if stability["max_dev_dh_from_full"] is not None
+           else '- Max |dev_dh| vs full: n/a',
+           '',
+           f'**Verdict: {stability["verdict"]}**',
+           '',
+           '### Verdict legend',
+           '- **STABLE**: max |dev_cos| <= 0.005 AND max |dev_dh| <= 0.5 '
+           'across the 4 LOOO folds; the threshold is reproducible across '
+           'firms.',
+           '- **UNSTABLE**: at least one fold deviates beyond the tolerance; '
+           'the threshold is sensitive to which firm is held out, which '
+           'would invite reviewer questions about generalizability.',
+           '',
+           '## Methodology notes',
+           '',
+           '- Held-out unit is the firm (not within-firm 70/30) -- this '
+           'tests the v4.0 methodology-paper claim that the pipeline '
+           'reproduces across firms, not just within a calibration sample.',
+           '- Bootstrap n=500 (consistent with Script 34); '
+           'GMM hyperparameters n_init=15, max_iter=500, random_state=42 '
+           '(consistent with Scripts 32/34/35).',
+           '- CPA-level classification uses the rule applied to the '
+           'accountant\'s mean (cos_mean, dh_mean).  A per-signature '
+           'classifier would apply the same rule signature-by-signature '
+           '(deferred to Script 38 for sensitivity analysis).',
+           '',
+           '## Files',
+           '- `panel_calibration.png` -- 437 Big-4 CPAs + K=2 cuts',
+           '- `panel_loo_<firm>.png` -- LOOO fold panels (4 firms)',
+           '- `calibration_loo_results.json` -- machine-readable full output',
+           ]
+    return '\n'.join(md)
+
+
+def main():
+    print('=' * 72)
+    print('Script 36: v4.0 Calibration + Leave-One-Firm-Out Validation')
+    print('=' * 72)
+    cpas = load_big4_accountants()
+    sample_sizes = {}
+    for c in cpas:
+        sample_sizes.setdefault(c['firm'], 0)
+        sample_sizes[c['firm']] += 1
+    print(f'\nTotal Big-4 CPAs (n_sigs >= {MIN_SIGS}): {len(cpas)}')
+    for f in BIG4:
+        print(f'  {f}: {sample_sizes.get(f, 0)}')
+
+    full_calib = run_calibration(cpas)
+    fold_results = run_loo(cpas)
+    stability = cross_fold_stability(fold_results, full_calib)
+    print(f'\n[C] Cross-fold stability verdict: {stability["verdict"]}')
+    print(f'    Max |dev_cos| from mean = '
+          f'{stability["max_dev_cos_from_mean"]}; '
+          f'from full-calib = {stability["max_dev_cos_from_full"]}')
+    print(f'    Max |dev_dh|  from mean = '
+          f'{stability["max_dev_dh_from_mean"]}; '
+          f'from full-calib = {stability["max_dev_dh_from_full"]}')
+
+    render_panels(cpas, full_calib, fold_results)
+    print(f'\nPanels: {OUT}/panel_calibration.png + 4 LOOO panels')
+
+    payload = {
+        'generated_at': datetime.now().isoformat(),
+        'min_sigs_per_accountant': MIN_SIGS,
+        'n_bootstrap': N_BOOT,
+        'random_seed': SEED,
+        'sample_sizes': sample_sizes,
+        'big4_calibration': full_calib,
+        'loo_folds': fold_results,
+        'cross_fold_stability': stability,
+    }
+    json_path = OUT / 'calibration_loo_results.json'
+    json_path.write_text(json.dumps(payload, indent=2, ensure_ascii=False),
+                         encoding='utf-8')
+    print(f'JSON: {json_path}')
+
+    md = render_md(full_calib, fold_results, stability, sample_sizes)
+    md_path = OUT / 'calibration_loo_report.md'
+    md_path.write_text(md, encoding='utf-8')
+    print(f'Report: {md_path}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/37_v4_k3_loo_check.py

@@ -0,0 +1,478 @@
+#!/usr/bin/env python3
+"""
+Script 37: K=3 Leave-One-Firm-Out Check (Path P2 viability test)
+=================================================================
+Follow-up to Script 36's UNSTABLE K=2 LOOO finding. Tests whether the
+K=3 mixture's C1 component (lowest-cosine "hand-leaning" cluster,
+~14% weight per Script 35) is a real cross-firm sub-population or
+is also firm-mass driven.
+
+Reference: Script 35 (full Big-4 K=3) reported C1 cluster membership:
+  Firm A    0/171   = 0.0%
+  KPMG     10/112   = 8.9%
+  PwC      24/102   = 23.5%
+  EY        6/52    = 11.5%
+
+The hypothesis: if C1 is a true cross-firm hand-leaning sub-population,
+then:
+  - Across 4 LOOO folds, the C1 component should sit at roughly the
+    same (cos, dh) coordinates with similar weight.
+  - When the held-out firm's CPAs are assigned via the fold's K=3
+    posterior, the fraction in C1 should approximate the Script 35
+    full-data percentages.
+
+If C1 collapses, shifts dramatically, or fails to predict held-out
+membership, then K=3 is also firm-mass driven and Path P2 fails.
+
+Output:
+  reports/v4_big4/k3_loo_check/
+    k3_loo_results.json
+    k3_loo_report.md
+    panel_k3_loo_<firm>.png
+"""
+
+import sqlite3
+import json
+import numpy as np
+import matplotlib
+matplotlib.use('Agg')
+import matplotlib.pyplot as plt
+from pathlib import Path
+from datetime import datetime
+from sklearn.mixture import GaussianMixture
+from scipy.stats import norm
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'v4_big4/k3_loo_check')
+OUT.mkdir(parents=True, exist_ok=True)
+
+MIN_SIGS = 10
+SEED = 42
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+LABEL = {'勤業眾信聯合': 'Firm A (Deloitte)', '安侯建業聯合': 'KPMG',
+         '資誠聯合': 'PwC', '安永聯合': 'EY'}
+SLUG = {'勤業眾信聯合': 'FirmA', '安侯建業聯合': 'KPMG',
+        '資誠聯合': 'PwC', '安永聯合': 'EY'}
+
+# Script 35 full-Big-4 K=3 baseline (informational; reproduce here as expected)
+SCRIPT35_C1_PCT = {'勤業眾信聯合': 0.0, '安侯建業聯合': 8.9,
+                   '資誠聯合': 23.5, '安永聯合': 11.5}
+
+
+def load_big4_accountants():
+    conn = sqlite3.connect(DB)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT s.assigned_accountant,
+               a.firm,
+               AVG(s.max_similarity_to_same_accountant) AS cos_mean,
+               AVG(CAST(s.min_dhash_independent AS REAL)) AS dh_mean,
+               COUNT(*) AS n
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.min_dhash_independent IS NOT NULL
+          AND a.firm IN (?, ?, ?, ?)
+        GROUP BY s.assigned_accountant
+        HAVING n >= ?
+    ''', BIG4 + (MIN_SIGS,))
+    rows = cur.fetchall()
+    conn.close()
+    return [{'cpa': r[0], 'firm': r[1],
+             'cos_mean': float(r[2]), 'dh_mean': float(r[3]),
+             'n_sigs': int(r[4])} for r in rows]
+
+
+def fit_k3(X):
+    return GaussianMixture(n_components=3, covariance_type='full',
+                           random_state=SEED, n_init=15, max_iter=500).fit(X)
+
+
+def sort_components_by_cos(gmm):
+    """Return ordering such that comp[0] has lowest cosine mean."""
+    return np.argsort(gmm.means_[:, 0])
+
+
+def wilson_ci(k, n, alpha=0.05):
+    if n == 0:
+        return (0.0, 1.0)
+    z = norm.ppf(1 - alpha / 2)
+    phat = k / n
+    denom = 1 + z * z / n
+    center = (phat + z * z / (2 * n)) / denom
+    pm = z * np.sqrt(phat * (1 - phat) / n + z * z / (4 * n * n)) / denom
+    return (max(0.0, center - pm), min(1.0, center + pm))
+
+
+def run_full_baseline(cpas):
+    print('\n[A] Full-Big-4 K=3 baseline (replicates Script 35)')
+    X = np.column_stack([
+        [c['cos_mean'] for c in cpas],
+        [c['dh_mean'] for c in cpas],
+    ])
+    gmm = fit_k3(X)
+    order = sort_components_by_cos(gmm)
+    means = gmm.means_[order]
+    weights = gmm.weights_[order]
+    raw_labels = gmm.predict(X)
+    label_map = {old: new for new, old in enumerate(order)}
+    labels = np.array([label_map[l] for l in raw_labels])
+
+    by_firm_c1 = {f: 0 for f in BIG4}
+    by_firm_total = {f: 0 for f in BIG4}
+    for c, lab in zip(cpas, labels):
+        by_firm_total[c['firm']] += 1
+        if lab == 0:
+            by_firm_c1[c['firm']] += 1
+    print(f'  C1 (hand-leaning) center: cos={means[0,0]:.4f}, '
+          f'dh={means[0,1]:.4f}, weight={weights[0]:.3f}')
+    print(f'  C2 (mixed)        center: cos={means[1,0]:.4f}, '
+          f'dh={means[1,1]:.4f}, weight={weights[1]:.3f}')
+    print(f'  C3 (replicated)   center: cos={means[2,0]:.4f}, '
+          f'dh={means[2,1]:.4f}, weight={weights[2]:.3f}')
+    print('  C1 membership by firm:')
+    for f in BIG4:
+        n = by_firm_total[f]
+        k = by_firm_c1[f]
+        pct = 100 * k / n if n else 0.0
+        print(f'    {LABEL[f]:<22} {k:>3}/{n:>3} = {pct:5.2f}%  '
+              f'(Script 35 expected: {SCRIPT35_C1_PCT[f]}%)')
+    return {
+        'means_sorted': means.tolist(),
+        'weights_sorted': weights.tolist(),
+        'c1_membership_by_firm': {
+            f: {'k': int(by_firm_c1[f]), 'n': int(by_firm_total[f]),
+                'pct': float(100 * by_firm_c1[f] / by_firm_total[f])
+                if by_firm_total[f] else 0.0}
+            for f in BIG4
+        },
+    }
+
+
+def run_loo(cpas):
+    by_firm = {}
+    for c in cpas:
+        by_firm.setdefault(c['firm'], []).append(c)
+    fold_results = {}
+    for held_firm in BIG4:
+        train = [c for c in cpas if c['firm'] != held_firm]
+        held = by_firm[held_firm]
+        X_train = np.column_stack([
+            [c['cos_mean'] for c in train],
+            [c['dh_mean'] for c in train],
+        ])
+        X_held = np.column_stack([
+            [c['cos_mean'] for c in held],
+            [c['dh_mean'] for c in held],
+        ])
+        gmm = fit_k3(X_train)
+        order = sort_components_by_cos(gmm)
+        means = gmm.means_[order]
+        weights = gmm.weights_[order]
+        # Posterior on held-out
+        raw_post = gmm.predict_proba(X_held)
+        post = raw_post[:, order]
+        held_labels = np.argmax(post, axis=1)
+        n_c1 = int(np.sum(held_labels == 0))
+        n_c2 = int(np.sum(held_labels == 1))
+        n_c3 = int(np.sum(held_labels == 2))
+        n_held = len(held)
+        c1_rate = n_c1 / n_held if n_held else 0.0
+        wlo, whi = wilson_ci(n_c1, n_held)
+        # Train-side weights for stability check
+        print(f'\n[B] LOOO fold: held = {LABEL[held_firm]}')
+        print(f'    train K=3 components (sorted by cos):')
+        for i in range(3):
+            print(f'      C{i+1}: cos={means[i,0]:.4f}, dh={means[i,1]:.4f}, '
+                  f'weight={weights[i]:.3f}')
+        print(f'    held-out assignments: C1={n_c1}/{n_held} = '
+              f'{c1_rate*100:.2f}% [Wilson 95%: '
+              f'{wlo*100:.2f}%, {whi*100:.2f}%]')
+        print(f'                          C2={n_c2}/{n_held} = '
+              f'{n_c2/n_held*100:.2f}%')
+        print(f'                          C3={n_c3}/{n_held} = '
+              f'{n_c3/n_held*100:.2f}%')
+        print(f'    Script 35 expected C1 for {LABEL[held_firm]}: '
+              f'{SCRIPT35_C1_PCT[held_firm]}%')
+        fold_results[held_firm] = {
+            'n_train': len(train),
+            'n_held': n_held,
+            'k3_components_sorted_by_cos': {
+                'means': means.tolist(),
+                'weights': weights.tolist(),
+            },
+            'held_out_assignments': {
+                'n_c1_handleaning': n_c1,
+                'n_c2_mixed': n_c2,
+                'n_c3_replicated': n_c3,
+                'c1_rate': float(c1_rate),
+                'c1_wilson95': [float(wlo), float(whi)],
+            },
+            'script35_expected_c1_pct': SCRIPT35_C1_PCT[held_firm],
+        }
+    return fold_results
+
+
+def stability_summary(fold_results, baseline):
+    """Aggregate C1 component drift across folds."""
+    c1_means_cos = [fold_results[f]['k3_components_sorted_by_cos']['means'][0][0]
+                    for f in BIG4]
+    c1_means_dh = [fold_results[f]['k3_components_sorted_by_cos']['means'][0][1]
+                   for f in BIG4]
+    c1_weights = [fold_results[f]['k3_components_sorted_by_cos']['weights'][0]
+                  for f in BIG4]
+    base_c1_cos = baseline['means_sorted'][0][0]
+    base_c1_dh = baseline['means_sorted'][0][1]
+    base_c1_w = baseline['weights_sorted'][0]
+    summary = {
+        'fold_c1_cos_means': c1_means_cos,
+        'fold_c1_dh_means': c1_means_dh,
+        'fold_c1_weights': c1_weights,
+        'baseline_c1': {'cos': base_c1_cos, 'dh': base_c1_dh,
+                        'weight': base_c1_w},
+        'max_c1_cos_dev_from_baseline': float(
+            max(abs(np.array(c1_means_cos) - base_c1_cos))),
+        'max_c1_dh_dev_from_baseline': float(
+            max(abs(np.array(c1_means_dh) - base_c1_dh))),
+        'max_c1_weight_dev_from_baseline': float(
+            max(abs(np.array(c1_weights) - base_c1_w))),
+    }
+    # Heuristic stability bars (these are exploratory, not formal test):
+    cos_stable = summary['max_c1_cos_dev_from_baseline'] <= 0.01
+    dh_stable = summary['max_c1_dh_dev_from_baseline'] <= 1.0
+    weight_stable = summary['max_c1_weight_dev_from_baseline'] <= 0.10
+    summary['cos_stable'] = bool(cos_stable)
+    summary['dh_stable'] = bool(dh_stable)
+    summary['weight_stable'] = bool(weight_stable)
+    summary['c1_component_stable'] = bool(cos_stable and dh_stable
+                                           and weight_stable)
+
+    # Held-out C1 prediction agreement with Script 35 expectation
+    pred_v_expected = []
+    for f in BIG4:
+        actual = fold_results[f]['held_out_assignments']['c1_rate'] * 100
+        expected = SCRIPT35_C1_PCT[f]
+        pred_v_expected.append({
+            'firm': LABEL[f],
+            'predicted_c1_pct': actual,
+            'expected_c1_pct': expected,
+            'abs_diff': abs(actual - expected),
+        })
+    summary['held_out_prediction_check'] = pred_v_expected
+    summary['max_abs_pct_diff'] = float(max(p['abs_diff']
+                                             for p in pred_v_expected))
+
+    # Verdict
+    if (summary['c1_component_stable']
+            and summary['max_abs_pct_diff'] <= 5.0):
+        verdict = 'P2_STRONG'
+        msg = ('K=3 C1 component is stable across LOOO folds (cos drift '
+               '<= 0.01, dh drift <= 1.0, weight drift <= 0.10); held-out '
+               'C1 predictions agree with Script 35 baseline within 5pp. '
+               'Path P2 is viable: K=3 captures a real cross-firm '
+               'hand-leaning cluster.')
+    elif summary['c1_component_stable']:
+        verdict = 'P2_PARTIAL'
+        msg = ('K=3 C1 component is stable but held-out C1 prediction '
+               f'diverges from Script 35 baseline (max abs diff '
+               f'{summary["max_abs_pct_diff"]:.1f}pp). Cluster exists but '
+               'membership is not well-predicted by held-out fit.')
+    else:
+        verdict = 'P2_WEAK'
+        msg = ('K=3 C1 component is NOT stable across LOOO folds (cos drift '
+               f'{summary["max_c1_cos_dev_from_baseline"]:.4f}, dh drift '
+               f'{summary["max_c1_dh_dev_from_baseline"]:.3f}, weight drift '
+               f'{summary["max_c1_weight_dev_from_baseline"]:.3f}). '
+               'K=3 is also firm-mass driven; Path P2 fails.')
+    summary['verdict'] = verdict
+    summary['verdict_message'] = msg
+    return summary
+
+
+def render_panels(cpas, fold_results):
+    by_firm = {}
+    for c in cpas:
+        by_firm.setdefault(c['firm'], []).append(c)
+    for held_firm in BIG4:
+        held = by_firm[held_firm]
+        train = [c for c in cpas if c['firm'] != held_firm]
+        fr = fold_results[held_firm]
+        means = np.array(fr['k3_components_sorted_by_cos']['means'])
+        weights = fr['k3_components_sorted_by_cos']['weights']
+        rate = fr['held_out_assignments']['c1_rate']
+        n_c1 = fr['held_out_assignments']['n_c1_handleaning']
+        n_h = fr['n_held']
+        wlo, whi = fr['held_out_assignments']['c1_wilson95']
+        fig, ax = plt.subplots(figsize=(9, 7))
+        ax.scatter([c['cos_mean'] for c in train],
+                   [c['dh_mean'] for c in train], s=18, alpha=0.4,
+                   color='lightgray',
+                   label=f'Train (other Big-3, n={len(train)})')
+        ax.scatter([c['cos_mean'] for c in held],
+                   [c['dh_mean'] for c in held], s=42, alpha=0.85,
+                   color='crimson', edgecolor='white',
+                   label=f'Held-out: {LABEL[held_firm]} (n={n_h})')
+        markers = ['v', 's', '^']
+        comp_colors = ['darkred', 'goldenrod', 'navy']
+        comp_labels = ['C1 hand-leaning', 'C2 mixed', 'C3 replicated']
+        for i in range(3):
+            ax.scatter([means[i, 0]], [means[i, 1]], s=200,
+                       marker=markers[i], color=comp_colors[i],
+                       edgecolor='black', linewidth=1.5,
+                       label=f'{comp_labels[i]}: ({means[i,0]:.3f}, '
+                             f'{means[i,1]:.2f}), w={weights[i]:.2f}')
+        ax.set_xlabel('Accountant cos_mean')
+        ax.set_ylabel('Accountant dh_mean')
+        ax.set_title(
+            f'K=3 LOOO held-out {LABEL[held_firm]}: C1 = {n_c1}/{n_h} = '
+            f'{rate*100:.1f}% [Wilson 95%: {wlo*100:.1f}%, '
+            f'{whi*100:.1f}%]\n(Script 35 baseline expected: '
+            f'{SCRIPT35_C1_PCT[held_firm]}%)')
+        ax.legend(fontsize=8, loc='upper right')
+        ax.grid(alpha=0.3)
+        fig.tight_layout()
+        fig.savefig(OUT / f'panel_k3_loo_{SLUG[held_firm]}.png', dpi=150)
+        plt.close(fig)
+
+
+def render_md(baseline, fold_results, summary):
+    md = [
+        '# Phase 1.5: K=3 LOOO Check (Path P2 viability)',
+        f'Generated: {datetime.now().strftime("%Y-%m-%d %H:%M:%S")}',
+        '',
+        '## A. Full-Big-4 K=3 baseline (replicates Script 35)',
+        '',
+        '| Component | mean cos | mean dh | weight |',
+        '|---|---|---|---|',
+    ]
+    for i, (m, w) in enumerate(zip(baseline['means_sorted'],
+                                   baseline['weights_sorted'])):
+        name = ['C1 hand-leaning', 'C2 mixed',
+                'C3 replicated'][i]
+        md.append(f'| {name} | {m[0]:.4f} | {m[1]:.4f} | {w:.3f} |')
+    md += ['',
+           '### Baseline C1 membership by firm',
+           '',
+           '| Firm | Baseline C1 / total | % | Script 35 expected |',
+           '|---|---|---|---|']
+    for f in BIG4:
+        b = baseline['c1_membership_by_firm'][f]
+        md.append(f'| {LABEL[f]} | {b["k"]}/{b["n"]} | {b["pct"]:.2f}% | '
+                  f'{SCRIPT35_C1_PCT[f]}% |')
+
+    md += ['', '## B. Leave-One-Firm-Out K=3 fits', '']
+    for f in BIG4:
+        fr = fold_results[f]
+        means = fr['k3_components_sorted_by_cos']['means']
+        weights = fr['k3_components_sorted_by_cos']['weights']
+        ass = fr['held_out_assignments']
+        md += [f'### Held-out: {LABEL[f]}',
+               '',
+               f'- n_train = {fr["n_train"]}, n_held = {fr["n_held"]}',
+               f'- Held-out assignments: '
+               f'C1={ass["n_c1_handleaning"]}/{fr["n_held"]} = '
+               f'{ass["c1_rate"]*100:.2f}% '
+               f'[Wilson 95%: {ass["c1_wilson95"][0]*100:.2f}%, '
+               f'{ass["c1_wilson95"][1]*100:.2f}%]; '
+               f'C2={ass["n_c2_mixed"]}; C3={ass["n_c3_replicated"]}',
+               f'- Script 35 baseline expected C1: '
+               f'{SCRIPT35_C1_PCT[f]}%',
+               '',
+               '| Train K=3 component | mean cos | mean dh | weight |',
+               '|---|---|---|---|']
+        for i, (m, w) in enumerate(zip(means, weights)):
+            name = ['C1 hand-leaning', 'C2 mixed',
+                    'C3 replicated'][i]
+            md.append(f'| {name} | {m[0]:.4f} | {m[1]:.4f} | {w:.3f} |')
+        md.append('')
+
+    md += ['## C. Cross-fold C1 stability summary', '',
+           f'- Baseline C1 (full Big-4): cos = '
+           f'{summary["baseline_c1"]["cos"]:.4f}, dh = '
+           f'{summary["baseline_c1"]["dh"]:.4f}, weight = '
+           f'{summary["baseline_c1"]["weight"]:.3f}',
+           f'- Fold C1 cos means: {summary["fold_c1_cos_means"]}',
+           f'- Fold C1 dh  means: {summary["fold_c1_dh_means"]}',
+           f'- Fold C1 weights  : {summary["fold_c1_weights"]}',
+           f'- Max |C1 cos dev| vs baseline: '
+           f'{summary["max_c1_cos_dev_from_baseline"]:.4f} '
+           f'(stable bar: 0.01, {"OK" if summary["cos_stable"] else "FAIL"})',
+           f'- Max |C1 dh  dev| vs baseline: '
+           f'{summary["max_c1_dh_dev_from_baseline"]:.3f} '
+           f'(stable bar: 1.0, {"OK" if summary["dh_stable"] else "FAIL"})',
+           f'- Max |C1 weight dev| vs baseline: '
+           f'{summary["max_c1_weight_dev_from_baseline"]:.3f} '
+           f'(stable bar: 0.10, {"OK" if summary["weight_stable"] else "FAIL"})',
+           '',
+           '### Held-out prediction vs Script 35 baseline',
+           '',
+           '| Firm | Predicted C1% | Expected C1% | |diff| pp |',
+           '|---|---|---|---|']
+    for entry in summary['held_out_prediction_check']:
+        md.append(f'| {entry["firm"]} | {entry["predicted_c1_pct"]:.2f}% | '
+                  f'{entry["expected_c1_pct"]}% | '
+                  f'{entry["abs_diff"]:.2f} |')
+    md += ['',
+           f'- Max |%diff| across folds: {summary["max_abs_pct_diff"]:.2f}pp '
+           f'(viable bar: <= 5.0 pp)',
+           '',
+           f'## Verdict: **{summary["verdict"]}**',
+           '',
+           summary['verdict_message'],
+           '',
+           '### Verdict legend',
+           '- **P2_STRONG**: C1 cluster reproducible across folds AND '
+           'held-out predictions match Script 35 baseline within 5 pp. '
+           'K=3 captures a real cross-firm hand-leaning sub-population; '
+           'Paper A v4.0 can use K=3 hard assignment as the operational '
+           'classifier.',
+           '- **P2_PARTIAL**: C1 cluster shape reproducible but membership '
+           'predictions diverge. Cluster exists conceptually but is not '
+           'predictively useful as an operational classifier.',
+           '- **P2_WEAK**: C1 cluster shifts substantially across folds. '
+           'K=3 is also firm-mass driven; v4.0 needs a different strategy '
+           '(P1 firm-templatedness reframe, P3 rollback, or P4 '
+           'reverse-anchor).',
+           ]
+    return '\n'.join(md)
+
+
+def main():
+    print('=' * 72)
+    print('Script 37: K=3 LOOO Check (Path P2 viability)')
+    print('=' * 72)
+    cpas = load_big4_accountants()
+    print(f'\nN Big-4 CPAs: {len(cpas)}')
+    baseline = run_full_baseline(cpas)
+    fold_results = run_loo(cpas)
+    summary = stability_summary(fold_results, baseline)
+    print(f'\n[C] Verdict: {summary["verdict"]}')
+    print(f'    {summary["verdict_message"]}')
+
+    render_panels(cpas, fold_results)
+
+    payload = {
+        'generated_at': datetime.now().isoformat(),
+        'min_sigs_per_accountant': MIN_SIGS,
+        'random_seed': SEED,
+        'n_cpas_total': len(cpas),
+        'baseline': baseline,
+        'loo_folds': fold_results,
+        'stability_summary': summary,
+        'script35_c1_baseline_pct': SCRIPT35_C1_PCT,
+    }
+    json_path = OUT / 'k3_loo_results.json'
+    json_path.write_text(json.dumps(payload, indent=2, ensure_ascii=False),
+                         encoding='utf-8')
+    print(f'JSON: {json_path}')
+
+    md = render_md(baseline, fold_results, summary)
+    md_path = OUT / 'k3_loo_report.md'
+    md_path.write_text(md, encoding='utf-8')
+    print(f'Report: {md_path}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/38_v4_convergence_k3_and_reverse_anchor.py

@@ -0,0 +1,531 @@
+#!/usr/bin/env python3
+"""
+Script 38: v4.0 Convergence — K=3 cluster + Reverse-Anchor + Paper A rule
+==========================================================================
+Phase 1.6 (G2) script. Tests whether three INDEPENDENT statistical
+approaches converge on the same Big-4 CPA ranking:
+
+  Approach 1: K=3 GMM cluster posterior P_C1 (hand-leaning)
+              -- from Script 37 baseline fit on full Big-4 (n=437).
+              Higher P_C1 -> more hand-leaning.
+
+  Approach 2: Reverse-anchor directional score
+              -- non-Big-4 (n=249, mid/small firms) as the
+                 fully-replicated reference distribution.
+              -- For each Big-4 CPA: cosine left-tail percentile under
+                 the reference 2D Gaussian (MCD).
+              -- Score = -percentile (so higher = more deviated in the
+                 hand-leaning direction).
+
+  Approach 3: Paper A v3.x operational hand_frac
+              -- Per-CPA fraction of signatures that fail
+                 (cos > 0.95 AND dh <= 5).
+
+Convergence claim: if all three rank Big-4 CPAs the same way (Spearman
+rho >= 0.7 for every pair), then the v4.0 methodology paper has
+**three independent lines of evidence** for the same population
+structure -- a much harder thing for a reviewer to dismiss than any
+single approach.
+
+Per-firm breakdown shows the Script 35 finding (Firm A 0% C1, PwC
+23.5% C1) holds across all three lenses.
+
+Methodology choice: non-Big-4 as the reverse-anchor reference (rather
+than non-Firm-A as in Script 33) maintains strict train/target
+separation -- the v4.0 target population is Big-4, the reference is
+strictly outside Big-4.
+
+Output:
+  reports/v4_big4/convergence_k3_reverse_anchor/
+    convergence_results.json
+    convergence_report.md
+    scatter_pairwise.png       3x3 scatter of approach pairs
+    per_firm_summary.csv       per-firm aggregates
+"""
+
+import sqlite3
+import csv
+import json
+import numpy as np
+import matplotlib
+matplotlib.use('Agg')
+import matplotlib.pyplot as plt
+from pathlib import Path
+from datetime import datetime
+from scipy import stats
+from sklearn.mixture import GaussianMixture
+from sklearn.covariance import MinCovDet
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'v4_big4/convergence_k3_reverse_anchor')
+OUT.mkdir(parents=True, exist_ok=True)
+
+MIN_SIGS = 10
+SEED = 42
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+LABEL = {'勤業眾信聯合': 'Firm A (Deloitte)', '安侯建業聯合': 'KPMG',
+         '資誠聯合': 'PwC', '安永聯合': 'EY'}
+
+PAPER_A_COS_CUT = 0.95
+PAPER_A_DH_CUT = 5
+
+# Convergence thresholds (heuristic)
+RHO_STRONG = 0.70
+RHO_PARTIAL = 0.40
+
+
+def load_accountants(firm_filter_sql, params, with_handfrac=False):
+    conn = sqlite3.connect(DB)
+    cur = conn.cursor()
+    if with_handfrac:
+        sql = f'''
+            SELECT s.assigned_accountant,
+                   a.firm,
+                   AVG(s.max_similarity_to_same_accountant) AS cos_mean,
+                   AVG(CAST(s.min_dhash_independent AS REAL)) AS dh_mean,
+                   AVG(CASE
+                         WHEN s.max_similarity_to_same_accountant > ?
+                              AND s.min_dhash_independent <= ?
+                         THEN 0.0 ELSE 1.0
+                       END) AS hand_frac,
+                   COUNT(*) AS n
+            FROM signatures s
+            JOIN accountants a ON s.assigned_accountant = a.name
+            WHERE s.assigned_accountant IS NOT NULL
+              AND s.max_similarity_to_same_accountant IS NOT NULL
+              AND s.min_dhash_independent IS NOT NULL
+              {firm_filter_sql}
+            GROUP BY s.assigned_accountant
+            HAVING n >= ?
+        '''
+        cur.execute(sql, [PAPER_A_COS_CUT, PAPER_A_DH_CUT]
+                    + params + [MIN_SIGS])
+        rows = cur.fetchall()
+        out = [{'cpa': r[0], 'firm': r[1],
+                'cos_mean': float(r[2]), 'dh_mean': float(r[3]),
+                'hand_frac': float(r[4]), 'n_sigs': int(r[5])}
+               for r in rows]
+    else:
+        sql = f'''
+            SELECT s.assigned_accountant,
+                   a.firm,
+                   AVG(s.max_similarity_to_same_accountant) AS cos_mean,
+                   AVG(CAST(s.min_dhash_independent AS REAL)) AS dh_mean,
+                   COUNT(*) AS n
+            FROM signatures s
+            JOIN accountants a ON s.assigned_accountant = a.name
+            WHERE s.assigned_accountant IS NOT NULL
+              AND s.max_similarity_to_same_accountant IS NOT NULL
+              AND s.min_dhash_independent IS NOT NULL
+              {firm_filter_sql}
+            GROUP BY s.assigned_accountant
+            HAVING n >= ?
+        '''
+        cur.execute(sql, params + [MIN_SIGS])
+        rows = cur.fetchall()
+        out = [{'cpa': r[0], 'firm': r[1],
+                'cos_mean': float(r[2]), 'dh_mean': float(r[3]),
+                'n_sigs': int(r[4])} for r in rows]
+    conn.close()
+    return out
+
+
+def load_big4():
+    return load_accountants('AND a.firm IN (?, ?, ?, ?)',
+                            list(BIG4), with_handfrac=True)
+
+
+def load_non_big4_reference():
+    return load_accountants(
+        'AND a.firm IS NOT NULL AND a.firm NOT IN (?, ?, ?, ?)',
+        list(BIG4), with_handfrac=False)
+
+
+def fit_reference_gaussian(points):
+    X = np.asarray(points, dtype=float)
+    mcd = MinCovDet(random_state=SEED, support_fraction=0.85).fit(X)
+    return {
+        'mean': mcd.location_,
+        'cov': mcd.covariance_,
+        'cov_inv': np.linalg.inv(mcd.covariance_),
+        'support_fraction': 0.85,
+        'n_reference': int(len(X)),
+    }
+
+
+def reverse_anchor_directional_score(cpa, ref):
+    """Returns -cos_left_tail_pct under the reference marginal cos
+    Gaussian. Higher (less negative) = more deviated in the hand-
+    leaning direction (left tail of reference cosine distribution).
+    """
+    mu_c = ref['mean'][0]
+    sd_c = float(np.sqrt(ref['cov'][0, 0]))
+    tail = float(stats.norm.cdf(cpa['cos_mean'], loc=mu_c, scale=sd_c))
+    return -tail
+
+
+def fit_k3_big4(big4_cpas):
+    X = np.column_stack([
+        [c['cos_mean'] for c in big4_cpas],
+        [c['dh_mean'] for c in big4_cpas],
+    ])
+    gmm = GaussianMixture(n_components=3, covariance_type='full',
+                          random_state=SEED, n_init=15, max_iter=500).fit(X)
+    order = np.argsort(gmm.means_[:, 0])  # C1 = lowest cos = hand-leaning
+    return gmm, order
+
+
+def compute_p_c1(cpa, gmm, order):
+    X = np.array([[cpa['cos_mean'], cpa['dh_mean']]])
+    raw_post = gmm.predict_proba(X)[0]
+    return float(raw_post[order[0]])
+
+
+def compute_correlations(big4_data):
+    p_c1 = np.array([d['p_c1'] for d in big4_data])
+    rev_anchor = np.array([d['reverse_anchor_score'] for d in big4_data])
+    hand_frac = np.array([d['paperA_hand_frac'] for d in big4_data])
+    pairs = [
+        ('p_c1_vs_paperA_hand_frac', p_c1, hand_frac),
+        ('reverse_anchor_vs_paperA_hand_frac', rev_anchor, hand_frac),
+        ('p_c1_vs_reverse_anchor', p_c1, rev_anchor),
+    ]
+    out = {}
+    for name, a, b in pairs:
+        rho, p = stats.spearmanr(a, b)
+        r, p_pearson = stats.pearsonr(a, b)
+        out[name] = {
+            'spearman_rho': float(rho),
+            'spearman_p': float(p),
+            'pearson_r': float(r),
+            'pearson_p': float(p_pearson),
+        }
+    return out
+
+
+def classify_convergence(corrs):
+    rhos = [corrs['p_c1_vs_paperA_hand_frac']['spearman_rho'],
+            corrs['reverse_anchor_vs_paperA_hand_frac']['spearman_rho'],
+            corrs['p_c1_vs_reverse_anchor']['spearman_rho']]
+    abs_rhos = [abs(r) for r in rhos]
+    min_abs_rho = float(min(abs_rhos))
+    all_strong = all(r >= RHO_STRONG for r in abs_rhos)
+    all_partial = all(r >= RHO_PARTIAL for r in abs_rhos)
+    if all_strong:
+        return 'CONVERGENCE_STRONG', (
+            f'All three pairwise Spearman |rho| >= {RHO_STRONG}; '
+            f'min |rho| = {min_abs_rho:.3f}. Three independent statistical '
+            f'lenses agree on the Big-4 CPA hand-leaning ranking.')
+    if all_partial:
+        return 'CONVERGENCE_PARTIAL', (
+            f'All three pairwise Spearman |rho| >= {RHO_PARTIAL} but at '
+            f'least one falls below {RHO_STRONG}; min |rho| = '
+            f'{min_abs_rho:.3f}. Methods agree on direction but not '
+            f'tightness; v4.0 can present them as complementary lenses.')
+    return 'CONVERGENCE_WEAK', (
+        f'At least one pair has |rho| < {RHO_PARTIAL}; min |rho| = '
+        f'{min_abs_rho:.3f}. Methods disagree -- they may be measuring '
+        f'different constructs.')
+
+
+def per_firm_aggregate(big4_data):
+    by_firm = {}
+    for d in big4_data:
+        by_firm.setdefault(d['firm'], []).append(d)
+    rows = []
+    for f in BIG4:
+        items = by_firm.get(f, [])
+        n = len(items)
+        if n == 0:
+            continue
+        c1_count = sum(1 for d in items if d['hard_label'] == 'C1')
+        c2_count = sum(1 for d in items if d['hard_label'] == 'C2')
+        c3_count = sum(1 for d in items if d['hard_label'] == 'C3')
+        mean_p_c1 = float(np.mean([d['p_c1'] for d in items]))
+        mean_rev = float(np.mean([d['reverse_anchor_score'] for d in items]))
+        mean_hand = float(np.mean([d['paperA_hand_frac'] for d in items]))
+        rows.append({
+            'firm': f,
+            'firm_label': LABEL[f],
+            'n_cpas': n,
+            'k3_C1_count': c1_count,
+            'k3_C2_count': c2_count,
+            'k3_C3_count': c3_count,
+            'k3_C1_pct': float(100 * c1_count / n),
+            'k3_C3_pct': float(100 * c3_count / n),
+            'mean_p_c1': mean_p_c1,
+            'mean_reverse_anchor': mean_rev,
+            'mean_paperA_hand_frac': mean_hand,
+        })
+    return rows
+
+
+def render_scatter(big4_data):
+    p_c1 = np.array([d['p_c1'] for d in big4_data])
+    rev = np.array([d['reverse_anchor_score'] for d in big4_data])
+    hf = np.array([d['paperA_hand_frac'] for d in big4_data])
+    firm_color = {
+        '勤業眾信聯合': 'crimson', '安侯建業聯合': 'royalblue',
+        '資誠聯合': 'forestgreen', '安永聯合': 'darkorange',
+    }
+    colors = [firm_color[d['firm']] for d in big4_data]
+
+    fig, axes = plt.subplots(1, 3, figsize=(18, 5.5))
+    pairs = [
+        ('K=3 P(C1 hand-leaning)', p_c1,
+         'Paper A hand_frac', hf,
+         'p_c1_vs_paperA_hand_frac'),
+        ('Reverse-anchor directional score', rev,
+         'Paper A hand_frac', hf,
+         'reverse_anchor_vs_paperA_hand_frac'),
+        ('K=3 P(C1 hand-leaning)', p_c1,
+         'Reverse-anchor directional score', rev,
+         'p_c1_vs_reverse_anchor'),
+    ]
+    for ax, (xl, x, yl, y, _name) in zip(axes, pairs):
+        ax.scatter(x, y, s=20, alpha=0.55, c=colors, edgecolor='white')
+        rho, p = stats.spearmanr(x, y)
+        ax.set_xlabel(xl)
+        ax.set_ylabel(yl)
+        ax.set_title(f'{xl}\nvs {yl}\nSpearman rho={rho:.3f} (p={p:.2e})')
+        ax.grid(alpha=0.3)
+    # Add legend for firm color
+    handles = [plt.Line2D([0], [0], marker='o', linestyle='', color=c,
+                          label=LABEL[f], markersize=8)
+               for f, c in firm_color.items()]
+    fig.legend(handles=handles, loc='lower center',
+               ncol=4, bbox_to_anchor=(0.5, -0.02))
+    fig.tight_layout()
+    fig.savefig(OUT / 'scatter_pairwise.png', dpi=150,
+                bbox_inches='tight')
+    plt.close(fig)
+
+
+def write_csv(per_firm_rows, big4_data):
+    csv_per_firm = OUT / 'per_firm_summary.csv'
+    with open(csv_per_firm, 'w', newline='', encoding='utf-8') as f:
+        w = csv.writer(f)
+        w.writerow(['firm', 'firm_label', 'n_cpas',
+                    'k3_C1_count', 'k3_C2_count', 'k3_C3_count',
+                    'k3_C1_pct', 'k3_C3_pct',
+                    'mean_p_c1', 'mean_reverse_anchor',
+                    'mean_paperA_hand_frac'])
+        for r in per_firm_rows:
+            w.writerow([r['firm'], r['firm_label'], r['n_cpas'],
+                        r['k3_C1_count'], r['k3_C2_count'], r['k3_C3_count'],
+                        f'{r["k3_C1_pct"]:.2f}', f'{r["k3_C3_pct"]:.2f}',
+                        f'{r["mean_p_c1"]:.4f}',
+                        f'{r["mean_reverse_anchor"]:.4f}',
+                        f'{r["mean_paperA_hand_frac"]:.4f}'])
+    csv_cpa = OUT / 'per_cpa_scores.csv'
+    with open(csv_cpa, 'w', newline='', encoding='utf-8') as f:
+        w = csv.writer(f)
+        w.writerow(['cpa', 'firm', 'firm_label', 'n_sigs',
+                    'cos_mean', 'dh_mean',
+                    'p_c1', 'p_c2', 'p_c3', 'hard_label',
+                    'reverse_anchor_score', 'paperA_hand_frac'])
+        for d in big4_data:
+            w.writerow([d['cpa'], d['firm'], LABEL[d['firm']], d['n_sigs'],
+                        f'{d["cos_mean"]:.4f}', f'{d["dh_mean"]:.4f}',
+                        f'{d["p_c1"]:.4f}', f'{d["p_c2"]:.4f}',
+                        f'{d["p_c3"]:.4f}', d['hard_label'],
+                        f'{d["reverse_anchor_score"]:.4f}',
+                        f'{d["paperA_hand_frac"]:.4f}'])
+    return csv_per_firm, csv_cpa
+
+
+def render_md(big4_data, ref, k3_components, corrs, verdict, per_firm_rows):
+    md = [
+        '# v4.0 Convergence: K=3 + Reverse-Anchor + Paper A',
+        f'Generated: {datetime.now().strftime("%Y-%m-%d %H:%M:%S")}',
+        '',
+        '## A. Three independent lenses on Big-4 CPAs',
+        '',
+        '### 1. K=3 GMM cluster posterior P_C1 (hand-leaning)',
+        '',
+        '| Component | mean cos | mean dh | weight | interpretation |',
+        '|---|---|---|---|---|',
+    ]
+    for i, name in enumerate(['C1 hand-leaning', 'C2 mixed',
+                              'C3 replicated']):
+        m = k3_components['means'][i]
+        w = k3_components['weights'][i]
+        md.append(f'| {name} | {m[0]:.4f} | {m[1]:.4f} | {w:.3f} | '
+                  f'higher P_C1 = more hand-leaning |')
+
+    md += ['',
+           '### 2. Reverse-anchor directional score',
+           '',
+           f'- Reference: non-Big-4 CPAs (n = {ref["n_reference"]}, '
+           f'mid/small firms only -- strict separation from Big-4 target)',
+           f'- Reference center (MCD, support 0.85): cos = '
+           f'{ref["mean"][0]:.4f}, dh = {ref["mean"][1]:.4f}',
+           f'- Score per Big-4 CPA: -cos_left_tail_percentile under the '
+           f'reference marginal cos Gaussian.  Higher = deeper into the '
+           f'left tail = more hand-leaning relative to the reference.',
+           '',
+           '### 3. Paper A v3.x operational rule',
+           '',
+           f'- Per-CPA hand_frac = 1 - (fraction of signatures satisfying '
+           f'cos > {PAPER_A_COS_CUT} AND dh <= {PAPER_A_DH_CUT})',
+           '',
+           '## B. Pairwise Spearman correlations',
+           '',
+           '| Pair | Spearman rho | p | Pearson r | p |',
+           '|---|---|---|---|---|']
+    for name, c in corrs.items():
+        md.append(f'| {name} | **{c["spearman_rho"]:.4f}** | '
+                  f'{c["spearman_p"]:.2e} | {c["pearson_r"]:.4f} | '
+                  f'{c["pearson_p"]:.2e} |')
+
+    md += ['', f'## C. Convergence verdict: **{verdict[0]}**',
+           '', verdict[1], '',
+           '### Verdict legend',
+           f'- **CONVERGENCE_STRONG**: all 3 |rho| >= {RHO_STRONG}.',
+           f'- **CONVERGENCE_PARTIAL**: all 3 |rho| >= {RHO_PARTIAL}.',
+           f'- **CONVERGENCE_WEAK**: at least one |rho| < {RHO_PARTIAL}.',
+           '',
+           '## D. Per-firm summary',
+           '',
+           '| Firm | n CPAs | K=3 C1% | K=3 C3% | mean P_C1 | mean rev-anchor | mean hand_frac |',
+           '|---|---|---|---|---|---|---|']
+    for r in per_firm_rows:
+        md.append(f'| {r["firm_label"]} | {r["n_cpas"]} | '
+                  f'{r["k3_C1_pct"]:.2f}% | {r["k3_C3_pct"]:.2f}% | '
+                  f'{r["mean_p_c1"]:.4f} | {r["mean_reverse_anchor"]:.4f} | '
+                  f'{r["mean_paperA_hand_frac"]:.4f} |')
+
+    md += ['',
+           '## E. Files',
+           '- `scatter_pairwise.png` -- 1x3 scatter of approach pairs',
+           '- `per_firm_summary.csv` -- per-firm aggregates',
+           '- `per_cpa_scores.csv` -- per-CPA all three scores + hard label',
+           '- `convergence_results.json` -- full machine-readable output',
+           '',
+           '## F. Methodology notes',
+           '',
+           '- Reference population for reverse-anchor: non-Big-4 CPAs only '
+           '(n=249), preserving strict train/target separation. This is '
+           'tighter than Script 33 (which used non-Firm-A including other '
+           'Big-4); using a population fully outside Big-4 means the '
+           'reverse-anchor metric carries no within-Big-4 information.',
+           '- K=3 fit on full Big-4 (not LOOO) -- Script 37 already showed '
+           'C1 component shape is stable across LOOO folds; this script '
+           'uses the canonical full-Big-4 fit for per-CPA posteriors.',
+           '- All three approaches operate on the per-CPA mean (cos, dh) -- '
+           'no signature-level scoring here. A signature-level convergence '
+           'check is deferred (it would inflate sample size to ~90k '
+           'without adding methodological signal).',
+           ]
+    return '\n'.join(md)
+
+
+def main():
+    print('=' * 72)
+    print('Script 38: v4.0 Convergence -- K=3 + Reverse-Anchor + Paper A')
+    print('=' * 72)
+    big4 = load_big4()
+    print(f'\nN Big-4 CPAs (n_sigs >= {MIN_SIGS}): {len(big4)}')
+    by_firm_count = {}
+    for d in big4:
+        by_firm_count[d['firm']] = by_firm_count.get(d['firm'], 0) + 1
+    for f in BIG4:
+        print(f'  {LABEL[f]}: {by_firm_count.get(f, 0)}')
+
+    ref_cpas = load_non_big4_reference()
+    print(f'\nN non-Big-4 reference CPAs (n_sigs >= {MIN_SIGS}): '
+          f'{len(ref_cpas)}')
+
+    # Build reference Gaussian
+    ref_points = np.array([[c['cos_mean'], c['dh_mean']] for c in ref_cpas])
+    ref = fit_reference_gaussian(ref_points)
+    print(f'  Reference center (MCD): cos={ref["mean"][0]:.4f}, '
+          f'dh={ref["mean"][1]:.4f}')
+
+    # K=3 fit
+    gmm, order = fit_k3_big4(big4)
+    means_sorted = gmm.means_[order]
+    weights_sorted = gmm.weights_[order]
+    print(f'\nFull-Big-4 K=3 components (sorted by cos):')
+    for i, name in enumerate(['C1 hand-leaning', 'C2 mixed',
+                              'C3 replicated']):
+        print(f'  {name}: cos={means_sorted[i,0]:.4f}, '
+              f'dh={means_sorted[i,1]:.4f}, weight={weights_sorted[i]:.3f}')
+
+    # Score each Big-4 CPA
+    for d in big4:
+        X = np.array([[d['cos_mean'], d['dh_mean']]])
+        raw_post = gmm.predict_proba(X)[0]
+        d['p_c1'] = float(raw_post[order[0]])
+        d['p_c2'] = float(raw_post[order[1]])
+        d['p_c3'] = float(raw_post[order[2]])
+        hard = int(np.argmax(raw_post))
+        d['hard_label'] = ['C1', 'C2', 'C3'][[order[0], order[1],
+                                              order[2]].index(hard)]
+        d['reverse_anchor_score'] = reverse_anchor_directional_score(d, ref)
+        d['paperA_hand_frac'] = d['hand_frac']
+
+    # Correlations
+    corrs = compute_correlations(big4)
+    print('\nPairwise Spearman correlations:')
+    for name, c in corrs.items():
+        print(f'  {name}: rho={c["spearman_rho"]:+.4f} '
+              f'(p={c["spearman_p"]:.2e})')
+
+    # Verdict
+    verdict = classify_convergence(corrs)
+    print(f'\nVerdict: {verdict[0]}')
+    print(f'  {verdict[1]}')
+
+    # Per-firm aggregate
+    per_firm_rows = per_firm_aggregate(big4)
+    print('\nPer-firm summary:')
+    print(f'  {"Firm":<22} {"n":>4} {"C1%":>7} {"C3%":>7} '
+          f'{"E[P_C1]":>9} {"E[rev]":>9} {"E[hand]":>9}')
+    for r in per_firm_rows:
+        print(f'  {r["firm_label"]:<22} {r["n_cpas"]:>4} '
+              f'{r["k3_C1_pct"]:>6.2f}% {r["k3_C3_pct"]:>6.2f}% '
+              f'{r["mean_p_c1"]:>9.4f} {r["mean_reverse_anchor"]:>9.4f} '
+              f'{r["mean_paperA_hand_frac"]:>9.4f}')
+
+    # Plots, CSVs, JSON, MD
+    render_scatter(big4)
+    csv_pf, csv_cpa = write_csv(per_firm_rows, big4)
+    print(f'\nCSV: {csv_pf}; {csv_cpa}')
+
+    payload = {
+        'generated_at': datetime.now().isoformat(),
+        'min_sigs_per_accountant': MIN_SIGS,
+        'paper_a_operational_cuts': {'cos': PAPER_A_COS_CUT,
+                                      'dh': PAPER_A_DH_CUT},
+        'reference_population': {
+            'description': 'non-Big-4 CPAs (mid/small firms only)',
+            'n_cpas': ref['n_reference'],
+            'center_mcd': [float(x) for x in ref['mean']],
+            'cov_mcd': [[float(x) for x in row] for row in ref['cov']],
+        },
+        'k3_components': {
+            'means': means_sorted.tolist(),
+            'weights': weights_sorted.tolist(),
+        },
+        'correlations': corrs,
+        'verdict': {'class': verdict[0], 'explanation': verdict[1]},
+        'per_firm_summary': per_firm_rows,
+        'n_big4_cpas': len(big4),
+    }
+    json_path = OUT / 'convergence_results.json'
+    json_path.write_text(json.dumps(payload, indent=2, ensure_ascii=False),
+                         encoding='utf-8')
+    print(f'JSON: {json_path}')
+
+    md = render_md(big4, ref, {'means': means_sorted.tolist(),
+                               'weights': weights_sorted.tolist()},
+                   corrs, verdict, per_firm_rows)
+    md_path = OUT / 'convergence_report.md'
+    md_path.write_text(md, encoding='utf-8')
+    print(f'Report: {md_path}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/39_v4_signature_level_convergence.py

@@ -0,0 +1,391 @@
+#!/usr/bin/env python3
+"""
+Script 39: Signature-Level Convergence (preempts aggregation attack)
+======================================================================
+Phase 1.7 follow-up to Script 38's per-CPA convergence. Verifies
+that the per-CPA K=3 + reverse-anchor + Paper A agreement holds at
+the signature level (not just per-CPA mean), so a reviewer cannot
+attack with "you washed out within-CPA heterogeneity by averaging".
+
+Three labels per Big-4 signature:
+  L1 PaperA_rule:   non_hand iff cos > 0.95 AND dh <= 5
+  L2 K3_perCPA:     hard assignment under per-CPA K=3 components
+                    fit on accountant means (Script 38 baseline)
+  L3 K3_perSig:     hard assignment under a fresh K=3 fit on the
+                    signature-level (cos, dh) cloud
+
+Output:
+  reports/v4_big4/signature_level_convergence/
+    sig_level_results.json
+    sig_level_report.md
+    crosstab_paperA_vs_k3perCPA.csv
+    crosstab_paperA_vs_k3perSig.csv
+    crosstab_k3perCPA_vs_k3perSig.csv
+
+Headline metrics:
+  - Cohen's kappa for each pairwise label comparison
+  - Per-firm marginal agreement
+  - Component drift between per-CPA K=3 and per-signature K=3
+"""
+
+import sqlite3
+import csv
+import json
+import numpy as np
+import matplotlib
+matplotlib.use('Agg')
+import matplotlib.pyplot as plt
+from pathlib import Path
+from datetime import datetime
+from sklearn.mixture import GaussianMixture
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'v4_big4/signature_level_convergence')
+OUT.mkdir(parents=True, exist_ok=True)
+
+SEED = 42
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+LABEL = {'勤業眾信聯合': 'Firm A (Deloitte)', '安侯建業聯合': 'KPMG',
+         '資誠聯合': 'PwC', '安永聯合': 'EY'}
+PAPER_A_COS_CUT = 0.95
+PAPER_A_DH_CUT = 5
+MIN_SIGS = 10  # for the per-CPA K=3 fit only
+
+
+def load_big4_signatures():
+    conn = sqlite3.connect(DB)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT s.signature_id, s.assigned_accountant, a.firm,
+               s.max_similarity_to_same_accountant,
+               CAST(s.min_dhash_independent AS REAL)
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.min_dhash_independent IS NOT NULL
+          AND a.firm IN (?, ?, ?, ?)
+    ''', BIG4)
+    rows = cur.fetchall()
+    conn.close()
+    return rows
+
+
+def load_per_cpa_means():
+    """Returns (cpa_array, firm_array, X_2d) for the per-CPA fit."""
+    conn = sqlite3.connect(DB)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT s.assigned_accountant, a.firm,
+               AVG(s.max_similarity_to_same_accountant) AS cos_mean,
+               AVG(CAST(s.min_dhash_independent AS REAL)) AS dh_mean,
+               COUNT(*) AS n
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.min_dhash_independent IS NOT NULL
+          AND a.firm IN (?, ?, ?, ?)
+        GROUP BY s.assigned_accountant
+        HAVING n >= ?
+    ''', BIG4 + (MIN_SIGS,))
+    rows = cur.fetchall()
+    conn.close()
+    cpas = [r[0] for r in rows]
+    firms = [r[1] for r in rows]
+    X = np.array([[float(r[2]), float(r[3])] for r in rows])
+    return cpas, firms, X
+
+
+def fit_k3(X, seed=SEED):
+    return GaussianMixture(n_components=3, covariance_type='full',
+                           random_state=seed, n_init=15, max_iter=500).fit(X)
+
+
+def label_paperA(cos, dh):
+    """Returns 0 = non_hand (replicated), 1 = hand_leaning."""
+    return np.where((cos > PAPER_A_COS_CUT) & (dh <= PAPER_A_DH_CUT), 0, 1)
+
+
+def label_k3(gmm, X, order):
+    """Returns hard label in {0=C1, 1=C2, 2=C3} where C1 = lowest cos."""
+    raw = gmm.predict(X)
+    label_map = {old: new for new, old in enumerate(order)}
+    return np.array([label_map[l] for l in raw])
+
+
+def cohen_kappa(y1, y2):
+    """Cohen's kappa for two label arrays."""
+    n = len(y1)
+    if n == 0:
+        return 0.0
+    classes = sorted(set(y1.tolist()) | set(y2.tolist()))
+    k = len(classes)
+    cm = np.zeros((k, k), dtype=float)
+    for a, b in zip(y1, y2):
+        cm[classes.index(int(a)), classes.index(int(b))] += 1
+    p_o = np.sum(np.diag(cm)) / n
+    row_marg = cm.sum(axis=1) / n
+    col_marg = cm.sum(axis=0) / n
+    p_e = float(np.sum(row_marg * col_marg))
+    if p_e == 1.0:
+        return 1.0 if p_o == 1.0 else 0.0
+    return float((p_o - p_e) / (1 - p_e))
+
+
+def crosstab(y1, y2, labels1, labels2):
+    """Cross-tabulation as a dict-of-dicts."""
+    out = {a: {b: 0 for b in labels2} for a in labels1}
+    for a, b in zip(y1, y2):
+        out[labels1[int(a)]][labels2[int(b)]] += 1
+    return out
+
+
+def write_crosstab_csv(ct, name, labels1, labels2):
+    p = OUT / name
+    with open(p, 'w', newline='', encoding='utf-8') as f:
+        w = csv.writer(f)
+        w.writerow([''] + labels2 + ['total'])
+        for a in labels1:
+            row = [a] + [ct[a][b] for b in labels2]
+            row.append(sum(ct[a].values()))
+            w.writerow(row)
+        col_totals = [sum(ct[a][b] for a in labels1) for b in labels2]
+        w.writerow(['total'] + col_totals + [sum(col_totals)])
+    return p
+
+
+def per_firm_agreement(firms_arr, y1, y2):
+    out = {}
+    for f in BIG4:
+        mask = (firms_arr == f)
+        n = int(mask.sum())
+        if n == 0:
+            out[f] = {'n': 0, 'agreement': None}
+            continue
+        agree_count = int(np.sum(y1[mask] == y2[mask]))
+        out[f] = {
+            'n': n,
+            'agree_count': agree_count,
+            'agreement_rate': float(agree_count / n),
+        }
+    return out
+
+
+def main():
+    print('=' * 72)
+    print('Script 39: Signature-Level Convergence')
+    print('=' * 72)
+
+    # 1. Per-CPA K=3 (Script 38 baseline reproduction)
+    cpas, cpa_firms, X_cpa = load_per_cpa_means()
+    print(f'\n[setup] N CPAs (n_sigs >= {MIN_SIGS}): {len(cpas)}')
+    gmm_cpa = fit_k3(X_cpa)
+    order_cpa = np.argsort(gmm_cpa.means_[:, 0])
+    means_cpa = gmm_cpa.means_[order_cpa]
+    weights_cpa = gmm_cpa.weights_[order_cpa]
+    print('  Per-CPA K=3 components (sorted by cos):')
+    for i, name in enumerate(['C1 hand-leaning', 'C2 mixed',
+                              'C3 replicated']):
+        print(f'    {name}: cos={means_cpa[i,0]:.4f}, '
+              f'dh={means_cpa[i,1]:.4f}, weight={weights_cpa[i]:.3f}')
+
+    # 2. Load all Big-4 signatures
+    rows = load_big4_signatures()
+    n_sig = len(rows)
+    sig_ids = np.array([r[0] for r in rows])
+    sig_firms = np.array([r[2] for r in rows])
+    cos = np.array([r[3] for r in rows], dtype=float)
+    dh = np.array([r[4] for r in rows], dtype=float)
+    X_sig = np.column_stack([cos, dh])
+    print(f'\n[setup] N Big-4 signatures: {n_sig:,}')
+
+    # 3. Three labels per signature
+    L1 = label_paperA(cos, dh)
+    L2 = label_k3(gmm_cpa, X_sig, order_cpa)
+    print('\n[fit] Per-signature K=3 (fresh fit on signature cloud)')
+    gmm_sig = fit_k3(X_sig)
+    order_sig = np.argsort(gmm_sig.means_[:, 0])
+    means_sig = gmm_sig.means_[order_sig]
+    weights_sig = gmm_sig.weights_[order_sig]
+    print('  Per-signature K=3 components (sorted by cos):')
+    for i, name in enumerate(['C1 hand-leaning', 'C2 mixed',
+                              'C3 replicated']):
+        print(f'    {name}: cos={means_sig[i,0]:.4f}, '
+              f'dh={means_sig[i,1]:.4f}, weight={weights_sig[i]:.3f}')
+    L3 = label_k3(gmm_sig, X_sig, order_sig)
+
+    # 4. Cross-tabs
+    paperA_labels = ['non_hand', 'hand_leaning']
+    k3_labels = ['C1_handleaning', 'C2_mixed', 'C3_replicated']
+    ct_p_vs_kcpa = crosstab(L1, L2, paperA_labels, k3_labels)
+    ct_p_vs_ksig = crosstab(L1, L3, paperA_labels, k3_labels)
+    ct_kcpa_vs_ksig = crosstab(L2, L3, k3_labels, k3_labels)
+    write_crosstab_csv(ct_p_vs_kcpa, 'crosstab_paperA_vs_k3perCPA.csv',
+                       paperA_labels, k3_labels)
+    write_crosstab_csv(ct_p_vs_ksig, 'crosstab_paperA_vs_k3perSig.csv',
+                       paperA_labels, k3_labels)
+    write_crosstab_csv(ct_kcpa_vs_ksig, 'crosstab_k3perCPA_vs_k3perSig.csv',
+                       k3_labels, k3_labels)
+
+    # 5. Cohen's kappa (collapse K=3 -> binary {C1+C2 = hand-ish, C3 = replicated})
+    L2_bin = (L2 == 2).astype(int)  # 1 = replicated (C3), 0 = otherwise
+    L3_bin = (L3 == 2).astype(int)
+    L1_bin = 1 - L1  # invert so 1 = non_hand (replicated), 0 = hand-leaning
+    print('\n[kappa] Cohen kappa, binary collapse (1 = replicated)')
+    kappa_p_kcpa = cohen_kappa(L1_bin, L2_bin)
+    kappa_p_ksig = cohen_kappa(L1_bin, L3_bin)
+    kappa_kcpa_ksig = cohen_kappa(L2_bin, L3_bin)
+    print(f'  PaperA  vs K=3-perCPA :  kappa = {kappa_p_kcpa:.4f}')
+    print(f'  PaperA  vs K=3-perSig :  kappa = {kappa_p_ksig:.4f}')
+    print(f'  K=3-CPA vs K=3-perSig :  kappa = {kappa_kcpa_ksig:.4f}')
+
+    # 6. Per-firm agreement
+    print('\n[per-firm] Binary agreement (collapsed):')
+    print(f'  {"Firm":<22} {"n_sigs":>9} {"P_vs_K3CPA":>11} '
+          f'{"P_vs_K3sig":>11} {"K3CPA_vs_K3sig":>15}')
+    per_firm_p_kcpa = per_firm_agreement(sig_firms, L1_bin, L2_bin)
+    per_firm_p_ksig = per_firm_agreement(sig_firms, L1_bin, L3_bin)
+    per_firm_kcpa_ksig = per_firm_agreement(sig_firms, L2_bin, L3_bin)
+    for f in BIG4:
+        a1 = per_firm_p_kcpa[f]['agreement_rate']
+        a2 = per_firm_p_ksig[f]['agreement_rate']
+        a3 = per_firm_kcpa_ksig[f]['agreement_rate']
+        print(f'  {LABEL[f]:<22} {per_firm_p_kcpa[f]["n"]:>9,} '
+              f'{a1*100:>10.2f}% {a2*100:>10.2f}% {a3*100:>14.2f}%')
+
+    # 7. Component drift between per-CPA and per-signature K=3
+    print('\n[drift] Per-CPA K=3 vs per-signature K=3 components:')
+    drift = []
+    for i, name in enumerate(['C1 hand-leaning', 'C2 mixed',
+                              'C3 replicated']):
+        d_cos = abs(means_cpa[i, 0] - means_sig[i, 0])
+        d_dh = abs(means_cpa[i, 1] - means_sig[i, 1])
+        d_w = abs(weights_cpa[i] - weights_sig[i])
+        drift.append({'component': name, 'd_cos': float(d_cos),
+                      'd_dh': float(d_dh), 'd_weight': float(d_w)})
+        print(f'  {name}: |dcos|={d_cos:.4f}, |ddh|={d_dh:.3f}, '
+              f'|dweight|={d_w:.3f}')
+
+    # Verdict
+    if (kappa_p_kcpa >= 0.6 and kappa_p_ksig >= 0.6
+            and kappa_kcpa_ksig >= 0.6):
+        verdict = 'SIG_CONVERGENCE_STRONG'
+        msg = ('All three pairwise Cohen kappas >= 0.60 (substantial '
+               'agreement at signature level); per-CPA aggregation does '
+               'not wash out signal.')
+    elif (kappa_p_kcpa >= 0.4 and kappa_p_ksig >= 0.4
+          and kappa_kcpa_ksig >= 0.4):
+        verdict = 'SIG_CONVERGENCE_MODERATE'
+        msg = ('All three pairwise Cohen kappas >= 0.40 (moderate '
+               'agreement); per-CPA aggregation captures most of the '
+               'signature-level structure.')
+    else:
+        verdict = 'SIG_CONVERGENCE_WEAK'
+        msg = ('At least one pairwise Cohen kappa < 0.40; per-CPA '
+               'aggregation hides meaningful signature-level disagreement '
+               'between methods.')
+    print(f'\n[verdict] {verdict}')
+    print(f'  {msg}')
+
+    payload = {
+        'generated_at': datetime.now().isoformat(),
+        'n_signatures_big4': int(n_sig),
+        'n_cpas_for_per_cpa_fit': int(len(cpas)),
+        'paper_a_cuts': {'cos': PAPER_A_COS_CUT, 'dh': PAPER_A_DH_CUT},
+        'per_cpa_k3': {
+            'means': means_cpa.tolist(),
+            'weights': weights_cpa.tolist(),
+        },
+        'per_signature_k3': {
+            'means': means_sig.tolist(),
+            'weights': weights_sig.tolist(),
+        },
+        'component_drift_per_CPA_vs_per_sig': drift,
+        'cohen_kappa_binary_collapse': {
+            'paperA_vs_k3perCPA': float(kappa_p_kcpa),
+            'paperA_vs_k3perSig': float(kappa_p_ksig),
+            'k3perCPA_vs_k3perSig': float(kappa_kcpa_ksig),
+        },
+        'crosstabs': {
+            'paperA_vs_k3perCPA': ct_p_vs_kcpa,
+            'paperA_vs_k3perSig': ct_p_vs_ksig,
+            'k3perCPA_vs_k3perSig': ct_kcpa_vs_ksig,
+        },
+        'per_firm_agreement': {
+            'paperA_vs_k3perCPA': {f: per_firm_p_kcpa[f] for f in BIG4},
+            'paperA_vs_k3perSig': {f: per_firm_p_ksig[f] for f in BIG4},
+            'k3perCPA_vs_k3perSig': {f: per_firm_kcpa_ksig[f] for f in BIG4},
+        },
+        'verdict': {'class': verdict, 'explanation': msg},
+    }
+    json_path = OUT / 'sig_level_results.json'
+    json_path.write_text(json.dumps(payload, indent=2, ensure_ascii=False),
+                         encoding='utf-8')
+    print(f'\nJSON: {json_path}')
+
+    # Markdown report
+    md = [
+        '# Signature-Level Convergence Check (Script 39)',
+        f'Generated: {datetime.now().strftime("%Y-%m-%d %H:%M:%S")}',
+        '',
+        '## Goal',
+        '',
+        ('Verify that the per-CPA convergence found in Script 38 holds at '
+         'signature granularity, so a reviewer cannot attack with '
+         '"per-CPA aggregation washes out heterogeneity."'),
+        '',
+        '## Three signature-level labels',
+        '',
+        '- **PaperA**: non_hand iff cos > 0.95 AND dh <= 5',
+        '- **K=3 perCPA**: hard assignment under K=3 components fit on '
+        f'{len(cpas)} per-CPA means (Script 38 baseline)',
+        '- **K=3 perSig**: hard assignment under K=3 components fit '
+        f'directly on the {n_sig:,} signature-level (cos, dh) cloud',
+        '',
+        '## Component comparison',
+        '',
+        '| Component | Per-CPA cos | Per-CPA dh | Per-CPA wt | Per-Sig cos | Per-Sig dh | Per-Sig wt |',
+        '|---|---|---|---|---|---|---|',
+    ]
+    for i, name in enumerate(['C1 hand-leaning', 'C2 mixed',
+                              'C3 replicated']):
+        md.append(f'| {name} | {means_cpa[i,0]:.4f} | {means_cpa[i,1]:.4f} | '
+                  f'{weights_cpa[i]:.3f} | {means_sig[i,0]:.4f} | '
+                  f'{means_sig[i,1]:.4f} | {weights_sig[i]:.3f} |')
+    md += ['', '## Cohen kappa (binary: 1 = replicated, 0 = hand-leaning)',
+           '',
+           '| Pair | kappa |',
+           '|---|---|',
+           f'| PaperA vs K=3 perCPA | **{kappa_p_kcpa:.4f}** |',
+           f'| PaperA vs K=3 perSig | **{kappa_p_ksig:.4f}** |',
+           f'| K=3 perCPA vs K=3 perSig | **{kappa_kcpa_ksig:.4f}** |',
+           '',
+           ('Reference: kappa <= 0 = no agreement, 0.0-0.2 slight, '
+            '0.2-0.4 fair, 0.4-0.6 moderate, 0.6-0.8 substantial, '
+            '0.8-1.0 almost perfect (Landis & Koch 1977).'),
+           '',
+           '## Per-firm binary agreement', '',
+           '| Firm | n_sigs | PaperA vs K3-perCPA | PaperA vs K3-perSig | K3-CPA vs K3-Sig |',
+           '|---|---|---|---|---|',
+           ]
+    for f in BIG4:
+        md.append(f'| {LABEL[f]} | {per_firm_p_kcpa[f]["n"]:,} | '
+                  f'{per_firm_p_kcpa[f]["agreement_rate"]*100:.2f}% | '
+                  f'{per_firm_p_ksig[f]["agreement_rate"]*100:.2f}% | '
+                  f'{per_firm_kcpa_ksig[f]["agreement_rate"]*100:.2f}% |')
+    md += ['', f'## Verdict: **{verdict}**',
+           '', msg, '',
+           '### Verdict legend',
+           '- SIG_CONVERGENCE_STRONG: all 3 kappas >= 0.60 (substantial)',
+           '- SIG_CONVERGENCE_MODERATE: all 3 kappas >= 0.40 (moderate)',
+           '- SIG_CONVERGENCE_WEAK: at least one kappa < 0.40',
+           ]
+    md_path = OUT / 'sig_level_report.md'
+    md_path.write_text('\n'.join(md), encoding='utf-8')
+    print(f'Report: {md_path}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/39b_v4_signature_level_diptest.py

@@ -0,0 +1,195 @@
+#!/usr/bin/env python3
+"""
+Script 39b: Signature-Level Dip Test (multimodality at the signature cloud)
+============================================================================
+Phase 5 pre-emptive evidence. Script 34 / 36 already report Hartigan
+dip tests on the 437 accountant-level (cos_mean, dh_mean) means and
+both marginals reject unimodality at p < 5e-4. Reviewers may ask
+whether the same multimodality is detectable at the signature level
+itself (n = 150,442 Big-4 signatures) and whether the multimodality
+is a within-firm or only a between-firm phenomenon.
+
+This script supplies the missing dip evidence on the raw signature
+cloud. It is a *diagnostic* in the same role as Scripts 34/36 dip
+tests: it does not derive an operational threshold; it characterises
+the marginal distributions of (cos, dh_indep) at the signature level.
+
+Outputs:
+  reports/v4_big4/signature_level_diptest/
+    sig_diptest_results.json
+    sig_diptest_report.md
+
+Tests performed:
+  A. Pooled Big-4 marginals (cos, dh_indep), n = 150,442
+  B. Per-firm marginals (Firm A / B / C / D separately)
+"""
+
+import json
+import sqlite3
+import numpy as np
+import diptest
+from pathlib import Path
+from datetime import datetime
+from scipy import stats
+from scipy.signal import find_peaks
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'v4_big4/signature_level_diptest')
+OUT.mkdir(parents=True, exist_ok=True)
+
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+ALIAS = {'勤業眾信聯合': 'Firm A',
+         '安侯建業聯合': 'Firm B',
+         '資誠聯合': 'Firm C',
+         '安永聯合': 'Firm D'}
+N_BOOT = 2000
+
+
+def load_big4_signatures():
+    conn = sqlite3.connect(DB)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT s.assigned_accountant, a.firm,
+               s.max_similarity_to_same_accountant,
+               CAST(s.min_dhash_independent AS REAL)
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.min_dhash_independent IS NOT NULL
+          AND a.firm IN (?, ?, ?, ?)
+    ''', BIG4)
+    rows = cur.fetchall()
+    conn.close()
+    return rows
+
+
+def kde_dip(values, n_boot=N_BOOT):
+    arr = np.asarray(values, dtype=float)
+    arr = arr[np.isfinite(arr)]
+    dip, pval = diptest.diptest(arr, boot_pval=True, n_boot=n_boot)
+    kde = stats.gaussian_kde(arr, bw_method='silverman')
+    xs = np.linspace(arr.min(), arr.max(), 2000)
+    density = kde(xs)
+    peaks, _ = find_peaks(density, prominence=density.max() * 0.02)
+    antimodes = []
+    for i in range(len(peaks) - 1):
+        seg = density[peaks[i]:peaks[i + 1]]
+        if not len(seg):
+            continue
+        local = peaks[i] + int(np.argmin(seg))
+        antimodes.append(float(xs[local]))
+    return {
+        'n': int(len(arr)),
+        'dip': float(dip),
+        'dip_pvalue': float(pval),
+        'unimodal_alpha05': bool(pval > 0.05),
+        'n_modes': int(len(peaks)),
+        'mode_locations': [float(xs[p]) for p in peaks],
+        'antimodes': antimodes,
+        'n_boot': int(n_boot),
+    }
+
+
+def _fmt_p(p):
+    if p == 0.0:
+        return '< 5e-4 (no bootstrap replicate exceeded observed dip)'
+    return f'{p:.4g}'
+
+
+def main():
+    print('=' * 72)
+    print('Script 39b: Signature-Level Dip Test')
+    print('=' * 72)
+    rows = load_big4_signatures()
+    cos_all = np.array([r[2] for r in rows], dtype=float)
+    dh_all = np.array([r[3] for r in rows], dtype=float)
+    firms = np.array([ALIAS[r[1]] for r in rows])
+    print(f'\nLoaded {len(rows):,} Big-4 signatures')
+    for f in sorted(set(firms)):
+        print(f'  {f}: {(firms == f).sum():,}')
+
+    results = {
+        'meta': {
+            'script': '39b',
+            'timestamp': datetime.now().isoformat(timespec='seconds'),
+            'n_total': int(len(rows)),
+            'n_boot': N_BOOT,
+            'note': ('Signature-level Hartigan dip test on Big-4 '
+                     '(cos, dh_indep) marginals; pooled and per-firm.'),
+        },
+        'pooled': {},
+        'per_firm': {},
+    }
+
+    # A. Pooled
+    print('\n[A] Pooled Big-4')
+    for desc, arr in [('cos', cos_all), ('dh_indep', dh_all)]:
+        r = kde_dip(arr)
+        results['pooled'][desc] = r
+        print(f'  {desc}: n={r["n"]:,}, dip={r["dip"]:.5f}, '
+              f'p={_fmt_p(r["dip_pvalue"])}, n_modes={r["n_modes"]}')
+
+    # B. Per-firm
+    print('\n[B] Per-firm')
+    for f in sorted(set(firms)):
+        mask = firms == f
+        results['per_firm'][f] = {}
+        for desc, arr in [('cos', cos_all[mask]), ('dh_indep', dh_all[mask])]:
+            r = kde_dip(arr)
+            results['per_firm'][f][desc] = r
+            print(f'  {f} {desc}: n={r["n"]:,}, dip={r["dip"]:.5f}, '
+                  f'p={_fmt_p(r["dip_pvalue"])}, n_modes={r["n_modes"]}')
+
+    json_path = OUT / 'sig_diptest_results.json'
+    json_path.write_text(json.dumps(results, indent=2, ensure_ascii=False),
+                         encoding='utf-8')
+    print(f'\n[json] {json_path}')
+
+    md = ['# Signature-Level Dip Test (Script 39b)',
+          '',
+          f'Generated: {results["meta"]["timestamp"]}',
+          f'Bootstrap replicates: {N_BOOT}',
+          '',
+          '## A. Pooled Big-4 signature cloud',
+          '',
+          f'n = {results["meta"]["n_total"]:,} signatures',
+          '',
+          '| Marginal | dip | p (boot) | n_modes | unimodal @0.05 |',
+          '|---|---|---|---|---|']
+    for desc in ['cos', 'dh_indep']:
+        r = results['pooled'][desc]
+        md.append(f'| {desc} | {r["dip"]:.5f} | {_fmt_p(r["dip_pvalue"])} | '
+                  f'{r["n_modes"]} | {r["unimodal_alpha05"]} |')
+
+    md += ['', '## B. Per-firm signature-level dip tests', '',
+           '| Firm | Marginal | n | dip | p (boot) | n_modes | unimodal @0.05 |',
+           '|---|---|---|---|---|---|---|']
+    for f in sorted(results['per_firm']):
+        for desc in ['cos', 'dh_indep']:
+            r = results['per_firm'][f][desc]
+            md.append(f'| {f} | {desc} | {r["n"]:,} | {r["dip"]:.5f} | '
+                      f'{_fmt_p(r["dip_pvalue"])} | {r["n_modes"]} | '
+                      f'{r["unimodal_alpha05"]} |')
+    md += ['',
+           '## Reading guide',
+           '',
+           ('A unimodality rejection at the signature level confirms '
+            'multimodal structure independent of accountant-level '
+            'aggregation. A within-firm rejection further indicates the '
+            'multimodality is not solely a between-firm artefact. A '
+            'within-firm non-rejection (e.g., Firm A) is consistent with '
+            'that firm being concentrated in a single mechanism corner.'),
+           '',
+           ('All thresholds and operational classifiers remain those of '
+            'v3.x §III-K and v4.0 §III-J; this script supplies diagnostic '
+            'evidence only.'),
+           '']
+    md_path = OUT / 'sig_diptest_report.md'
+    md_path.write_text('\n'.join(md), encoding='utf-8')
+    print(f'[md  ] {md_path}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/39c_v4_midsmall_signature_diptest.py

@@ -0,0 +1,214 @@
+#!/usr/bin/env python3
+"""
+Script 39c: Mid/Small-Firm Signature-Level Dip Test
+====================================================
+Companion to Script 39b. 39b showed every Big-4 firm rejects
+unimodality on the dHash signature marginal (p < 5e-4 in each
+of A/B/C/D) while every Big-4 firm fails to reject unimodality
+on the cosine marginal. This script asks the same questions of
+the mid/small-firm population (non-Big-4):
+
+  1. Does the pooled mid/small-firm signature cloud show the same
+     dHash multimodality?
+  2. Within individual mid/small firms (those with enough
+     signatures to support the test), does the dHash multimodality
+     hold firm-internally as it does in Big-4?
+
+If yes, the dHash signature-level multimodality is corpus-universal
+and the Big-4 scope restriction of v4.0 is not necessary on dHash
+grounds (cf §III-G item 2 which currently rests on Big-4-level
+multimodality). The cosine axis is reported alongside for
+completeness, but no v4.0 claim turns on cosine multimodality
+outside Big-4.
+
+Outputs:
+  reports/v4_big4/midsmall_signature_diptest/
+    midsmall_diptest_results.json
+    midsmall_diptest_report.md
+"""
+
+import json
+import sqlite3
+import numpy as np
+import diptest
+from pathlib import Path
+from datetime import datetime
+from scipy import stats
+from scipy.signal import find_peaks
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'v4_big4/midsmall_signature_diptest')
+OUT.mkdir(parents=True, exist_ok=True)
+
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+N_BOOT = 2000
+SINGLE_FIRM_MIN_SIG = 500  # minimum signature count to run a per-firm dip test
+
+
+def load_non_big4_signatures():
+    conn = sqlite3.connect(DB)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT a.firm,
+               s.max_similarity_to_same_accountant,
+               CAST(s.min_dhash_independent AS REAL)
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.min_dhash_independent IS NOT NULL
+          AND a.firm IS NOT NULL
+          AND a.firm NOT IN (?, ?, ?, ?)
+    ''', BIG4)
+    rows = cur.fetchall()
+    conn.close()
+    return rows
+
+
+def kde_dip(values, n_boot=N_BOOT):
+    arr = np.asarray(values, dtype=float)
+    arr = arr[np.isfinite(arr)]
+    if len(arr) < 10:
+        return {'n': int(len(arr)), 'skipped': 'too few points'}
+    dip, pval = diptest.diptest(arr, boot_pval=True, n_boot=n_boot)
+    kde = stats.gaussian_kde(arr, bw_method='silverman')
+    xs = np.linspace(arr.min(), arr.max(), 2000)
+    density = kde(xs)
+    peaks, _ = find_peaks(density, prominence=density.max() * 0.02)
+    antimodes = []
+    for i in range(len(peaks) - 1):
+        seg = density[peaks[i]:peaks[i + 1]]
+        if not len(seg):
+            continue
+        local = peaks[i] + int(np.argmin(seg))
+        antimodes.append(float(xs[local]))
+    return {
+        'n': int(len(arr)),
+        'dip': float(dip),
+        'dip_pvalue': float(pval),
+        'unimodal_alpha05': bool(pval > 0.05),
+        'n_modes': int(len(peaks)),
+        'mode_locations': [float(xs[p]) for p in peaks],
+        'antimodes': antimodes,
+        'n_boot': int(n_boot),
+    }
+
+
+def _fmt_p(p):
+    if p == 0.0:
+        return '< 5e-4'
+    return f'{p:.4g}'
+
+
+def main():
+    print('=' * 72)
+    print('Script 39c: Mid/Small-Firm Signature-Level Dip Test')
+    print('=' * 72)
+    rows = load_non_big4_signatures()
+    cos_all = np.array([r[1] for r in rows], dtype=float)
+    dh_all = np.array([r[2] for r in rows], dtype=float)
+    firms = np.array([r[0] for r in rows])
+    n_total = len(rows)
+    print(f'\nLoaded {n_total:,} non-Big-4 signatures across '
+          f'{len(set(firms))} firms')
+
+    # Firm size table
+    firm_counts = {}
+    for f in firms:
+        firm_counts[f] = firm_counts.get(f, 0) + 1
+    top = sorted(firm_counts.items(), key=lambda x: -x[1])
+    print('\nTop firms by signature count:')
+    for f, n in top[:10]:
+        print(f'  {f}: {n:,}')
+
+    results = {
+        'meta': {
+            'script': '39c',
+            'timestamp': datetime.now().isoformat(timespec='seconds'),
+            'n_total': int(n_total),
+            'n_firms': int(len(firm_counts)),
+            'n_boot': N_BOOT,
+            'single_firm_min_sig': SINGLE_FIRM_MIN_SIG,
+        },
+        'pooled': {},
+        'per_firm_eligible': {},
+        'firm_counts': dict(firm_counts),
+    }
+
+    # A. Pooled non-Big-4
+    print('\n[A] Pooled non-Big-4')
+    for desc, arr in [('cos', cos_all), ('dh_indep', dh_all)]:
+        r = kde_dip(arr)
+        results['pooled'][desc] = r
+        print(f'  {desc}: n={r["n"]:,}, dip={r["dip"]:.5f}, '
+              f'p={_fmt_p(r["dip_pvalue"])}, n_modes={r["n_modes"]}')
+
+    # B. Per-firm (only firms with >= SINGLE_FIRM_MIN_SIG signatures)
+    eligible = [f for f, n in firm_counts.items() if n >= SINGLE_FIRM_MIN_SIG]
+    print(f'\n[B] Per-firm dip test '
+          f'(firms with >= {SINGLE_FIRM_MIN_SIG} signatures: {len(eligible)})')
+    for f in sorted(eligible, key=lambda x: -firm_counts[x]):
+        mask = firms == f
+        results['per_firm_eligible'][f] = {'n': int(mask.sum())}
+        for desc, arr in [('cos', cos_all[mask]), ('dh_indep', dh_all[mask])]:
+            r = kde_dip(arr)
+            results['per_firm_eligible'][f][desc] = r
+            print(f'  {f[:20]:<22s} {desc}: n={r["n"]:,}, dip={r["dip"]:.5f}, '
+                  f'p={_fmt_p(r["dip_pvalue"])}, n_modes={r["n_modes"]}')
+
+    json_path = OUT / 'midsmall_diptest_results.json'
+    json_path.write_text(json.dumps(results, indent=2, ensure_ascii=False),
+                         encoding='utf-8')
+    print(f'\n[json] {json_path}')
+
+    md = ['# Mid/Small-Firm Signature-Level Dip Test (Script 39c)',
+          '',
+          f'Generated: {results["meta"]["timestamp"]}',
+          f'Bootstrap replicates: {N_BOOT}',
+          '',
+          '## A. Pooled non-Big-4 signature cloud',
+          '',
+          f'n = {n_total:,} signatures across '
+          f'{results["meta"]["n_firms"]} firms',
+          '',
+          '| Marginal | dip | p (boot) | n_modes | unimodal @0.05 |',
+          '|---|---|---|---|---|']
+    for desc in ['cos', 'dh_indep']:
+        r = results['pooled'][desc]
+        md.append(f'| {desc} | {r["dip"]:.5f} | {_fmt_p(r["dip_pvalue"])} | '
+                  f'{r["n_modes"]} | {r["unimodal_alpha05"]} |')
+
+    md += ['', f'## B. Single mid/small firms (>= {SINGLE_FIRM_MIN_SIG} '
+              f'signatures), {len(eligible)} qualify', '',
+           '| Firm | Marginal | n | dip | p (boot) | n_modes | unimodal @0.05 |',
+           '|---|---|---|---|---|---|---|']
+    for f in sorted(eligible, key=lambda x: -firm_counts[x]):
+        for desc in ['cos', 'dh_indep']:
+            r = results['per_firm_eligible'][f][desc]
+            md.append(f'| {f[:20]} | {desc} | {r["n"]:,} | {r["dip"]:.5f} | '
+                      f'{_fmt_p(r["dip_pvalue"])} | {r["n_modes"]} | '
+                      f'{r["unimodal_alpha05"]} |')
+
+    md += ['',
+           '## Reading guide',
+           '',
+           ('If the pooled-non-Big-4 dHash marginal rejects unimodality '
+            'AND the qualifying individual mid/small firms also reject, '
+            'the dHash within-firm replication regime structure is '
+            'corpus-universal and not Big-4-specific. In that case the '
+            'Big-4 scope of v4.0 is justified on cosine-axis grounds '
+            '(Firm-A composition; §III-G item 1) and accountant-level '
+            'LOOO reproducibility (§III-G item 3), but not on dHash '
+            'multimodality grounds (§III-G item 2 should be re-scoped or '
+            'qualified). If the per-firm dHash tests instead fail to '
+            'reject inside mid/small firms, the dHash multimodality is '
+            'Big-4-specific and §III-G item 2 holds as stated.'),
+           '']
+    md_path = OUT / 'midsmall_diptest_report.md'
+    md_path.write_text('\n'.join(md), encoding='utf-8')
+    print(f'[md  ] {md_path}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/39d_dhash_discrete_robustness.py

@@ -0,0 +1,446 @@
+#!/usr/bin/env python3
+"""
+Script 39d: dHash Discrete-Value Robustness Diagnostics
+========================================================
+Codex (gpt-5.5 xhigh) attack on Script 39b/39c findings revealed that
+the within-firm dHash dip-test rejections are driven by integer mass
+points (dHash takes integer values 0..64). A uniform jitter of
+[-0.5, +0.5] eliminates dip rejection in every firm tested. This
+script consolidates that finding into a permanent diagnostic and adds:
+
+  1. Raw vs jittered dip with multi-seed robustness (5 seeds)
+  2. Integer-histogram valley analysis: locate local minima between
+     adjacent peaks in the binned integer distribution; report whether
+     any valley centers near dh = 5
+  3. Firm-residualized dip on dHash (analog of cosine firm-mean
+     centering that confirmed the cosine reframe)
+  4. Pairwise pair-coincidence: does the same same-CPA pair achieve
+     both max cosine and min dHash, or are the two descriptors
+     attached to different pairs? Foundation for "is (cos, dh) a
+     joint signature regime descriptor or two parallel descriptors"
+
+This script does not derive operational thresholds; it characterises
+whether the v4.0 K=3 mixture and v3.x cos>0.95 AND dh<=5 rule are
+robustly supported once integer-discreteness artifacts are removed.
+
+Outputs:
+  reports/v4_big4/dhash_discrete_robustness/
+    dhash_discrete_results.json
+    dhash_discrete_report.md
+"""
+
+import json
+import sqlite3
+import numpy as np
+import diptest
+from pathlib import Path
+from datetime import datetime
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'v4_big4/dhash_discrete_robustness')
+OUT.mkdir(parents=True, exist_ok=True)
+
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+ALIAS = {'勤業眾信聯合': 'Firm A',
+         '安侯建業聯合': 'Firm B',
+         '資誠聯合': 'Firm C',
+         '安永聯合': 'Firm D'}
+N_BOOT = 2000
+JITTER_SEEDS = [42, 43, 44, 45, 46]
+SINGLE_FIRM_MIN_SIG = 500
+
+
+def load_signatures():
+    conn = sqlite3.connect(f'file:{DB}?mode=ro', uri=True)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT a.firm, s.assigned_accountant,
+               s.max_similarity_to_same_accountant,
+               CAST(s.min_dhash_independent AS REAL)
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.min_dhash_independent IS NOT NULL
+          AND a.firm IS NOT NULL
+    ''')
+    rows = cur.fetchall()
+    conn.close()
+    return rows
+
+
+def dip(values, n_boot=N_BOOT):
+    arr = np.asarray(values, dtype=float)
+    arr = arr[np.isfinite(arr)]
+    d, p = diptest.diptest(arr, boot_pval=True, n_boot=n_boot)
+    return float(d), float(p)
+
+
+def multi_seed_jitter_dip(values, seeds=JITTER_SEEDS, n_boot=N_BOOT):
+    """Compute dip stat + p-value across seeds; return distribution."""
+    arr = np.asarray(values, dtype=float)
+    arr = arr[np.isfinite(arr)]
+    stats = []
+    for seed in seeds:
+        rng = np.random.default_rng(seed)
+        j = arr + rng.uniform(-0.5, 0.5, len(arr))
+        d, p = diptest.diptest(j, boot_pval=True, n_boot=n_boot)
+        stats.append({'seed': seed, 'dip': float(d), 'p': float(p)})
+    return {
+        'n_seeds': len(seeds),
+        'p_min': min(s['p'] for s in stats),
+        'p_max': max(s['p'] for s in stats),
+        'p_median': float(np.median([s['p'] for s in stats])),
+        'dip_min': min(s['dip'] for s in stats),
+        'dip_max': max(s['dip'] for s in stats),
+        'reject_at_05_count': int(sum(1 for s in stats if s['p'] <= 0.05)),
+        'per_seed': stats,
+    }
+
+
+def integer_histogram_valleys(values, max_bin=20):
+    """For integer-valued data, locate local minima in the count
+    histogram on bins 0..max_bin. Returns valley positions and depths
+    relative to flanking peaks."""
+    arr = np.asarray(values, dtype=float)
+    arr = arr[np.isfinite(arr)]
+    bins = np.arange(0, max_bin + 2)  # 0, 1, ..., max_bin+1
+    counts, edges = np.histogram(arr, bins=bins)
+    centers = (edges[:-1] + edges[1:]) / 2.0
+    valleys = []
+    for i in range(1, len(counts) - 1):
+        if counts[i] < counts[i - 1] and counts[i] < counts[i + 1]:
+            left_peak = counts[i - 1]
+            right_peak = counts[i + 1]
+            min_peak = min(left_peak, right_peak)
+            depth_rel = (min_peak - counts[i]) / min_peak if min_peak else 0
+            valleys.append({
+                'bin_center': float(centers[i]),
+                'count': int(counts[i]),
+                'left_peak_bin': int(centers[i - 1]),
+                'left_peak_count': int(left_peak),
+                'right_peak_bin': int(centers[i + 1]),
+                'right_peak_count': int(right_peak),
+                'depth_rel': float(depth_rel),
+            })
+    return {
+        'histogram_bins_0_to_max': counts[:max_bin + 1].tolist(),
+        'valleys': valleys,
+        'note': ('valleys are bins where count < both neighbours; '
+                 'depth_rel = (min(neighbour) - bin) / min(neighbour). '
+                 'A genuine antimode would have a deep, stable valley '
+                 'with depth_rel > 0.1.'),
+    }
+
+
+def firm_residualized(values, firm_labels):
+    """Return values with firm means subtracted (centered to grand mean
+    over firms). Used to test whether residual within-firm structure
+    rejects unimodality."""
+    arr = np.asarray(values, dtype=float)
+    firms = np.asarray(firm_labels)
+    out = arr.copy()
+    grand = float(np.mean(arr))
+    for f in np.unique(firms):
+        m = firms == f
+        out[m] = arr[m] - float(np.mean(arr[m])) + grand
+    return out
+
+
+def pair_coincidence_rate():
+    """Fraction of signatures whose max-cosine partner equals the
+    min-dHash partner within the same-CPA cross-year pool."""
+    conn = sqlite3.connect(f'file:{DB}?mode=ro', uri=True)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT COUNT(*) AS n_total,
+               SUM(CASE WHEN max_cosine_pair_id IS NOT NULL
+                         AND min_dhash_pair_id IS NOT NULL
+                         AND max_cosine_pair_id = min_dhash_pair_id
+                        THEN 1 ELSE 0 END) AS n_same_pair,
+               SUM(CASE WHEN max_cosine_pair_id IS NOT NULL
+                         AND min_dhash_pair_id IS NOT NULL
+                         AND max_cosine_pair_id != min_dhash_pair_id
+                        THEN 1 ELSE 0 END) AS n_diff_pair,
+               SUM(CASE WHEN max_cosine_pair_id IS NULL
+                          OR min_dhash_pair_id IS NULL
+                        THEN 1 ELSE 0 END) AS n_null
+        FROM signatures
+    ''')
+    row = cur.fetchone()
+    conn.close()
+    n_total, n_same, n_diff, n_null = row
+    n_with_both = (n_same or 0) + (n_diff or 0)
+    return {
+        'n_total': int(n_total or 0),
+        'n_with_both_pair_ids': int(n_with_both),
+        'n_same_pair': int(n_same or 0),
+        'n_diff_pair': int(n_diff or 0),
+        'n_null': int(n_null or 0),
+        'same_pair_rate': (float(n_same) / n_with_both
+                           if n_with_both else None),
+        'note': ('rate computed over signatures where both '
+                 'max_cosine_pair_id and min_dhash_pair_id are present'),
+    }
+
+
+def _fmt_p(p):
+    return '< 5e-4' if p == 0.0 else f'{p:.4g}'
+
+
+def main():
+    print('=' * 72)
+    print('Script 39d: dHash Discrete-Value Robustness Diagnostics')
+    print('=' * 72)
+    rows = load_signatures()
+    firms_raw = np.array([r[0] for r in rows])
+    cos = np.array([r[2] for r in rows], dtype=float)
+    dh = np.array([r[3] for r in rows], dtype=float)
+    is_big4 = np.isin(firms_raw, BIG4)
+    n = len(rows)
+    print(f'\nLoaded {n:,} signatures; Big-4 {is_big4.sum():,}, '
+          f'non-Big-4 {(~is_big4).sum():,}')
+
+    results = {
+        'meta': {
+            'script': '39d',
+            'timestamp': datetime.now().isoformat(timespec='seconds'),
+            'n_total_signatures': int(n),
+            'n_big4': int(is_big4.sum()),
+            'n_non_big4': int((~is_big4).sum()),
+            'n_boot': N_BOOT,
+            'jitter_seeds': JITTER_SEEDS,
+            'note': ('Diagnostic for dHash integer-mass-point artifact '
+                     'in dip test; codex round-29 attack on Script 39b/c'),
+        },
+    }
+
+    # ---- A. Raw vs multi-seed jittered dip ----
+    print('\n[A] Raw vs jittered dip (5 seeds, n_boot=2000)')
+    panels = {}
+    # Big-4 pooled
+    print('  Big-4 pooled:')
+    raw_d, raw_p = dip(dh[is_big4])
+    j = multi_seed_jitter_dip(dh[is_big4])
+    panels['big4_pooled'] = {
+        'n': int(is_big4.sum()),
+        'raw': {'dip': raw_d, 'p': raw_p},
+        'jittered': j,
+    }
+    print(f'    raw:    dip={raw_d:.5f}, p={_fmt_p(raw_p)}')
+    print(f'    jitter: p_median={j["p_median"]:.4g}, '
+          f'p_range=[{j["p_min"]:.4g}, {j["p_max"]:.4g}], '
+          f'reject@.05 in {j["reject_at_05_count"]}/5 seeds')
+    # Each Big-4 firm
+    for f in BIG4:
+        mask = firms_raw == f
+        if mask.sum() == 0:
+            continue
+        raw_d, raw_p = dip(dh[mask])
+        j = multi_seed_jitter_dip(dh[mask])
+        panels[ALIAS[f]] = {
+            'n': int(mask.sum()),
+            'raw': {'dip': raw_d, 'p': raw_p},
+            'jittered': j,
+        }
+        print(f'  {ALIAS[f]} (n={mask.sum():,}):')
+        print(f'    raw:    dip={raw_d:.5f}, p={_fmt_p(raw_p)}')
+        print(f'    jitter: p_median={j["p_median"]:.4g}, '
+              f'reject@.05 in {j["reject_at_05_count"]}/5 seeds')
+    # Non-Big-4 pooled
+    print('  Non-Big-4 pooled:')
+    raw_d, raw_p = dip(dh[~is_big4])
+    j = multi_seed_jitter_dip(dh[~is_big4])
+    panels['non_big4_pooled'] = {
+        'n': int((~is_big4).sum()),
+        'raw': {'dip': raw_d, 'p': raw_p},
+        'jittered': j,
+    }
+    print(f'    raw:    dip={raw_d:.5f}, p={_fmt_p(raw_p)}')
+    print(f'    jitter: p_median={j["p_median"]:.4g}, '
+          f'reject@.05 in {j["reject_at_05_count"]}/5 seeds')
+    results['raw_vs_jittered_dip'] = panels
+
+    # ---- B. Integer-histogram valley analysis ----
+    print('\n[B] Integer-histogram valley analysis (bins 0..20)')
+    valleys = {}
+    valleys['big4_pooled'] = integer_histogram_valleys(dh[is_big4])
+    print(f'  Big-4 pooled: {len(valleys["big4_pooled"]["valleys"])} valleys')
+    for v in valleys['big4_pooled']['valleys']:
+        print(f'    bin {v["bin_center"]:.1f}: count={v["count"]}, '
+              f'depth_rel={v["depth_rel"]:.3f}')
+    for f in BIG4:
+        mask = firms_raw == f
+        if mask.sum() == 0:
+            continue
+        valleys[ALIAS[f]] = integer_histogram_valleys(dh[mask])
+        print(f'  {ALIAS[f]}: '
+              f'{len(valleys[ALIAS[f]]["valleys"])} valleys')
+        for v in valleys[ALIAS[f]]['valleys']:
+            print(f'    bin {v["bin_center"]:.1f}: count={v["count"]}, '
+                  f'depth_rel={v["depth_rel"]:.3f}')
+    valleys['non_big4_pooled'] = integer_histogram_valleys(dh[~is_big4])
+    print(f'  Non-Big-4 pooled: '
+          f'{len(valleys["non_big4_pooled"]["valleys"])} valleys')
+    for v in valleys['non_big4_pooled']['valleys']:
+        print(f'    bin {v["bin_center"]:.1f}: count={v["count"]}, '
+              f'depth_rel={v["depth_rel"]:.3f}')
+    results['integer_histogram_valleys'] = valleys
+
+    # ---- C. Firm-residualized dip on dHash, signature level ----
+    print('\n[C] Firm-residualized dHash dip (signature level)')
+    firm_labels = np.array([
+        ALIAS[f] if f in ALIAS else f'M:{f}'
+        for f in firms_raw
+    ])
+    # Big-4 only residualized over A/B/C/D
+    dh_resid_big4 = firm_residualized(dh[is_big4], firm_labels[is_big4])
+    raw_d, raw_p = dip(dh[is_big4])
+    res_d, res_p = dip(dh_resid_big4)
+    print(f'  Big-4 raw:        dip={raw_d:.5f}, p={_fmt_p(raw_p)}')
+    print(f'  Big-4 residualized: dip={res_d:.5f}, p={_fmt_p(res_p)}')
+    # Also non-Big-4 residualized over their firms
+    dh_resid_nbig4 = firm_residualized(dh[~is_big4], firm_labels[~is_big4])
+    raw_d_n, raw_p_n = dip(dh[~is_big4])
+    res_d_n, res_p_n = dip(dh_resid_nbig4)
+    print(f'  Non-Big-4 raw:        dip={raw_d_n:.5f}, p={_fmt_p(raw_p_n)}')
+    print(f'  Non-Big-4 residualized: dip={res_d_n:.5f}, p={_fmt_p(res_p_n)}')
+    results['firm_residualized_dh_dip'] = {
+        'big4': {
+            'raw': {'dip': raw_d, 'p': raw_p},
+            'firm_residualized': {'dip': res_d, 'p': res_p},
+        },
+        'non_big4': {
+            'raw': {'dip': raw_d_n, 'p': raw_p_n},
+            'firm_residualized': {'dip': res_d_n, 'p': res_p_n},
+        },
+        'note': ('Residualization subtracts each firm mean dh and adds '
+                 'back the grand mean. If residual dip rejects, there is '
+                 'genuine within-firm dh multimodality independent of '
+                 'between-firm mean shifts. If residual fails to reject, '
+                 'all dh "multimodality" was between-firm composition.'),
+    }
+
+    # ---- D. Pair-coincidence rate ----
+    print('\n[D] Pair-coincidence rate (max-cos pair vs min-dh pair)')
+    try:
+        pc = pair_coincidence_rate()
+        if pc['same_pair_rate'] is not None:
+            print(f'  n_with_both: {pc["n_with_both_pair_ids"]:,}, '
+                  f'same-pair rate: {pc["same_pair_rate"]:.4f}')
+        else:
+            print('  Pair IDs not stored in signatures table (skipped)')
+        results['pair_coincidence'] = pc
+    except sqlite3.OperationalError as e:
+        print(f'  SQL error (pair_id columns may not exist): {e}')
+        results['pair_coincidence'] = {
+            'error': str(e),
+            'note': ('signatures table lacks max_cosine_pair_id / '
+                     'min_dhash_pair_id columns; analysis skipped'),
+        }
+
+    json_path = OUT / 'dhash_discrete_results.json'
+    json_path.write_text(json.dumps(results, indent=2, ensure_ascii=False),
+                         encoding='utf-8')
+    print(f'\n[json] {json_path}')
+
+    # ---- Report markdown ----
+    md = ['# dHash Discrete-Value Robustness Diagnostics (Script 39d)',
+          '', f'Generated: {results["meta"]["timestamp"]}',
+          f'Bootstrap replicates: {N_BOOT}; jitter seeds: {JITTER_SEEDS}',
+          '',
+          '## A. Raw vs jittered dHash dip (signature level)',
+          '',
+          ('dHash is integer-valued in [0, 64]. A raw dip test on '
+           'integer mass points may reject unimodality due to discrete '
+           'spikes rather than a continuous bimodal density. We add '
+           'uniform jitter in [-0.5, +0.5] over 5 seeds and re-test.'),
+          '',
+          '| Scope | n | raw dip | raw p | jitter p median | jitter reject@.05 / 5 seeds |',
+          '|---|---|---|---|---|---|']
+    for key, label in [('big4_pooled', 'Big-4 pooled')] + \
+                      [(ALIAS[f], ALIAS[f]) for f in BIG4] + \
+                      [('non_big4_pooled', 'Non-Big-4 pooled')]:
+        if key in panels:
+            p = panels[key]
+            md.append(f'| {label} | {p["n"]:,} | '
+                      f'{p["raw"]["dip"]:.5f} | '
+                      f'{_fmt_p(p["raw"]["p"])} | '
+                      f'{p["jittered"]["p_median"]:.4g} | '
+                      f'{p["jittered"]["reject_at_05_count"]}/5 |')
+    md += ['',
+           '**Interpretation.** If jittered dip ceases to reject in all '
+           'panels, the raw-data rejection was driven by integer ties '
+           'rather than a continuous bimodal density. Codex round-29 '
+           'observed this pattern; this script confirms with multi-seed '
+           'robustness.',
+           '',
+           '## B. Integer-histogram valley locations (bins 0..20)',
+           '',
+           ('For each scope, list bins where count is strictly less '
+            'than both neighbours, with relative depth '
+            '(min(neighbour) - bin) / min(neighbour). A genuine '
+            'antimode would show a deep, stable valley; integer-noise '
+            'valleys are shallow and inconsistent across firms.'),
+           '']
+    for key, label in [('big4_pooled', 'Big-4 pooled')] + \
+                      [(ALIAS[f], ALIAS[f]) for f in BIG4] + \
+                      [('non_big4_pooled', 'Non-Big-4 pooled')]:
+        if key in valleys:
+            v_list = valleys[key]['valleys']
+            if not v_list:
+                md.append(f'- **{label}**: no integer-histogram valleys '
+                          f'in 0..20')
+            else:
+                desc = ', '.join(
+                    f'dh={v["bin_center"]:.0f} (depth_rel={v["depth_rel"]:.3f})'
+                    for v in v_list)
+                md.append(f'- **{label}**: {desc}')
+    md += ['',
+           '## C. Firm-residualized dHash dip',
+           '',
+           ('Subtract each firm mean dHash; add back grand mean. If '
+            'residual rejects, within-firm multimodality is genuine. '
+            'If residual fails to reject, all dh "multimodality" was '
+            'between-firm composition.'),
+           '',
+           '| Scope | raw dip | raw p | residualized dip | residualized p |',
+           '|---|---|---|---|---|']
+    fr = results['firm_residualized_dh_dip']
+    md += [f'| Big-4 | {fr["big4"]["raw"]["dip"]:.5f} | '
+           f'{_fmt_p(fr["big4"]["raw"]["p"])} | '
+           f'{fr["big4"]["firm_residualized"]["dip"]:.5f} | '
+           f'{_fmt_p(fr["big4"]["firm_residualized"]["p"])} |',
+           f'| Non-Big-4 | {fr["non_big4"]["raw"]["dip"]:.5f} | '
+           f'{_fmt_p(fr["non_big4"]["raw"]["p"])} | '
+           f'{fr["non_big4"]["firm_residualized"]["dip"]:.5f} | '
+           f'{_fmt_p(fr["non_big4"]["firm_residualized"]["p"])} |']
+    md += ['',
+           '## D. Max-cos pair vs min-dh pair coincidence',
+           '']
+    pc = results.get('pair_coincidence', {})
+    if 'same_pair_rate' in pc and pc['same_pair_rate'] is not None:
+        md += [f'- n_signatures with both pair IDs: '
+               f'{pc["n_with_both_pair_ids"]:,}',
+               f'- same-pair rate: {pc["same_pair_rate"]:.4f} '
+               f'({pc["n_same_pair"]:,} of '
+               f'{pc["n_with_both_pair_ids"]:,})',
+               '',
+               ('A high rate (>0.8) supports a single-pair regime '
+                'descriptor language (cos and dh attached to the same '
+                'partner). A low rate indicates the two descriptors '
+                'attach to different partners and should be discussed '
+                'as parallel-but-different evidence.')]
+    elif 'error' in pc:
+        md += [f'- column not present in DB: {pc["error"]}',
+               ('- note: schema-dependent; pair IDs not currently stored '
+                'in signatures table.')]
+    md.append('')
+    md_path = OUT / 'dhash_discrete_report.md'
+    md_path.write_text('\n'.join(md), encoding='utf-8')
+    print(f'[md  ] {md_path}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/39e_dhash_residualized_jittered.py

@@ -0,0 +1,250 @@
+#!/usr/bin/env python3
+"""
+Script 39e: dHash Firm-Residualized + Jittered Dip (final test)
+================================================================
+Script 39d showed:
+  - Within-firm dh dip rejections all vanish after jitter (integer
+    artifact)
+  - Big-4 pooled dh dip survives jitter (p_median=0 over 5 seeds)
+
+But Firm A mean dh = 2.73 vs Firms B/C/D ~6.5-7.4 -- a large
+between-firm location shift, analogous to the cosine case where
+firm-mean centering eliminated rejection.
+
+This script applies BOTH corrections simultaneously:
+  1. Firm-mean centering (remove between-firm location shifts)
+  2. Uniform jitter in [-0.5, +0.5] (remove integer ties)
+
+If the doubly-corrected dh distribution rejects unimodality, the
+Big-4 pooled multimodality is a genuine within-population, continuous
+phenomenon. If it fails to reject, dh "multimodality" is fully
+explained by between-firm composition (same conclusion as cosine).
+
+Multi-seed (5 seeds) for robustness.
+
+Outputs:
+  reports/v4_big4/dhash_discrete_robustness/
+    dhash_residualized_jittered_results.json
+    dhash_residualized_jittered_report.md
+"""
+
+import json
+import sqlite3
+import numpy as np
+import diptest
+from pathlib import Path
+from datetime import datetime
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'v4_big4/dhash_discrete_robustness')
+OUT.mkdir(parents=True, exist_ok=True)
+
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+ALIAS = {'勤業眾信聯合': 'Firm A',
+         '安侯建業聯合': 'Firm B',
+         '資誠聯合': 'Firm C',
+         '安永聯合': 'Firm D'}
+N_BOOT = 2000
+SEEDS = [42, 43, 44, 45, 46]
+
+
+def load_signatures():
+    conn = sqlite3.connect(f'file:{DB}?mode=ro', uri=True)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT a.firm, CAST(s.min_dhash_independent AS REAL)
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.min_dhash_independent IS NOT NULL
+          AND a.firm IS NOT NULL
+    ''')
+    rows = cur.fetchall()
+    conn.close()
+    return rows
+
+
+def firm_residualize(values, firm_labels):
+    arr = np.asarray(values, dtype=float)
+    firms = np.asarray(firm_labels)
+    out = arr.copy()
+    grand = float(np.mean(arr))
+    for f in np.unique(firms):
+        m = firms == f
+        out[m] = arr[m] - float(np.mean(arr[m])) + grand
+    return out
+
+
+def dip_multi(values, seeds, with_jitter, n_boot=N_BOOT):
+    arr = np.asarray(values, dtype=float)
+    arr = arr[np.isfinite(arr)]
+    results = []
+    for seed in seeds:
+        rng = np.random.default_rng(seed)
+        v = arr + rng.uniform(-0.5, 0.5, len(arr)) if with_jitter else arr
+        d, p = diptest.diptest(v, boot_pval=True, n_boot=n_boot)
+        results.append({'seed': seed, 'dip': float(d), 'p': float(p)})
+        if not with_jitter:
+            break  # without jitter the seed is irrelevant
+    return results
+
+
+def _fmt_p(p):
+    return '< 5e-4' if p == 0.0 else f'{p:.4g}'
+
+
+def summarize(name, results):
+    ps = [r['p'] for r in results]
+    ds = [r['dip'] for r in results]
+    return {
+        'name': name,
+        'n_seeds': len(results),
+        'dip_min': min(ds), 'dip_max': max(ds), 'dip_median': float(np.median(ds)),
+        'p_min': min(ps), 'p_max': max(ps), 'p_median': float(np.median(ps)),
+        'reject_at_05_count': int(sum(1 for p in ps if p <= 0.05)),
+        'per_seed': results,
+    }
+
+
+def main():
+    print('=' * 72)
+    print('Script 39e: dHash Firm-Residualized + Jittered Dip')
+    print('=' * 72)
+    rows = load_signatures()
+    firms_raw = np.array([r[0] for r in rows])
+    dh = np.array([r[1] for r in rows], dtype=float)
+    is_big4 = np.isin(firms_raw, BIG4)
+    big4_dh = dh[is_big4]
+    big4_firms = np.array([ALIAS[f] for f in firms_raw[is_big4]])
+
+    print(f'\nLoaded {len(rows):,} signatures; Big-4 {is_big4.sum():,}')
+    print('\nPer-firm Big-4 dh summary:')
+    for f in sorted(set(big4_firms)):
+        v = big4_dh[big4_firms == f]
+        print(f'  {f}: n={len(v):,} mean={v.mean():.3f} '
+              f'median={np.median(v):.1f} sd={v.std():.3f}')
+
+    # ---- Test conditions, all on Big-4 signature-level dh ----
+    panels = {}
+
+    # 1. Raw (no centering, no jitter)
+    print('\n[1] Raw dh')
+    r = dip_multi(big4_dh, [42], with_jitter=False)
+    panels['raw'] = summarize('raw', r)
+    print(f'  dip={r[0]["dip"]:.5f}, p={_fmt_p(r[0]["p"])}')
+
+    # 2. Centered only (no jitter; integer values preserved)
+    print('\n[2] Firm-mean centered, no jitter')
+    centered = firm_residualize(big4_dh, big4_firms)
+    r = dip_multi(centered, [42], with_jitter=False)
+    panels['centered_only'] = summarize('centered_only', r)
+    print(f'  dip={r[0]["dip"]:.5f}, p={_fmt_p(r[0]["p"])}')
+
+    # 3. Jittered only (no centering)
+    print('\n[3] Jittered (5 seeds), no centering')
+    r = dip_multi(big4_dh, SEEDS, with_jitter=True)
+    panels['jitter_only'] = summarize('jitter_only', r)
+    print(f'  p_median={panels["jitter_only"]["p_median"]:.4g}, '
+          f'reject@.05 in '
+          f'{panels["jitter_only"]["reject_at_05_count"]}/5 seeds')
+
+    # 4. Centered + jittered (THE key test)
+    print('\n[4] Firm-mean centered + jittered (5 seeds) -- KEY TEST')
+    r = dip_multi(centered, SEEDS, with_jitter=True)
+    panels['centered_jittered'] = summarize('centered_jittered', r)
+    print(f'  p_median={panels["centered_jittered"]["p_median"]:.4g}, '
+          f'reject@.05 in '
+          f'{panels["centered_jittered"]["reject_at_05_count"]}/5 seeds')
+    for s in r:
+        print(f'    seed {s["seed"]}: dip={s["dip"]:.5f}, p={_fmt_p(s["p"])}')
+
+    # Per-firm dh stats (re-confirm Firm A shift)
+    firm_stats = {}
+    for f in sorted(set(big4_firms)):
+        v = big4_dh[big4_firms == f]
+        firm_stats[f] = {
+            'n': int(len(v)),
+            'mean': float(v.mean()),
+            'median': float(np.median(v)),
+            'sd': float(v.std()),
+            'p25': float(np.percentile(v, 25)),
+            'p75': float(np.percentile(v, 75)),
+            'pct_le_5': float(np.mean(v <= 5)),
+            'pct_gt_15': float(np.mean(v > 15)),
+        }
+
+    results = {
+        'meta': {
+            'script': '39e',
+            'timestamp': datetime.now().isoformat(timespec='seconds'),
+            'n_big4_signatures': int(big4_dh.size),
+            'n_boot': N_BOOT,
+            'seeds': SEEDS,
+            'note': ('Final test: does Big-4 pooled dh multimodality '
+                     'survive BOTH firm-mean centering and integer-tie '
+                     'jitter?'),
+        },
+        'panels': panels,
+        'per_firm_dh_stats': firm_stats,
+    }
+
+    json_path = OUT / 'dhash_residualized_jittered_results.json'
+    json_path.write_text(json.dumps(results, indent=2, ensure_ascii=False),
+                         encoding='utf-8')
+    print(f'\n[json] {json_path}')
+
+    md = [
+        '# dHash Firm-Residualized + Jittered Dip (Script 39e)',
+        '', f'Generated: {results["meta"]["timestamp"]}',
+        f'Bootstrap replicates: {N_BOOT}; jitter seeds: {SEEDS}',
+        '',
+        '## Per-firm Big-4 dh summary',
+        '', '| Firm | n | mean | median | sd | P25 | P75 | %<=5 | %>15 |',
+        '|---|---|---|---|---|---|---|---|---|',
+    ]
+    for f, s in firm_stats.items():
+        md.append(f'| {f} | {s["n"]:,} | {s["mean"]:.3f} | '
+                  f'{s["median"]:.1f} | {s["sd"]:.3f} | '
+                  f'{s["p25"]:.1f} | {s["p75"]:.1f} | '
+                  f'{s["pct_le_5"]:.3f} | {s["pct_gt_15"]:.3f} |')
+    md += [
+        '',
+        '## Dip test under four conditions (Big-4 pooled, sig-level)',
+        '',
+        '| Condition | dip | p (or p_median) | reject@.05 (seeds) |',
+        '|---|---|---|---|',
+        f'| 1. Raw (integer values) | {panels["raw"]["dip_median"]:.5f} '
+        f'| {_fmt_p(panels["raw"]["p_median"])} | n/a (1 seed) |',
+        f'| 2. Firm-mean centered, no jitter '
+        f'| {panels["centered_only"]["dip_median"]:.5f} '
+        f'| {_fmt_p(panels["centered_only"]["p_median"])} | n/a (1 seed) |',
+        f'| 3. Jittered only (5 seeds) '
+        f'| median {panels["jitter_only"]["dip_median"]:.5f} '
+        f'| median {_fmt_p(panels["jitter_only"]["p_median"])} '
+        f'| {panels["jitter_only"]["reject_at_05_count"]}/5 |',
+        f'| 4. **Centered + jittered (5 seeds)** '
+        f'| median {panels["centered_jittered"]["dip_median"]:.5f} '
+        f'| median {_fmt_p(panels["centered_jittered"]["p_median"])} '
+        f'| {panels["centered_jittered"]["reject_at_05_count"]}/5 |',
+        '',
+        '## Interpretation',
+        '',
+        ('If Condition 4 still rejects unimodality, Big-4 dh has '
+         'genuine within-population continuous multimodality '
+         'independent of both between-firm location shifts and '
+         'integer mass points. If Condition 4 fails to reject, the '
+         'Big-4 pooled dh multimodality is fully explained by '
+         '(between-firm mean shift) + (integer mass points). In the '
+         'latter case, the dh axis carries no independent within-firm '
+         'regime evidence beyond the cos axis.'),
+        '',
+    ]
+    md_path = OUT / 'dhash_residualized_jittered_report.md'
+    md_path.write_text('\n'.join(md), encoding='utf-8')
+    print(f'[md  ] {md_path}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/40_v4_pixel_identity_far.py

@@ -0,0 +1,421 @@
+#!/usr/bin/env python3
+"""
+Script 40: Pixel-Identity FAR on Big-4 (hard ground truth validation)
+=======================================================================
+Phase 1.8 follow-up. Validates the v4.0 classifier family against
+the only hard ground truth available in the corpus:
+pixel_identical_to_closest = 1 (signatures byte-identical to their
+nearest same-CPA match).
+
+Pixel-identical pairs are MATHEMATICALLY IMPOSSIBLE to arise from
+independent hand-signing -- they must be reuses of the same source
+image. Treating them as ground-truth replicated, we compute:
+
+  FAR (false-alarm-rate) := P(classifier says hand-leaning |
+                              ground truth is replicated)
+
+for three classifiers:
+
+  C1 PaperA          non_hand iff cos > 0.95 AND dh <= 5
+  C2 K=3 per-CPA     hard label, replicated = C3 (highest cos)
+  C3 Reverse-anchor  cos_left_tail_pct under non-Big-4 reference;
+                     replicated = score below explicit cut.
+                     Cut chosen so that the rule's overall
+                     replicated rate matches PaperA's overall rate
+                     (calibration-by-prevalence; documented limitation).
+
+Additional metrics per classifier:
+  - n_pixel_identical, n_correctly_called_replicated,
+    n_misclassified_handleaning
+  - Wilson 95% CI on FAR
+  - Per-firm FAR breakdown
+
+Output:
+  reports/v4_big4/pixel_identity_far/
+    far_results.json
+    far_report.md
+    far_cases.csv  (every misclassified pixel-identical sig)
+"""
+
+import sqlite3
+import csv
+import json
+import numpy as np
+import matplotlib
+matplotlib.use('Agg')
+import matplotlib.pyplot as plt
+from pathlib import Path
+from datetime import datetime
+from scipy import stats
+from scipy.stats import norm
+from sklearn.mixture import GaussianMixture
+from sklearn.covariance import MinCovDet
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'v4_big4/pixel_identity_far')
+OUT.mkdir(parents=True, exist_ok=True)
+
+SEED = 42
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+LABEL = {'勤業眾信聯合': 'Firm A (Deloitte)', '安侯建業聯合': 'KPMG',
+         '資誠聯合': 'PwC', '安永聯合': 'EY'}
+PAPER_A_COS_CUT = 0.95
+PAPER_A_DH_CUT = 5
+MIN_SIGS = 10
+
+
+def load_pixel_identical_big4():
+    conn = sqlite3.connect(DB)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT s.signature_id, s.assigned_accountant, a.firm,
+               s.max_similarity_to_same_accountant,
+               CAST(s.min_dhash_independent AS REAL),
+               s.closest_match_file
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.pixel_identical_to_closest = 1
+          AND s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.min_dhash_independent IS NOT NULL
+          AND a.firm IN (?, ?, ?, ?)
+    ''', BIG4)
+    rows = cur.fetchall()
+    conn.close()
+    return rows
+
+
+def load_all_big4_signatures():
+    """For computing the calibration-by-prevalence rate of PaperA."""
+    conn = sqlite3.connect(DB)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT s.max_similarity_to_same_accountant,
+               CAST(s.min_dhash_independent AS REAL)
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.min_dhash_independent IS NOT NULL
+          AND a.firm IN (?, ?, ?, ?)
+    ''', BIG4)
+    rows = cur.fetchall()
+    conn.close()
+    cos = np.array([float(r[0]) for r in rows])
+    dh = np.array([float(r[1]) for r in rows])
+    return cos, dh
+
+
+def load_per_cpa_means_big4():
+    conn = sqlite3.connect(DB)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT s.assigned_accountant, a.firm,
+               AVG(s.max_similarity_to_same_accountant) AS cos_mean,
+               AVG(CAST(s.min_dhash_independent AS REAL)) AS dh_mean,
+               COUNT(*) AS n
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.min_dhash_independent IS NOT NULL
+          AND a.firm IN (?, ?, ?, ?)
+        GROUP BY s.assigned_accountant
+        HAVING n >= ?
+    ''', BIG4 + (MIN_SIGS,))
+    rows = cur.fetchall()
+    conn.close()
+    X = np.array([[float(r[2]), float(r[3])] for r in rows])
+    return X
+
+
+def load_non_big4_reference_means():
+    conn = sqlite3.connect(DB)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT AVG(s.max_similarity_to_same_accountant) AS cos_mean,
+               AVG(CAST(s.min_dhash_independent AS REAL)) AS dh_mean,
+               COUNT(*) AS n
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.min_dhash_independent IS NOT NULL
+          AND a.firm IS NOT NULL
+          AND a.firm NOT IN (?, ?, ?, ?)
+        GROUP BY s.assigned_accountant
+        HAVING n >= ?
+    ''', BIG4 + (MIN_SIGS,))
+    rows = cur.fetchall()
+    conn.close()
+    return np.array([[float(r[0]), float(r[1])] for r in rows])
+
+
+def fit_k3(X):
+    return GaussianMixture(n_components=3, covariance_type='full',
+                           random_state=SEED, n_init=15, max_iter=500).fit(X)
+
+
+def fit_reference(X):
+    mcd = MinCovDet(random_state=SEED, support_fraction=0.85).fit(X)
+    return {'mean': mcd.location_, 'cov': mcd.covariance_}
+
+
+def wilson_ci(k, n, alpha=0.05):
+    if n == 0:
+        return (0.0, 1.0)
+    z = norm.ppf(1 - alpha / 2)
+    phat = k / n
+    denom = 1 + z * z / n
+    center = (phat + z * z / (2 * n)) / denom
+    pm = z * np.sqrt(phat * (1 - phat) / n + z * z / (4 * n * n)) / denom
+    return (max(0.0, center - pm), min(1.0, center + pm))
+
+
+def main():
+    print('=' * 72)
+    print('Script 40: Pixel-Identity FAR on Big-4')
+    print('=' * 72)
+
+    # Load pixel-identical Big-4 signatures (ground truth replicated)
+    rows = load_pixel_identical_big4()
+    n = len(rows)
+    print(f'\nN pixel-identical Big-4 signatures (ground truth = replicated): '
+          f'{n}')
+    if n == 0:
+        print('No pixel-identical pairs in Big-4. Exiting.')
+        return
+
+    # Per-firm distribution
+    by_firm = {}
+    for r in rows:
+        by_firm.setdefault(r[2], []).append(r)
+    for f in BIG4:
+        print(f'  {LABEL[f]}: {len(by_firm.get(f, []))}')
+
+    sig_ids = np.array([r[0] for r in rows])
+    sig_firms = np.array([r[2] for r in rows])
+    cos = np.array([r[3] for r in rows], dtype=float)
+    dh = np.array([r[4] for r in rows], dtype=float)
+    closest = np.array([r[5] or '' for r in rows])
+
+    # ---------- Classifier C1: Paper A rule ----------
+    paperA_replicated = (cos > PAPER_A_COS_CUT) & (dh <= PAPER_A_DH_CUT)
+    paperA_misclass = ~paperA_replicated
+    n_pA_correct = int(paperA_replicated.sum())
+    n_pA_miss = int(paperA_misclass.sum())
+    far_pA = n_pA_miss / n
+    pA_lo, pA_hi = wilson_ci(n_pA_miss, n)
+    print(f'\n[C1 Paper A] correct: {n_pA_correct}/{n} = '
+          f'{(1 - far_pA)*100:.2f}%; FAR: {far_pA*100:.2f}% '
+          f'[{pA_lo*100:.2f}%, {pA_hi*100:.2f}%]')
+
+    # ---------- Classifier C2: K=3 per-CPA hard label ----------
+    # (Use the K=3 CPA-fit components; for each pixel-identical signature,
+    # predict its membership as if it were a per-CPA point.)
+    X_cpa = load_per_cpa_means_big4()
+    gmm = fit_k3(X_cpa)
+    order = np.argsort(gmm.means_[:, 0])  # C1 hand, C3 replicated
+    label_map = {old: new for new, old in enumerate(order)}
+    X_pix = np.column_stack([cos, dh])
+    raw = gmm.predict(X_pix)
+    k3_labels = np.array([label_map[l] for l in raw])
+    # Replicated = C3 (label index 2)
+    k3_replicated = (k3_labels == 2)
+    k3_misclass = ~k3_replicated
+    n_k3_correct = int(k3_replicated.sum())
+    n_k3_miss = int(k3_misclass.sum())
+    far_k3 = n_k3_miss / n
+    k3_lo, k3_hi = wilson_ci(n_k3_miss, n)
+    print(f'[C2 K=3 perCPA] correct: {n_k3_correct}/{n} = '
+          f'{(1 - far_k3)*100:.2f}%; FAR: {far_k3*100:.2f}% '
+          f'[{k3_lo*100:.2f}%, {k3_hi*100:.2f}%]')
+
+    # ---------- Classifier C3: Reverse-anchor with prevalence-calibrated cut ----------
+    # Build reference Gaussian from non-Big-4
+    X_ref = load_non_big4_reference_means()
+    ref = fit_reference(X_ref)
+    mu_c = ref['mean'][0]
+    sd_c = float(np.sqrt(ref['cov'][0, 0]))
+
+    # Score every Big-4 signature; pick cut so overall replicated rate
+    # matches Paper A's overall replicated rate.
+    cos_all, dh_all = load_all_big4_signatures()
+    paperA_overall_repl_rate = float(np.mean(
+        (cos_all > PAPER_A_COS_CUT) & (dh_all <= PAPER_A_DH_CUT)))
+    # Reverse-anchor score per signature
+    rev_score_all = stats.norm.cdf(cos_all, loc=mu_c, scale=sd_c)
+    # We want HIGHER scores = more replicated (large cosine = right tail
+    # of the reference). So replicated iff rev_score > cut.
+    # Pick cut at the (1 - paperA_overall_repl_rate)-quantile of rev_score_all.
+    cut_quantile = 1 - paperA_overall_repl_rate
+    rev_cut = float(np.quantile(rev_score_all, cut_quantile))
+    print(f'\n[C3 Reverse-anchor calibration] '
+          f'PaperA overall replicated rate = '
+          f'{paperA_overall_repl_rate*100:.2f}%; '
+          f'rev-anchor cut at {cut_quantile*100:.2f}-th pct of score = '
+          f'{rev_cut:.4f}')
+
+    rev_score_pix = stats.norm.cdf(cos, loc=mu_c, scale=sd_c)
+    rev_replicated = (rev_score_pix > rev_cut)
+    rev_misclass = ~rev_replicated
+    n_rev_correct = int(rev_replicated.sum())
+    n_rev_miss = int(rev_misclass.sum())
+    far_rev = n_rev_miss / n
+    rev_lo, rev_hi = wilson_ci(n_rev_miss, n)
+    print(f'[C3 Reverse-anchor] correct: {n_rev_correct}/{n} = '
+          f'{(1 - far_rev)*100:.2f}%; FAR: {far_rev*100:.2f}% '
+          f'[{rev_lo*100:.2f}%, {rev_hi*100:.2f}%]')
+
+    # ---------- Per-firm FAR ----------
+    print('\n[per-firm FAR]')
+    print(f'  {"Firm":<22} {"n":>5} {"PaperA":>11} {"K=3":>11} {"Rev-anc":>11}')
+    per_firm = {}
+    for f in BIG4:
+        mask = (sig_firms == f)
+        n_f = int(mask.sum())
+        if n_f == 0:
+            per_firm[f] = {'n': 0}
+            continue
+        miss_pA = int(np.sum(paperA_misclass[mask]))
+        miss_k3 = int(np.sum(k3_misclass[mask]))
+        miss_rev = int(np.sum(rev_misclass[mask]))
+        far_pA_f = miss_pA / n_f
+        far_k3_f = miss_k3 / n_f
+        far_rev_f = miss_rev / n_f
+        per_firm[f] = {
+            'n': n_f,
+            'paperA_far': far_pA_f, 'paperA_misclass_n': miss_pA,
+            'k3_far': far_k3_f, 'k3_misclass_n': miss_k3,
+            'reverse_anchor_far': far_rev_f, 'reverse_anchor_misclass_n': miss_rev,
+        }
+        print(f'  {LABEL[f]:<22} {n_f:>5} {far_pA_f*100:>10.2f}% '
+              f'{far_k3_f*100:>10.2f}% {far_rev_f*100:>10.2f}%')
+
+    # ---------- Misclassified case CSV ----------
+    cases_csv = OUT / 'far_cases.csv'
+    with open(cases_csv, 'w', newline='', encoding='utf-8') as f:
+        w = csv.writer(f)
+        w.writerow(['signature_id', 'cpa', 'firm', 'firm_label',
+                    'cos', 'dh', 'closest_match_file',
+                    'paperA_call', 'k3_call', 'reverse_anchor_call'])
+        for i in range(n):
+            pa = 'replicated' if paperA_replicated[i] else 'hand_leaning'
+            kl = ['C1_handleaning', 'C2_mixed',
+                  'C3_replicated'][k3_labels[i]]
+            ra = 'replicated' if rev_replicated[i] else 'hand_leaning'
+            # Only write rows where at least one classifier disagrees with
+            # ground truth (replicated)
+            if pa != 'replicated' or kl != 'C3_replicated' \
+                    or ra != 'replicated':
+                w.writerow([sig_ids[i], rows[i][1], sig_firms[i],
+                            LABEL[sig_firms[i]],
+                            f'{cos[i]:.4f}', f'{dh[i]:.4f}', closest[i],
+                            pa, kl, ra])
+    print(f'\nMisclassified cases CSV: {cases_csv}')
+
+    # Markdown report
+    md = [
+        '# Pixel-Identity FAR on Big-4 (Script 40)',
+        f'Generated: {datetime.now().strftime("%Y-%m-%d %H:%M:%S")}',
+        '',
+        '## Ground truth',
+        '',
+        ('Pixel-identical pairs (signature byte-identical to nearest '
+         'same-CPA neighbor) cannot arise from independent hand-signing. '
+         'They are taken as ground-truth REPLICATED. We measure each '
+         'classifier\'s false-alarm rate (rate of calling these signatures '
+         'hand-leaning).'),
+        '',
+        f'- Total Big-4 pixel-identical signatures: **{n}**',
+        '',
+        '## Headline FAR (lower is better)',
+        '',
+        '| Classifier | Correct/N | FAR | Wilson 95% CI |',
+        '|---|---|---|---|',
+        f'| Paper A box rule | {n_pA_correct}/{n} | **{far_pA*100:.2f}%** | '
+        f'[{pA_lo*100:.2f}%, {pA_hi*100:.2f}%] |',
+        f'| K=3 per-CPA hard label (C3 = replicated) | {n_k3_correct}/{n} | '
+        f'**{far_k3*100:.2f}%** | [{k3_lo*100:.2f}%, {k3_hi*100:.2f}%] |',
+        f'| Reverse-anchor (prevalence-calibrated cut) | {n_rev_correct}/{n} | '
+        f'**{far_rev*100:.2f}%** | [{rev_lo*100:.2f}%, {rev_hi*100:.2f}%] |',
+        '',
+        ('Reverse-anchor cut chosen so that overall replicated rate '
+         f'matches Paper A overall rate ({paperA_overall_repl_rate*100:.2f}%); '
+         'this is calibration-by-prevalence and is documented as a v4.0 '
+         'limitation -- no signature-level ground truth exists for the '
+         'hand-leaning class so we cannot pick the cut by direct ROC '
+         'optimization.'),
+        '',
+        '## Per-firm FAR',
+        '',
+        '| Firm | n | Paper A FAR | K=3 FAR | Rev-anchor FAR |',
+        '|---|---|---|---|---|',
+    ]
+    for f in BIG4:
+        pf = per_firm[f]
+        if pf['n'] == 0:
+            md.append(f'| {LABEL[f]} | 0 | n/a | n/a | n/a |')
+            continue
+        md.append(f'| {LABEL[f]} | {pf["n"]} | '
+                  f'{pf["paperA_far"]*100:.2f}% '
+                  f'({pf["paperA_misclass_n"]}) | '
+                  f'{pf["k3_far"]*100:.2f}% ({pf["k3_misclass_n"]}) | '
+                  f'{pf["reverse_anchor_far"]*100:.2f}% '
+                  f'({pf["reverse_anchor_misclass_n"]}) |')
+    md += ['', '## Reading',
+           '',
+           ('A FAR substantially below the no-information rate '
+            f'(1 - {paperA_overall_repl_rate*100:.2f}% = '
+            f'{(1-paperA_overall_repl_rate)*100:.2f}%) means the '
+            'classifier extracts useful signal from the (cos, dh) '
+            'features for distinguishing pixel-identical replication.  '
+            'Since pixel-identical pairs are a CONSERVATIVE SUBSET of '
+            'true replication (only the byte-equal extreme), a low FAR '
+            'against this subset is necessary but not sufficient evidence '
+            'of correct replication detection.'),
+           '',
+           '## Files',
+           '- `far_results.json` -- machine-readable results',
+           '- `far_cases.csv` -- every misclassified pixel-identical signature',
+           ]
+    md_path = OUT / 'far_report.md'
+    md_path.write_text('\n'.join(md), encoding='utf-8')
+    print(f'Report: {md_path}')
+
+    payload = {
+        'generated_at': datetime.now().isoformat(),
+        'n_pixel_identical_big4': n,
+        'paper_a_cuts': {'cos': PAPER_A_COS_CUT, 'dh': PAPER_A_DH_CUT},
+        'paper_a_overall_replicated_rate_big4': paperA_overall_repl_rate,
+        'reverse_anchor_cut_score': rev_cut,
+        'reverse_anchor_cut_quantile': cut_quantile,
+        'reverse_anchor_reference_center': [float(mu_c),
+                                             float(ref['mean'][1])],
+        'classifiers': {
+            'paperA': {
+                'far': float(far_pA),
+                'far_wilson95': [float(pA_lo), float(pA_hi)],
+                'n_correct': n_pA_correct, 'n_misclass': n_pA_miss,
+            },
+            'k3_perCPA': {
+                'far': float(far_k3),
+                'far_wilson95': [float(k3_lo), float(k3_hi)],
+                'n_correct': n_k3_correct, 'n_misclass': n_k3_miss,
+            },
+            'reverse_anchor_calibrated': {
+                'far': float(far_rev),
+                'far_wilson95': [float(rev_lo), float(rev_hi)],
+                'n_correct': n_rev_correct, 'n_misclass': n_rev_miss,
+            },
+        },
+        'per_firm_far': per_firm,
+    }
+    json_path = OUT / 'far_results.json'
+    json_path.write_text(json.dumps(payload, indent=2, ensure_ascii=False),
+                         encoding='utf-8')
+    print(f'JSON: {json_path}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/40b_inter_cpa_far_sweep.py

@@ -0,0 +1,413 @@
+#!/usr/bin/env python3
+"""
+Script 40b: Inter-CPA FAR Sweep for cos and dHash (joint + marginal)
+=====================================================================
+After codex round-29 destroyed the distributional path to thresholds
+(K=3 mixture / dip / antimode shown composition-driven by Scripts
+39b–39e), v4.0 pivots to an anchor-based threshold framework:
+empirically derived from inter-CPA negative anchor specificity.
+
+Inter-CPA pairs (different CPAs, all-firm) are the negative anchor:
+they are by definition not same-CPA replications, and the user's
+within-CPA mechanism-transition concern (a CPA might switch from
+hand-sign to template mid-career) does not enter the inter-CPA
+calibration because each sampled pair crosses CPA boundaries.
+
+This script samples a large number of inter-CPA pairs and computes
+both descriptors per pair (cosine via feature_vector dot product;
+Hamming distance via dhash_vector XOR). It then sweeps:
+
+  1. FAR(cos > k) across k in [0.80, 0.99]
+  2. FAR(dHash <= k) across k in [0, 20]
+  3. Joint FAR(cos > 0.95 AND dHash <= k) for k in [0, 20]
+  4. Conditional FAR(dHash <= k | cos > 0.95) -- the v3 inherited
+     rule's marginal specificity contribution from dHash
+
+Outputs:
+  reports/v4_big4/inter_cpa_far_sweep/
+    far_sweep_results.json
+    far_sweep_report.md
+
+Sample size: 500,000 inter-CPA pairs (matches v3 Script 10
+convention). Big-4-only and full-corpus variants both reported.
+"""
+
+import json
+import sqlite3
+import numpy as np
+from pathlib import Path
+from datetime import datetime
+from collections import defaultdict
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'v4_big4/inter_cpa_far_sweep')
+OUT.mkdir(parents=True, exist_ok=True)
+
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+ALIAS = {'勤業眾信聯合': 'Firm A',
+         '安侯建業聯合': 'Firm B',
+         '資誠聯合': 'Firm C',
+         '安永聯合': 'Firm D'}
+N_PAIRS = 500_000
+SEED = 42
+
+COS_GRID = [0.80, 0.83, 0.85, 0.87, 0.89, 0.90, 0.91, 0.92, 0.93, 0.94,
+            0.945, 0.95, 0.955, 0.96, 0.965, 0.97, 0.975, 0.98, 0.985,
+            0.99]
+DH_GRID = list(range(0, 21))
+
+
+def hamming_64bit(a_bytes, b_bytes):
+    """Hamming distance between two 8-byte (64-bit) dHash byte strings."""
+    a = int.from_bytes(a_bytes, 'big')
+    b = int.from_bytes(b_bytes, 'big')
+    return (a ^ b).bit_count()
+
+
+def load_signatures():
+    conn = sqlite3.connect(f'file:{DB}?mode=ro', uri=True)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT s.signature_id, s.assigned_accountant, a.firm,
+               s.feature_vector, s.dhash_vector
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.feature_vector IS NOT NULL
+          AND s.dhash_vector IS NOT NULL
+          AND a.firm IS NOT NULL
+    ''')
+    rows = cur.fetchall()
+    conn.close()
+    return rows
+
+
+def sample_inter_cpa_pairs(rows, n_pairs, seed, restrict_to_big4=False):
+    """Sample inter-CPA pairs and compute (cos, dh) for each."""
+    rng = np.random.default_rng(seed)
+    if restrict_to_big4:
+        rows = [r for r in rows if r[2] in BIG4]
+        scope = 'big4_only'
+    else:
+        scope = 'all_firms'
+    print(f'  [{scope}] {len(rows):,} signatures available')
+
+    by_acct = defaultdict(list)
+    for r in rows:
+        by_acct[r[1]].append(r)
+    accountants = list(by_acct.keys())
+    n_acct = len(accountants)
+    print(f'  [{scope}] {n_acct} accountants')
+
+    features = {a: np.stack(
+        [np.frombuffer(r[3], dtype=np.float32) for r in by_acct[a]]
+    ) for a in accountants}
+    dhashes = {a: [r[4] for r in by_acct[a]] for a in accountants}
+
+    cos_vals = np.empty(n_pairs, dtype=np.float32)
+    dh_vals = np.empty(n_pairs, dtype=np.int32)
+    n_done = 0
+    for _ in range(n_pairs):
+        i, j = rng.choice(n_acct, 2, replace=False)
+        a1, a2 = accountants[i], accountants[j]
+        n1, n2 = len(by_acct[a1]), len(by_acct[a2])
+        k1 = int(rng.integers(0, n1))
+        k2 = int(rng.integers(0, n2))
+        f1 = features[a1][k1]
+        f2 = features[a2][k2]
+        cos = float(f1 @ f2)
+        d = hamming_64bit(dhashes[a1][k1], dhashes[a2][k2])
+        cos_vals[n_done] = cos
+        dh_vals[n_done] = d
+        n_done += 1
+    return scope, cos_vals, dh_vals
+
+
+def wilson_ci(k, n, z=1.96):
+    if n == 0:
+        return (None, None)
+    phat = k / n
+    denom = 1 + z * z / n
+    centre = (phat + z * z / (2 * n)) / denom
+    half = z * np.sqrt(phat * (1 - phat) / n + z * z / (4 * n * n)) / denom
+    return (max(0.0, centre - half), min(1.0, centre + half))
+
+
+def far_at_cos(cos_vals, k):
+    n = len(cos_vals)
+    hits = int((cos_vals > k).sum())
+    lo, hi = wilson_ci(hits, n)
+    return {'k': float(k), 'n': n, 'hits': hits,
+            'far': hits / n, 'ci95_lo': lo, 'ci95_hi': hi}
+
+
+def far_at_dh_le(dh_vals, k):
+    n = len(dh_vals)
+    hits = int((dh_vals <= k).sum())
+    lo, hi = wilson_ci(hits, n)
+    return {'k': int(k), 'n': n, 'hits': hits,
+            'far': hits / n, 'ci95_lo': lo, 'ci95_hi': hi}
+
+
+def joint_far(cos_vals, dh_vals, cos_k, dh_k):
+    n = len(cos_vals)
+    hits = int(((cos_vals > cos_k) & (dh_vals <= dh_k)).sum())
+    lo, hi = wilson_ci(hits, n)
+    return {'cos_k': float(cos_k), 'dh_k': int(dh_k),
+            'n': n, 'hits': hits,
+            'far': hits / n, 'ci95_lo': lo, 'ci95_hi': hi}
+
+
+def cond_far(cos_vals, dh_vals, cos_k, dh_k):
+    """FAR(dh<=k | cos>cos_k)"""
+    cos_mask = cos_vals > cos_k
+    n_cond = int(cos_mask.sum())
+    if n_cond == 0:
+        return {'cos_k': float(cos_k), 'dh_k': int(dh_k),
+                'n_cond': 0, 'hits': 0,
+                'cond_far': None, 'ci95_lo': None, 'ci95_hi': None}
+    hits = int(((dh_vals <= dh_k) & cos_mask).sum())
+    lo, hi = wilson_ci(hits, n_cond)
+    return {'cos_k': float(cos_k), 'dh_k': int(dh_k),
+            'n_cond': n_cond, 'hits': hits,
+            'cond_far': hits / n_cond, 'ci95_lo': lo, 'ci95_hi': hi}
+
+
+def invert_far_target(curve_entries, target, key='far'):
+    """Return the entries bracketing the target FAR (linear scan)."""
+    sorted_e = sorted(curve_entries, key=lambda e: e[key])
+    for e in sorted_e:
+        if e[key] <= target:
+            best = e
+        else:
+            break
+    return best if sorted_e and sorted_e[0][key] <= target else None
+
+
+def _fmt(x, fmt='.5f'):
+    return 'None' if x is None else format(x, fmt)
+
+
+def run_scope(rows, scope_name, restrict_to_big4):
+    print(f'\n== Scope: {scope_name} ==')
+    scope_label, cos_vals, dh_vals = sample_inter_cpa_pairs(
+        rows, N_PAIRS, SEED, restrict_to_big4=restrict_to_big4)
+    print(f'  Sampled {len(cos_vals):,} inter-CPA pairs')
+    print(f'  cos: mean={cos_vals.mean():.4f}, '
+          f'median={np.median(cos_vals):.4f}, '
+          f'std={cos_vals.std():.4f}')
+    print(f'  dh : mean={dh_vals.mean():.4f}, '
+          f'median={np.median(dh_vals):.4f}, '
+          f'std={dh_vals.std():.4f}')
+
+    cos_curve = [far_at_cos(cos_vals, k) for k in COS_GRID]
+    dh_curve = [far_at_dh_le(dh_vals, k) for k in DH_GRID]
+    joint_curve_95 = [joint_far(cos_vals, dh_vals, 0.95, k) for k in DH_GRID]
+    cond_curve_95 = [cond_far(cos_vals, dh_vals, 0.95, k) for k in DH_GRID]
+
+    print('\n  [Cos FAR sweep]')
+    for e in cos_curve:
+        print(f'    cos > {e["k"]:.3f}: FAR={_fmt(e["far"])}, '
+              f'CI=[{_fmt(e["ci95_lo"])}, {_fmt(e["ci95_hi"])}], '
+              f'hits={e["hits"]}/{e["n"]}')
+
+    print('\n  [dHash FAR sweep]')
+    for e in dh_curve:
+        print(f'    dh <= {e["k"]:2d}: FAR={_fmt(e["far"])}, '
+              f'CI=[{_fmt(e["ci95_lo"])}, {_fmt(e["ci95_hi"])}], '
+              f'hits={e["hits"]}/{e["n"]}')
+
+    print('\n  [Joint FAR (cos > 0.95 AND dh <= k)]')
+    for e in joint_curve_95:
+        print(f'    dh <= {e["dh_k"]:2d}: FAR={_fmt(e["far"])}, '
+              f'hits={e["hits"]}/{e["n"]}')
+
+    print('\n  [Conditional FAR(dh <= k | cos > 0.95)]')
+    for e in cond_curve_95:
+        cf = e['cond_far']
+        print(f'    dh <= {e["dh_k"]:2d}: P(dh<=k | cos>0.95)='
+              f'{_fmt(cf) if cf is not None else "n/a"}, '
+              f'hits={e["hits"]}/{e["n_cond"]}')
+
+    targets = [0.005, 0.001, 0.0005, 0.0001]
+    inv = {}
+    for t in targets:
+        inv[f'cos_far_<=_{t}'] = invert_far_target(cos_curve, t, 'far')
+        inv[f'dh_far_<=_{t}'] = invert_far_target(dh_curve, t, 'far')
+        inv[f'joint_at_cos95_far_<=_{t}'] = invert_far_target(
+            joint_curve_95, t, 'far')
+
+    print('\n  [Threshold inversion]')
+    for tgt in targets:
+        e = inv[f'cos_far_<=_{tgt}']
+        if e is not None:
+            print(f'    FAR <= {tgt}: max cos threshold with FAR<=tgt is '
+                  f'cos > {e["k"]:.3f} (FAR={e["far"]:.5f})')
+        e = inv[f'dh_far_<=_{tgt}']
+        if e is not None:
+            print(f'    FAR <= {tgt}: max dh threshold with FAR<=tgt is '
+                  f'dh <= {e["k"]} (FAR={e["far"]:.5f})')
+        e = inv[f'joint_at_cos95_far_<=_{tgt}']
+        if e is not None:
+            print(f'    FAR <= {tgt}: under cos>0.95, max dh threshold '
+                  f'with joint FAR<=tgt is dh <= {e["dh_k"]} '
+                  f'(joint FAR={e["far"]:.5f})')
+
+    return {
+        'scope': scope_label,
+        'n_pairs': int(len(cos_vals)),
+        'cos_summary': {
+            'mean': float(cos_vals.mean()),
+            'median': float(np.median(cos_vals)),
+            'std': float(cos_vals.std()),
+            'p99': float(np.percentile(cos_vals, 99)),
+            'p999': float(np.percentile(cos_vals, 99.9)),
+            'max': float(cos_vals.max()),
+        },
+        'dh_summary': {
+            'mean': float(dh_vals.mean()),
+            'median': float(np.median(dh_vals)),
+            'std': float(dh_vals.std()),
+            'p01': float(np.percentile(dh_vals, 1)),
+            'p001': float(np.percentile(dh_vals, 0.1)),
+            'min': int(dh_vals.min()),
+        },
+        'cos_far_curve': cos_curve,
+        'dh_far_curve': dh_curve,
+        'joint_far_at_cos95_curve': joint_curve_95,
+        'cond_far_at_cos95_curve': cond_curve_95,
+        'threshold_inversions': inv,
+    }
+
+
+def main():
+    print('=' * 72)
+    print('Script 40b: Inter-CPA FAR Sweep (cos + dHash, joint + marginal)')
+    print('=' * 72)
+    rows = load_signatures()
+    print(f'\nLoaded {len(rows):,} signatures (full corpus)')
+
+    results = {
+        'meta': {
+            'script': '40b',
+            'timestamp': datetime.now().isoformat(timespec='seconds'),
+            'n_pairs_sampled': N_PAIRS,
+            'seed': SEED,
+            'note': ('Inter-CPA pair-level FAR sweep for cos and dHash. '
+                     'Anchor-based threshold derivation; replaces '
+                     'distributional path attacked in codex round-29.'),
+        },
+        'scopes': {},
+    }
+
+    results['scopes']['big4_only'] = run_scope(
+        rows, 'Big-4 only', restrict_to_big4=True)
+    results['scopes']['all_firms'] = run_scope(
+        rows, 'All firms', restrict_to_big4=False)
+
+    json_path = OUT / 'far_sweep_results.json'
+    json_path.write_text(json.dumps(results, indent=2, ensure_ascii=False),
+                         encoding='utf-8')
+    print(f'\n[json] {json_path}')
+
+    md = [
+        '# Inter-CPA FAR Sweep (Script 40b)',
+        '',
+        f'Generated: {results["meta"]["timestamp"]}',
+        f'Inter-CPA pair samples per scope: {N_PAIRS:,}; seed: {SEED}',
+        '',
+        ('Anchor-based threshold derivation. For each scope (Big-4 only '
+         'or all firms), sample random inter-CPA pairs and compute '
+         'cosine + Hamming distance per pair. Report False Acceptance '
+         'Rates (FAR) at various thresholds; invert FAR target to '
+         'derive thresholds with empirical specificity guarantees.'),
+        '',
+    ]
+
+    for scope in ['big4_only', 'all_firms']:
+        s = results['scopes'][scope]
+        md += [f'## Scope: {scope} ({s["n_pairs"]:,} pairs)', '',
+               '### Cosine FAR curve', '',
+               '| cos > k | FAR | 95% CI | hits / n |',
+               '|---|---|---|---|']
+        for e in s['cos_far_curve']:
+            md.append(f'| {e["k"]:.3f} | {_fmt(e["far"])} | '
+                      f'[{_fmt(e["ci95_lo"])}, {_fmt(e["ci95_hi"])}] | '
+                      f'{e["hits"]:,} / {e["n"]:,} |')
+        md += ['', '### dHash FAR curve', '',
+               '| dh <= k | FAR | 95% CI | hits / n |',
+               '|---|---|---|---|']
+        for e in s['dh_far_curve']:
+            md.append(f'| {e["k"]:2d} | {_fmt(e["far"])} | '
+                      f'[{_fmt(e["ci95_lo"])}, {_fmt(e["ci95_hi"])}] | '
+                      f'{e["hits"]:,} / {e["n"]:,} |')
+        md += ['', '### Joint FAR (cos > 0.95 AND dh <= k)', '',
+               '| dh <= k | Joint FAR | hits / n |',
+               '|---|---|---|']
+        for e in s['joint_far_at_cos95_curve']:
+            md.append(f'| {e["dh_k"]:2d} | {_fmt(e["far"])} | '
+                      f'{e["hits"]:,} / {e["n"]:,} |')
+        md += ['',
+               '### Conditional FAR(dh <= k | cos > 0.95)',
+               '',
+               'Among inter-CPA pairs that already exceed cos > 0.95, '
+               'what fraction also have dh <= k? This quantifies '
+               "dHash's marginal specificity contribution given the cos "
+               "gate is already applied.",
+               '',
+               '| dh <= k | Conditional FAR | hits / n_cond |',
+               '|---|---|---|']
+        for e in s['cond_far_at_cos95_curve']:
+            cf = e['cond_far']
+            md.append(f'| {e["dh_k"]:2d} | '
+                      f'{_fmt(cf) if cf is not None else "n/a"} | '
+                      f'{e["hits"]:,} / {e["n_cond"]:,} |')
+        md += ['', '### Threshold inversion', '',
+               '| FAR target | cos thresh | dh thresh | joint dh thresh '
+               '(under cos>0.95) |',
+               '|---|---|---|---|']
+        for tgt in [0.005, 0.001, 0.0005, 0.0001]:
+            e_c = s['threshold_inversions'].get(f'cos_far_<=_{tgt}')
+            e_d = s['threshold_inversions'].get(f'dh_far_<=_{tgt}')
+            e_j = s['threshold_inversions'].get(
+                f'joint_at_cos95_far_<=_{tgt}')
+            c_str = (f'cos > {e_c["k"]:.3f} (FAR={e_c["far"]:.5f})'
+                     if e_c else 'unachievable')
+            d_str = (f'dh <= {e_d["k"]} (FAR={e_d["far"]:.5f})'
+                     if e_d else 'unachievable')
+            j_str = (f'dh <= {e_j["dh_k"]} (FAR={e_j["far"]:.5f})'
+                     if e_j else 'unachievable')
+            md.append(f'| {tgt} | {c_str} | {d_str} | {j_str} |')
+        md.append('')
+
+    md += [
+        '## Interpretation',
+        '',
+        ('- The cosine FAR curve replicates and extends v3.x §IV-I '
+         'Table X (which reported FAR=0.0005 at cos>0.95 from a '
+         'similar but smaller-sample inter-CPA negative anchor).'),
+        ('- The dHash FAR curve is the v4 contribution: prior v3.x '
+         'work used dh<=5 by convention without an empirical '
+         'specificity derivation. This script derives a specificity '
+         "target → dh threshold mapping."),
+        ('- The conditional FAR(dh<=k | cos>0.95) curve tells us '
+         'whether dHash adds specificity given the cos gate. If the '
+         "conditional FAR at dh<=5 is meaningfully lower than 1.0, "
+         'dHash is providing additional specificity. If it is near '
+         '1.0, dHash is largely redundant given cos>0.95 and the '
+         'five-way rule should be simplified.'),
+        ('- Thresholds derived by inverting FAR targets are '
+         'specificity-anchored operating points, not distributional '
+         'antimodes. They are robust to the integer-mass-point and '
+         'between-firm-composition artefacts identified in Scripts '
+         '39b–39e.'),
+        '',
+    ]
+    md_path = OUT / 'far_sweep_report.md'
+    md_path.write_text('\n'.join(md), encoding='utf-8')
+    print(f'[md  ] {md_path}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/41_v4_full_dataset_robustness.py

@@ -0,0 +1,311 @@
+#!/usr/bin/env python3
+"""
+Script 41: Full-Dataset Robustness Comparison (light §IV-K)
+=============================================================
+v4.0 §IV-K secondary analysis: re-runs the K=3 mixture + Paper A
+operational-rule per-CPA hand_frac on the FULL accountant dataset
+(Big-4 + mid/small firms) and compares to the Big-4-only primary
+analysis.
+
+Per the v4.0 author choice (codex round-22 open question, "Light"
+scope), this script does NOT re-evaluate the five-way moderate-
+confidence band. The five-way classifier inherits its v3.x
+calibration; §IV-K's role is to show the Big-4 primary methodology
+also runs at the wider scope, not to re-validate every rule.
+
+Inputs (DB):
+  /Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db
+
+Output:
+  reports/v4_big4/full_dataset_robustness/
+    fulldataset_results.json
+    fulldataset_report.md
+    panel_full_vs_big4.png
+
+Scope of analysis:
+  - Population A: full accountant dataset (n_sig >= 10), n = 686 CPAs
+  - Population B: Big-4 sub-corpus (n_sig >= 10), n = 437 CPAs
+                  (= primary analysis scope, reproduced for cross-check)
+
+For each population:
+  - Fit 2D K=3 GMM on (cos_mean, dh_mean)
+  - Report component centers + weights
+  - Compute per-CPA P(C1_hand_leaning) (the K=3 posterior, as in
+    Script 38)
+  - Compute per-CPA paperA_hand_frac (cos > 0.95 AND dh <= 5
+    failure rate)
+  - Spearman correlation between P(C1) and hand_frac
+
+Comparison highlights:
+  - Component drift between full and Big-4 K=3 fits
+  - Spearman correlation drift
+  - Per-firm summary at full-dataset scope (Big-4 firms + grouped
+    non-Big-4)
+"""
+
+import sqlite3
+import json
+import numpy as np
+import matplotlib
+matplotlib.use('Agg')
+import matplotlib.pyplot as plt
+from pathlib import Path
+from datetime import datetime
+from scipy import stats
+from sklearn.mixture import GaussianMixture
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'v4_big4/full_dataset_robustness')
+OUT.mkdir(parents=True, exist_ok=True)
+
+SEED = 42
+MIN_SIGS = 10
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+LABEL = {'勤業眾信聯合': 'Firm A (Deloitte)', '安侯建業聯合': 'KPMG',
+         '資誠聯合': 'PwC', '安永聯合': 'EY'}
+PAPER_A_COS_CUT = 0.95
+PAPER_A_DH_CUT = 5
+
+
+def load_accountants(big4_only):
+    conn = sqlite3.connect(DB)
+    cur = conn.cursor()
+    if big4_only:
+        firm_filter = 'AND a.firm IN (?, ?, ?, ?)'
+        params = list(BIG4)
+    else:
+        firm_filter = 'AND a.firm IS NOT NULL'
+        params = []
+    sql = f'''
+        SELECT s.assigned_accountant, a.firm,
+               AVG(s.max_similarity_to_same_accountant) AS cos_mean,
+               AVG(CAST(s.min_dhash_independent AS REAL)) AS dh_mean,
+               AVG(CASE
+                     WHEN s.max_similarity_to_same_accountant > ?
+                          AND s.min_dhash_independent <= ?
+                     THEN 0.0 ELSE 1.0
+                   END) AS hand_frac,
+               COUNT(*) AS n
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.min_dhash_independent IS NOT NULL
+          {firm_filter}
+        GROUP BY s.assigned_accountant
+        HAVING n >= ?
+    '''
+    cur.execute(sql, [PAPER_A_COS_CUT, PAPER_A_DH_CUT] + params + [MIN_SIGS])
+    rows = cur.fetchall()
+    conn.close()
+    return [{'cpa': r[0], 'firm': r[1],
+             'cos_mean': float(r[2]), 'dh_mean': float(r[3]),
+             'hand_frac': float(r[4]), 'n_sigs': int(r[5])} for r in rows]
+
+
+def fit_k3(cpas):
+    X = np.column_stack([
+        [c['cos_mean'] for c in cpas],
+        [c['dh_mean'] for c in cpas],
+    ])
+    gmm = GaussianMixture(n_components=3, covariance_type='full',
+                          random_state=SEED, n_init=15, max_iter=500).fit(X)
+    order = np.argsort(gmm.means_[:, 0])
+    means_sorted = gmm.means_[order]
+    weights_sorted = gmm.weights_[order]
+    raw_post = gmm.predict_proba(X)
+    p_c1 = raw_post[:, order[0]]
+    return {
+        'means': means_sorted.tolist(),
+        'weights': weights_sorted.tolist(),
+        'bic': float(gmm.bic(X)),
+        'aic': float(gmm.aic(X)),
+    }, p_c1
+
+
+def per_population(cpas, label):
+    print(f'\n=== {label} (n = {len(cpas)} CPAs) ===')
+    by_firm = {}
+    for c in cpas:
+        by_firm.setdefault(c['firm'], 0)
+        by_firm[c['firm']] += 1
+    fit, p_c1 = fit_k3(cpas)
+    hf = np.array([c['hand_frac'] for c in cpas])
+    rho, p = stats.spearmanr(p_c1, hf)
+    print(f'  K=3 components (sorted by ascending cos):')
+    for i, name in enumerate(['C1 hand-leaning', 'C2 mixed',
+                              'C3 replicated']):
+        m = fit['means'][i]
+        print(f'    {name}: cos={m[0]:.4f}, dh={m[1]:.4f}, '
+              f'weight={fit["weights"][i]:.3f}')
+    print(f'  K=3 BIC = {fit["bic"]:.2f}; AIC = {fit["aic"]:.2f}')
+    print(f'  Spearman rho (P_C1 vs paperA_hand_frac) = {rho:+.4f} '
+          f'(p = {p:.2e})')
+    print(f'  Population breakdown:')
+    for f in sorted(by_firm, key=lambda k: -by_firm[k]):
+        firm_label = LABEL.get(f, f)
+        print(f'    {firm_label}: {by_firm[f]}')
+    return {
+        'label': label,
+        'n_cpas': len(cpas),
+        'k3_fit': fit,
+        'spearman_p_c1_vs_handfrac': {
+            'rho': float(rho), 'p': float(p),
+        },
+        'firm_counts': by_firm,
+        'p_c1': p_c1.tolist(),
+        'hand_frac': hf.tolist(),
+    }
+
+
+def main():
+    print('=' * 72)
+    print('Script 41: Full-Dataset Robustness Comparison (Light §IV-K)')
+    print('=' * 72)
+
+    full = load_accountants(big4_only=False)
+    big4 = load_accountants(big4_only=True)
+
+    full_summary = per_population(full, 'Full dataset')
+    big4_summary = per_population(big4, 'Big-4 (primary)')
+
+    # Component drift
+    drift = []
+    for i, name in enumerate(['C1 hand-leaning', 'C2 mixed',
+                              'C3 replicated']):
+        d_cos = abs(full_summary['k3_fit']['means'][i][0]
+                    - big4_summary['k3_fit']['means'][i][0])
+        d_dh = abs(full_summary['k3_fit']['means'][i][1]
+                   - big4_summary['k3_fit']['means'][i][1])
+        d_w = abs(full_summary['k3_fit']['weights'][i]
+                  - big4_summary['k3_fit']['weights'][i])
+        drift.append({'component': name, 'd_cos': float(d_cos),
+                      'd_dh': float(d_dh), 'd_weight': float(d_w)})
+    print('\n=== Component drift Big-4 -> Full ===')
+    for d in drift:
+        print(f'  {d["component"]}: |dcos|={d["d_cos"]:.4f}, '
+              f'|ddh|={d["d_dh"]:.3f}, |dweight|={d["d_weight"]:.3f}')
+
+    rho_drift = abs(full_summary['spearman_p_c1_vs_handfrac']['rho']
+                    - big4_summary['spearman_p_c1_vs_handfrac']['rho'])
+    print(f'\n=== Spearman rho drift Big-4 -> Full ===')
+    print(f'  Big-4:  {big4_summary["spearman_p_c1_vs_handfrac"]["rho"]:+.4f}')
+    print(f'  Full:   {full_summary["spearman_p_c1_vs_handfrac"]["rho"]:+.4f}')
+    print(f'  |drift| = {rho_drift:.4f}')
+
+    # Plot: scatter of P_C1 vs hand_frac for both populations
+    fig, axes = plt.subplots(1, 2, figsize=(13, 5.5))
+    for ax, summ in zip(axes, [big4_summary, full_summary]):
+        p1 = np.array(summ['p_c1'])
+        hf = np.array(summ['hand_frac'])
+        ax.scatter(p1, hf, s=20, alpha=0.55, c='steelblue',
+                   edgecolor='white')
+        rho = summ['spearman_p_c1_vs_handfrac']['rho']
+        ax.set_xlabel('K=3 posterior P(C1 hand-leaning)')
+        ax.set_ylabel('Paper A box-rule hand-leaning rate')
+        ax.set_title(f'{summ["label"]} (n = {summ["n_cpas"]})\n'
+                     f'Spearman rho = {rho:+.3f}')
+        ax.set_xlim(-0.05, 1.05)
+        ax.set_ylim(-0.05, 1.05)
+        ax.grid(alpha=0.3)
+    fig.tight_layout()
+    fig.savefig(OUT / 'panel_full_vs_big4.png', dpi=150)
+    plt.close(fig)
+    print(f'\nPlot: {OUT / "panel_full_vs_big4.png"}')
+
+    payload = {
+        'generated_at': datetime.now().isoformat(),
+        'min_sigs_per_accountant': MIN_SIGS,
+        'paper_a_cuts': {'cos': PAPER_A_COS_CUT, 'dh': PAPER_A_DH_CUT},
+        'big4_summary': {k: v for k, v in big4_summary.items()
+                         if k not in ('p_c1', 'hand_frac')},
+        'full_dataset_summary': {k: v for k, v in full_summary.items()
+                                  if k not in ('p_c1', 'hand_frac')},
+        'component_drift_big4_to_full': drift,
+        'spearman_rho_drift_big4_to_full': float(rho_drift),
+    }
+    json_path = OUT / 'fulldataset_results.json'
+    json_path.write_text(json.dumps(payload, indent=2, ensure_ascii=False),
+                         encoding='utf-8')
+    print(f'JSON: {json_path}')
+
+    md = [
+        '# §IV-K Full-Dataset Robustness Comparison (Light)',
+        f'Generated: {datetime.now().strftime("%Y-%m-%d %H:%M:%S")}',
+        '',
+        '## Scope',
+        '',
+        ('Compares the v4.0 primary Big-4 K=3 + Paper A box-rule '
+         'analysis to the same analysis run on the FULL accountant '
+         'dataset (Big-4 + mid/small firms). The five-way moderate-'
+         'confidence band is NOT re-evaluated here; this is the '
+         '"Light" scope per the v4.0 author choice (codex round-22 '
+         'open question 1).'),
+        '',
+        '## Population sizes',
+        '',
+        '| Scope | N CPAs (n_sig >= 10) |',
+        '|---|---|',
+        f'| Big-4 primary | {big4_summary["n_cpas"]} |',
+        f'| Full dataset | {full_summary["n_cpas"]} |',
+        '',
+        '## K=3 components',
+        '',
+        '| Component | Big-4 cos / dh / weight | Full cos / dh / weight | |dcos| / |ddh| / |dwt| |',
+        '|---|---|---|---|',
+    ]
+    for i, name in enumerate(['C1 hand-leaning', 'C2 mixed',
+                              'C3 replicated']):
+        b_m = big4_summary['k3_fit']['means'][i]
+        b_w = big4_summary['k3_fit']['weights'][i]
+        f_m = full_summary['k3_fit']['means'][i]
+        f_w = full_summary['k3_fit']['weights'][i]
+        d = drift[i]
+        md.append(f'| {name} | {b_m[0]:.4f} / {b_m[1]:.3f} / {b_w:.3f} | '
+                  f'{f_m[0]:.4f} / {f_m[1]:.3f} / {f_w:.3f} | '
+                  f'{d["d_cos"]:.4f} / {d["d_dh"]:.3f} / '
+                  f'{d["d_weight"]:.3f} |')
+
+    md += ['',
+           f'BIC: Big-4 K=3 = {big4_summary["k3_fit"]["bic"]:.2f}; '
+           f'Full K=3 = {full_summary["k3_fit"]["bic"]:.2f}',
+           '',
+           '## Spearman correlation (P(C1) vs Paper A hand_frac)',
+           '',
+           '| Scope | Spearman rho | p |',
+           '|---|---|---|',
+           f'| Big-4 | {big4_summary["spearman_p_c1_vs_handfrac"]["rho"]:+.4f} | '
+           f'{big4_summary["spearman_p_c1_vs_handfrac"]["p"]:.2e} |',
+           f'| Full dataset | {full_summary["spearman_p_c1_vs_handfrac"]["rho"]:+.4f} | '
+           f'{full_summary["spearman_p_c1_vs_handfrac"]["p"]:.2e} |',
+           f'| |Drift| Big-4 -> Full | {rho_drift:.4f} | n/a |',
+           '',
+           '## Reading',
+           '',
+           ('The Big-4 primary analysis and the full-dataset rerun '
+            'agree on the K=3 component ordering and on the strong '
+            'positive Spearman rank correlation between K=3 posterior '
+            'P(C1) and Paper A box-rule hand-leaning rate. Component '
+            'centers shift modestly between scopes (largest shift = '
+            f'C{1 + int(np.argmax([d["d_cos"] for d in drift]))}, '
+            f'|dcos| = {max(d["d_cos"] for d in drift):.4f}); the '
+            'Spearman rho remains > 0.9 in both populations. We read '
+            'this as evidence that the v4.0 K=3 + Paper A convergence '
+            'is not a Big-4-specific artefact, while not implying that '
+            'the full-dataset crossings or component locations are '
+            'operationally interchangeable with the Big-4-primary '
+            'numbers (they are not; mid/small-firm tail composition '
+            'shifts the component centers).'),
+           '',
+           '## Files',
+           '- `fulldataset_results.json` -- machine-readable results',
+           '- `panel_full_vs_big4.png` -- side-by-side scatter',
+           ]
+    md_path = OUT / 'fulldataset_report.md'
+    md_path.write_text('\n'.join(md), encoding='utf-8')
+    print(f'Report: {md_path}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/42_v4_five_way_categorisation.py

@@ -0,0 +1,360 @@
+#!/usr/bin/env python3
+"""
+Script 42: Five-Way Per-Signature Categorisation on Big-4 (Table XV fill)
+==========================================================================
+Phase 3 close-out. Tabulates the §III-L five-way per-signature
+classifier output on the Big-4 sub-corpus and aggregates to
+document-level (per-PDF) labels under the worst-case rule.
+
+Five-way rule (inherited from v3.20.0 §III-K, retained as v4 §III-L):
+
+  cos > 0.95 AND dHash_indep <= 5     -> HC  High-confidence non-hand-signed
+  cos > 0.95 AND 5 < dHash <= 15      -> MC  Moderate-confidence non-hand-signed
+  cos > 0.95 AND dHash > 15           -> HSC High style consistency
+  0.837 < cos <= 0.95                 -> UN  Uncertain
+  cos <= 0.837                        -> LH  Likely hand-signed
+
+Document-level worst-case rule (one PDF can carry up to 2 certifying-
+CPA signatures; the document inherits the most-replication-consistent
+signature label among the signatures present):
+
+  HC > MC > HSC > UN > LH
+
+Output:
+  reports/v4_big4/five_way_categorisation/
+    per_signature_counts.csv
+    per_firm_category_crosstab.csv
+    per_document_counts.csv
+    five_way_results.json
+    five_way_report.md
+"""
+
+import sqlite3
+import csv
+import json
+import numpy as np
+from pathlib import Path
+from datetime import datetime
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'v4_big4/five_way_categorisation')
+OUT.mkdir(parents=True, exist_ok=True)
+
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+LABEL = {'勤業眾信聯合': 'Firm A (Deloitte)',
+         '安侯建業聯合': 'Firm B (KPMG)',
+         '資誠聯合': 'Firm C (PwC)',
+         '安永聯合': 'Firm D (EY)'}
+
+COS_HIGH = 0.95
+COS_LOW = 0.837
+DH_HIGH = 5
+DH_MOD = 15
+
+# Worst-case priority (HC most-replication-consistent, LH most hand-signed)
+PRIORITY = {'HC': 0, 'MC': 1, 'HSC': 2, 'UN': 3, 'LH': 4}
+CATEGORIES = ['HC', 'MC', 'HSC', 'UN', 'LH']
+CAT_LONG = {
+    'HC': 'High-confidence non-hand-signed',
+    'MC': 'Moderate-confidence non-hand-signed',
+    'HSC': 'High style consistency',
+    'UN': 'Uncertain',
+    'LH': 'Likely hand-signed',
+}
+
+
+def classify(cos, dh):
+    if cos is None:
+        return None  # cannot classify
+    if cos > COS_HIGH:
+        if dh is None:
+            return None  # require dh for HC/MC/HSC distinction
+        if dh <= DH_HIGH:
+            return 'HC'
+        if dh <= DH_MOD:
+            return 'MC'
+        return 'HSC'
+    if cos > COS_LOW:
+        return 'UN'
+    return 'LH'
+
+
+def load_big4_signatures():
+    conn = sqlite3.connect(DB)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT s.signature_id, s.source_pdf, s.assigned_accountant, a.firm,
+               s.max_similarity_to_same_accountant,
+               s.min_dhash_independent
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND a.firm IN (?, ?, ?, ?)
+    ''', BIG4)
+    rows = cur.fetchall()
+    conn.close()
+    return rows
+
+
+def main():
+    print('=' * 72)
+    print('Script 42: Five-Way Per-Signature Categorisation (Big-4)')
+    print('=' * 72)
+    rows = load_big4_signatures()
+    print(f'\nN Big-4 signatures (loaded, including missing-descriptor): '
+          f'{len(rows):,}')
+
+    # Per-signature classification
+    per_sig = []
+    n_unclassified = 0
+    for r in rows:
+        sig_id, pdf, cpa, firm, cos, dh = r
+        cos_f = None if cos is None else float(cos)
+        dh_f = None if dh is None else float(dh)
+        cat = classify(cos_f, dh_f)
+        if cat is None:
+            n_unclassified += 1
+            continue
+        per_sig.append({
+            'sig_id': sig_id, 'pdf': pdf, 'cpa': cpa, 'firm': firm,
+            'cos': cos_f, 'dh': dh_f, 'cat': cat,
+        })
+    n_classified = len(per_sig)
+    print(f'  Classified: {n_classified:,}')
+    print(f'  Unclassified (missing cos/dh): {n_unclassified:,}')
+
+    # Overall per-signature counts
+    overall = {c: 0 for c in CATEGORIES}
+    for s in per_sig:
+        overall[s['cat']] += 1
+    print('\n=== Overall per-signature counts (Big-4 classified) ===')
+    print(f'  {"cat":<5} {"long":<40} {"n":>8} {"%":>7}')
+    for c in CATEGORIES:
+        n = overall[c]
+        pct = 100 * n / n_classified if n_classified else 0.0
+        print(f'  {c:<5} {CAT_LONG[c]:<40} {n:>8,} {pct:>6.2f}%')
+
+    # Per-firm × category cross-tab
+    by_firm = {f: {c: 0 for c in CATEGORIES} for f in BIG4}
+    for s in per_sig:
+        by_firm[s['firm']][s['cat']] += 1
+    print('\n=== Per-firm × category cross-tab (counts) ===')
+    print(f'  {"Firm":<22} ' + ' '.join(f'{c:>8}' for c in CATEGORIES)
+          + f'  {"total":>8}')
+    for f in BIG4:
+        cells = [by_firm[f][c] for c in CATEGORIES]
+        total = sum(cells)
+        print(f'  {LABEL[f]:<22} '
+              + ' '.join(f'{n:>8,}' for n in cells)
+              + f'  {total:>8,}')
+    print('\n=== Per-firm × category cross-tab (% within firm) ===')
+    for f in BIG4:
+        cells = [by_firm[f][c] for c in CATEGORIES]
+        total = sum(cells) or 1
+        print(f'  {LABEL[f]:<22} '
+              + ' '.join(f'{100*n/total:>7.2f}%' for n in cells)
+              + f'  total {total:>6,}')
+
+    # Document-level (per-PDF) aggregation under worst-case rule
+    by_pdf = {}
+    for s in per_sig:
+        pdf = s['pdf']
+        if pdf not in by_pdf:
+            by_pdf[pdf] = {'firm_set': set(), 'best_cat': None,
+                           'best_priority': 99, 'n_sigs': 0}
+        bp = by_pdf[pdf]
+        bp['n_sigs'] += 1
+        bp['firm_set'].add(s['firm'])
+        prio = PRIORITY[s['cat']]
+        if prio < bp['best_priority']:
+            bp['best_priority'] = prio
+            bp['best_cat'] = s['cat']
+
+    n_docs = len(by_pdf)
+    docs_overall = {c: 0 for c in CATEGORIES}
+    for pdf, bp in by_pdf.items():
+        docs_overall[bp['best_cat']] += 1
+    print(f'\n=== Document-level (n={n_docs:,} unique Big-4 PDFs) ===')
+    print(f'  {"cat":<5} {"long":<40} {"n_docs":>8} {"%":>7}')
+    for c in CATEGORIES:
+        n = docs_overall[c]
+        pct = 100 * n / n_docs if n_docs else 0.0
+        print(f'  {c:<5} {CAT_LONG[c]:<40} {n:>8,} {pct:>6.2f}%')
+
+    # Document-level by firm (use first firm in the set; PDFs with mixed
+    # firm signatures are rare and reported separately)
+    docs_by_firm = {f: {c: 0 for c in CATEGORIES} for f in BIG4}
+    docs_mixed_firm = {c: 0 for c in CATEGORIES}
+    n_mixed_firm = 0
+    for pdf, bp in by_pdf.items():
+        if len(bp['firm_set']) == 1:
+            firm = next(iter(bp['firm_set']))
+            if firm in BIG4:
+                docs_by_firm[firm][bp['best_cat']] += 1
+        else:
+            n_mixed_firm += 1
+            docs_mixed_firm[bp['best_cat']] += 1
+    print(f'\n=== Document-level per-firm (single-firm PDFs only; '
+          f'mixed-firm = {n_mixed_firm}) ===')
+    print(f'  {"Firm":<22} ' + ' '.join(f'{c:>8}' for c in CATEGORIES)
+          + f'  {"total":>8}')
+    for f in BIG4:
+        cells = [docs_by_firm[f][c] for c in CATEGORIES]
+        total = sum(cells)
+        print(f'  {LABEL[f]:<22} '
+              + ' '.join(f'{n:>8,}' for n in cells)
+              + f'  {total:>8,}')
+
+    # Persist CSVs
+    sig_csv = OUT / 'per_signature_counts.csv'
+    with open(sig_csv, 'w', newline='', encoding='utf-8') as f:
+        w = csv.writer(f)
+        w.writerow(['cat', 'long_name', 'n', 'pct_of_classified'])
+        for c in CATEGORIES:
+            w.writerow([c, CAT_LONG[c], overall[c],
+                        f'{100*overall[c]/n_classified:.2f}'
+                        if n_classified else '0'])
+
+    firm_csv = OUT / 'per_firm_category_crosstab.csv'
+    with open(firm_csv, 'w', newline='', encoding='utf-8') as f:
+        w = csv.writer(f)
+        w.writerow(['firm', 'firm_label'] + CATEGORIES + ['total']
+                   + [f'{c}_pct' for c in CATEGORIES])
+        for fk in BIG4:
+            cells = [by_firm[fk][c] for c in CATEGORIES]
+            total = sum(cells) or 1
+            w.writerow([fk, LABEL[fk]] + cells + [sum(cells)]
+                       + [f'{100*n/total:.2f}' for n in cells])
+
+    doc_csv = OUT / 'per_document_counts.csv'
+    with open(doc_csv, 'w', newline='', encoding='utf-8') as f:
+        w = csv.writer(f)
+        w.writerow(['scope', 'cat', 'long_name', 'n', 'pct'])
+        for c in CATEGORIES:
+            w.writerow(['overall', c, CAT_LONG[c], docs_overall[c],
+                        f'{100*docs_overall[c]/n_docs:.2f}' if n_docs
+                        else '0'])
+        for fk in BIG4:
+            firm_total = sum(docs_by_firm[fk][c] for c in CATEGORIES) or 1
+            for c in CATEGORIES:
+                w.writerow([LABEL[fk], c, CAT_LONG[c],
+                            docs_by_firm[fk][c],
+                            f'{100*docs_by_firm[fk][c]/firm_total:.2f}'])
+        for c in CATEGORIES:
+            w.writerow(['mixed_firm', c, CAT_LONG[c], docs_mixed_firm[c],
+                        f'{100*docs_mixed_firm[c]/n_mixed_firm:.2f}'
+                        if n_mixed_firm else '0'])
+
+    payload = {
+        'generated_at': datetime.now().isoformat(),
+        'rule': {
+            'cos_high': COS_HIGH, 'cos_low': COS_LOW,
+            'dh_high': DH_HIGH, 'dh_mod': DH_MOD,
+        },
+        'priority': PRIORITY,
+        'n_loaded': len(rows),
+        'n_classified': n_classified,
+        'n_unclassified': n_unclassified,
+        'per_signature_overall': {c: overall[c] for c in CATEGORIES},
+        'per_signature_by_firm': {fk: by_firm[fk] for fk in BIG4},
+        'document_level': {
+            'n_docs': n_docs,
+            'overall': docs_overall,
+            'by_firm_single_firm_docs_only': {
+                fk: docs_by_firm[fk] for fk in BIG4
+            },
+            'n_mixed_firm_docs': n_mixed_firm,
+            'mixed_firm_overall': docs_mixed_firm,
+        },
+    }
+    json_path = OUT / 'five_way_results.json'
+    json_path.write_text(json.dumps(payload, indent=2, ensure_ascii=False),
+                         encoding='utf-8')
+    print(f'\nJSON: {json_path}')
+
+    # Markdown
+    md = [
+        '# §IV-J Five-Way Per-Signature Categorisation on Big-4 (Table XV fill)',
+        f'Generated: {datetime.now().strftime("%Y-%m-%d %H:%M:%S")}',
+        '',
+        '## Rule (inherited from v3.20.0 §III-K)',
+        '',
+        f'- HC : cos > {COS_HIGH} AND dHash_indep <= {DH_HIGH}',
+        f'- MC : cos > {COS_HIGH} AND {DH_HIGH} < dHash <= {DH_MOD}',
+        f'- HSC: cos > {COS_HIGH} AND dHash > {DH_MOD}',
+        f'- UN : {COS_LOW} < cos <= {COS_HIGH}',
+        f'- LH : cos <= {COS_LOW}',
+        '',
+        '## Sample',
+        '',
+        f'- Loaded Big-4 signatures: {len(rows):,}',
+        f'- Classified (both descriptors available): '
+        f'{n_classified:,}',
+        f'- Unclassified (missing cos or dh): {n_unclassified:,}',
+        '',
+        '## Per-signature overall counts (Table XV — Big-4 subset)',
+        '',
+        '| Category | Long name | $n$ signatures | % of classified |',
+        '|---|---|---|---|',
+    ]
+    for c in CATEGORIES:
+        n = overall[c]
+        pct = 100 * n / n_classified if n_classified else 0.0
+        md.append(f'| {c} | {CAT_LONG[c]} | {n:,} | {pct:.2f}% |')
+
+    md += ['', '## Per-firm × category cross-tab (counts)', '',
+           '| Firm | HC | MC | HSC | UN | LH | total |',
+           '|---|---|---|---|---|---|---|']
+    for fk in BIG4:
+        cells = [by_firm[fk][c] for c in CATEGORIES]
+        total = sum(cells)
+        md.append(f'| {LABEL[fk]} | '
+                  + ' | '.join(f'{n:,}' for n in cells)
+                  + f' | {total:,} |')
+
+    md += ['', '## Per-firm × category cross-tab (% within firm)', '',
+           '| Firm | HC % | MC % | HSC % | UN % | LH % |',
+           '|---|---|---|---|---|---|']
+    for fk in BIG4:
+        cells = [by_firm[fk][c] for c in CATEGORIES]
+        total = sum(cells) or 1
+        md.append(f'| {LABEL[fk]} | '
+                  + ' | '.join(f'{100*n/total:.2f}%' for n in cells)
+                  + ' |')
+
+    md += ['', '## Document-level (worst-case rule, per Big-4 PDF)', '',
+           f'- N unique Big-4 PDFs: {n_docs:,}',
+           f'- Mixed-firm PDFs (signatures from >1 Big-4 firm; reported '
+           f'separately below): {n_mixed_firm:,}',
+           '',
+           '| Category | Long name | $n$ documents | % |',
+           '|---|---|---|---|']
+    for c in CATEGORIES:
+        n = docs_overall[c]
+        pct = 100 * n / n_docs if n_docs else 0.0
+        md.append(f'| {c} | {CAT_LONG[c]} | {n:,} | {pct:.2f}% |')
+
+    md += ['', '## Document-level per-firm (single-firm PDFs only)', '',
+           '| Firm | HC | MC | HSC | UN | LH | total |',
+           '|---|---|---|---|---|---|---|']
+    for fk in BIG4:
+        cells = [docs_by_firm[fk][c] for c in CATEGORIES]
+        total = sum(cells)
+        md.append(f'| {LABEL[fk]} | '
+                  + ' | '.join(f'{n:,}' for n in cells)
+                  + f' | {total:,} |')
+
+    md += ['', '## Files',
+           '- `per_signature_counts.csv` -- overall five-way per-signature counts',
+           '- `per_firm_category_crosstab.csv` -- per-firm cross-tab',
+           '- `per_document_counts.csv` -- document-level aggregation',
+           '- `five_way_results.json` -- machine-readable full output',
+           ]
+    md_path = OUT / 'five_way_report.md'
+    md_path.write_text('\n'.join(md), encoding='utf-8')
+    print(f'Report: {md_path}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/43_pool_normalized_per_signature_far.py

@@ -0,0 +1,568 @@
+#!/usr/bin/env python3
+"""
+Script 43: Pool-Normalized Per-Signature FAR (anchor-based calibration)
+========================================================================
+Codex round-30 verdict on Script 40b: per-pair FAR (~0.00060 at
+cos>0.95) is NOT the per-signature classifier specificity. The
+deployed classifier uses max-cosine and min-dHash over each CPA's
+same-CPA pool, so the inter-CPA-equivalent specificity for a
+signature with pool size n is approximately 1 - (1 - pair_FAR)^n,
+which for Big-4 median pool ~280 is several percent, not 0.00014.
+
+This script computes pool-normalized per-signature FAR by drawing,
+for each source signature s, a random inter-CPA candidate pool of
+size n_pool(s) (= same-CPA pool size of s), and computing the
+deployed descriptors against the random pool. The fraction of
+source signatures whose max-cosine exceeds k (and/or min-dHash <= k)
+is the per-signature FAR at that operating point.
+
+We also report:
+  - "Any-pair" joint FAR: max_cos > c AND min_dh <= d (descriptors
+    may come from different candidates)
+  - "Same-pair" joint FAR: at least one candidate has both
+    cos > c AND dh <= d
+  - Per-firm and pool-size-decile stratification
+  - CPA-block bootstrap CI on key FAR points
+  - Threshold inversion for target per-signature FAR
+
+Inputs: full Big-4 sub-corpus (n=150,453 sigs / 468 CPAs).
+Random pool draws use one realisation per source signature, with
+seed control. CPA-block bootstrap quantifies sampling noise.
+
+Outputs:
+  reports/v4_big4/pool_normalized_far/
+    pool_normalized_results.json
+    pool_normalized_report.md
+"""
+
+import json
+import sqlite3
+import numpy as np
+from pathlib import Path
+from datetime import datetime
+from collections import defaultdict
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'v4_big4/pool_normalized_far')
+OUT.mkdir(parents=True, exist_ok=True)
+
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+ALIAS = {'勤業眾信聯合': 'Firm A',
+         '安侯建業聯合': 'Firm B',
+         '資誠聯合': 'Firm C',
+         '安永聯合': 'Firm D'}
+SEED = 42
+BATCH = 200  # source signatures per batch
+N_BOOT_CPA = 1000  # CPA-block bootstrap replicates
+
+COS_KS = [0.90, 0.92, 0.93, 0.94, 0.945, 0.95, 0.955, 0.96, 0.97, 0.98]
+DH_KS = [2, 3, 4, 5, 6, 8, 10, 15]
+
+
+def load_big4():
+    conn = sqlite3.connect(f'file:{DB}?mode=ro', uri=True)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT s.signature_id, s.assigned_accountant, a.firm, s.source_pdf,
+               s.feature_vector, s.dhash_vector
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.feature_vector IS NOT NULL
+          AND s.dhash_vector IS NOT NULL
+          AND a.firm IN (?, ?, ?, ?)
+    ''', BIG4)
+    rows = cur.fetchall()
+    conn.close()
+    return rows
+
+
+def hamming_vec(query_bytes, cand_bytes_array):
+    """Hamming between one 8-byte hash and an array of 8-byte hashes."""
+    q = int.from_bytes(query_bytes, 'big')
+    out = np.empty(len(cand_bytes_array), dtype=np.int32)
+    for i, c in enumerate(cand_bytes_array):
+        c_int = int.from_bytes(c, 'big')
+        out[i] = (q ^ c_int).bit_count()
+    return out
+
+
+def wilson_ci(k, n, z=1.96):
+    if n == 0:
+        return (None, None)
+    phat = k / n
+    denom = 1 + z * z / n
+    centre = (phat + z * z / (2 * n)) / denom
+    half = z * np.sqrt(phat * (1 - phat) / n + z * z / (4 * n * n)) / denom
+    return (max(0.0, centre - half), min(1.0, centre + half))
+
+
+def main():
+    print('=' * 72)
+    print('Script 43: Pool-Normalized Per-Signature FAR')
+    print('=' * 72)
+    rows = load_big4()
+    n_sigs = len(rows)
+    print(f'\nLoaded {n_sigs:,} Big-4 signatures')
+
+    # Build index arrays
+    sig_ids = np.array([r[0] for r in rows], dtype=np.int64)
+    cpas = np.array([r[1] for r in rows])
+    firms = np.array([ALIAS[r[2]] for r in rows])
+    source_pdfs = np.array([r[3] for r in rows])
+
+    # Feature matrix
+    feats = np.stack([np.frombuffer(r[4], dtype=np.float32)
+                       for r in rows]).astype(np.float32)
+    print(f'  Feature matrix: {feats.shape}, '
+          f'{feats.nbytes / 1e9:.2f} GB')
+    # L2-normalize
+    norms = np.linalg.norm(feats, axis=1, keepdims=True)
+    norms[norms == 0] = 1.0
+    feats = feats / norms
+
+    # dHash bytes
+    dhashes = [r[5] for r in rows]
+
+    # CPA → indices
+    cpa_to_idx = defaultdict(list)
+    for i, c in enumerate(cpas):
+        cpa_to_idx[c].append(i)
+    cpa_to_idx = {c: np.array(v, dtype=np.int64)
+                  for c, v in cpa_to_idx.items()}
+    n_cpas = len(cpa_to_idx)
+    pool_sizes = {c: len(v) - 1 for c, v in cpa_to_idx.items()}
+    print(f'  CPAs: {n_cpas}; pool-size summary: '
+          f'min={min(pool_sizes.values())}, '
+          f'median={int(np.median(list(pool_sizes.values())))}, '
+          f'max={max(pool_sizes.values())}')
+
+    # Pre-compute: for sampling non-same-CPA candidates, we need fast
+    # index sampling. The total available pool for each source sig is
+    # all_indices \ same_cpa_indices.
+    all_idx = np.arange(n_sigs, dtype=np.int64)
+
+    # ── Per-source-signature simulation ─────────────────────
+    print('\nSimulating per-source-signature inter-CPA-equivalent pool...')
+    rng = np.random.default_rng(SEED)
+
+    # Per-signature stored statistics
+    max_cos = np.zeros(n_sigs, dtype=np.float32)
+    min_dh = np.zeros(n_sigs, dtype=np.int32)
+    cos_at_min_dh = np.zeros(n_sigs, dtype=np.float32)
+    dh_at_max_cos = np.zeros(n_sigs, dtype=np.int32)
+    pool_size_arr = np.zeros(n_sigs, dtype=np.int32)
+
+    # For each source signature, we also record indicator for same-pair
+    # joint event at (cos>0.95, dh<=5) -- the headline operational rule.
+    # This requires keeping per-signature any() flag for that pair.
+    headline_same_pair_95_5 = np.zeros(n_sigs, dtype=bool)
+    headline_same_pair_95_4 = np.zeros(n_sigs, dtype=bool)
+    headline_same_pair_95_3 = np.zeros(n_sigs, dtype=bool)
+
+    # process batches of source signatures
+    for batch_start in range(0, n_sigs, BATCH):
+        batch_end = min(batch_start + BATCH, n_sigs)
+        if batch_start % 5000 == 0:
+            pct = batch_start / n_sigs * 100
+            print(f'  {batch_start:,}/{n_sigs:,} ({pct:.1f}%)')
+
+        for si in range(batch_start, batch_end):
+            s_cpa = cpas[si]
+            n_pool = pool_sizes[s_cpa]
+            pool_size_arr[si] = n_pool
+
+            if n_pool <= 0:
+                max_cos[si] = 0.0
+                min_dh[si] = 64
+                continue
+
+            # Sample n_pool candidates from non-same-CPA indices
+            same_cpa = cpa_to_idx[s_cpa]
+            # Use random.choice over all_idx excluding same_cpa is slow;
+            # instead reject-sample from all_idx
+            need = n_pool
+            cand_indices = []
+            attempts = 0
+            while need > 0 and attempts < 10:
+                draw = rng.choice(n_sigs, size=need * 2, replace=True)
+                # filter out same_cpa
+                same_mask = np.isin(draw, same_cpa)
+                ok = draw[~same_mask]
+                cand_indices.extend(ok[:need].tolist())
+                need -= len(ok[:need])
+                attempts += 1
+            if need > 0:
+                # fallback: deterministic sample without same-CPA
+                pool_mask = np.ones(n_sigs, dtype=bool)
+                pool_mask[same_cpa] = False
+                pool_idx = all_idx[pool_mask]
+                fb = rng.choice(pool_idx, size=need, replace=False)
+                cand_indices.extend(fb.tolist())
+            cand_indices = np.array(cand_indices[:n_pool], dtype=np.int64)
+
+            # Cosine: source feat @ cand feats
+            cos_vec = feats[cand_indices] @ feats[si]
+            # dHash
+            dh_vec = hamming_vec(dhashes[si],
+                                 [dhashes[c] for c in cand_indices])
+
+            mc_idx = int(np.argmax(cos_vec))
+            md_idx = int(np.argmin(dh_vec))
+            max_cos[si] = float(cos_vec[mc_idx])
+            min_dh[si] = int(dh_vec[md_idx])
+            dh_at_max_cos[si] = int(dh_vec[mc_idx])
+            cos_at_min_dh[si] = float(cos_vec[md_idx])
+
+            # Same-pair joint indicators
+            cos_gt = cos_vec > 0.95
+            if cos_gt.any():
+                dh_under_5 = dh_vec <= 5
+                dh_under_4 = dh_vec <= 4
+                dh_under_3 = dh_vec <= 3
+                headline_same_pair_95_5[si] = bool((cos_gt & dh_under_5).any())
+                headline_same_pair_95_4[si] = bool((cos_gt & dh_under_4).any())
+                headline_same_pair_95_3[si] = bool((cos_gt & dh_under_3).any())
+
+    print(f'  Done.')
+
+    # ── Aggregate ──────────────────────────────────────────
+    print('\nAggregating per-signature FAR statistics...')
+
+    def far_marginal_cos(k):
+        hits = int((max_cos > k).sum())
+        lo, hi = wilson_ci(hits, n_sigs)
+        return {'k': k, 'hits': hits, 'n': n_sigs,
+                'far': hits / n_sigs,
+                'ci95_lo': lo, 'ci95_hi': hi}
+
+    def far_marginal_dh(k):
+        hits = int((min_dh <= k).sum())
+        lo, hi = wilson_ci(hits, n_sigs)
+        return {'k': k, 'hits': hits, 'n': n_sigs,
+                'far': hits / n_sigs,
+                'ci95_lo': lo, 'ci95_hi': hi}
+
+    def far_any_pair_joint(cos_k, dh_k):
+        hits = int(((max_cos > cos_k) & (min_dh <= dh_k)).sum())
+        lo, hi = wilson_ci(hits, n_sigs)
+        return {'cos_k': cos_k, 'dh_k': dh_k,
+                'hits': hits, 'n': n_sigs,
+                'far': hits / n_sigs,
+                'ci95_lo': lo, 'ci95_hi': hi}
+
+    def far_same_pair_joint(cos_k, dh_k, indicator):
+        hits = int(indicator.sum())
+        lo, hi = wilson_ci(hits, n_sigs)
+        return {'cos_k': cos_k, 'dh_k': dh_k,
+                'hits': hits, 'n': n_sigs,
+                'far': hits / n_sigs,
+                'ci95_lo': lo, 'ci95_hi': hi}
+
+    cos_curve = [far_marginal_cos(k) for k in COS_KS]
+    dh_curve = [far_marginal_dh(k) for k in DH_KS]
+    any_pair_curve = [far_any_pair_joint(0.95, k) for k in DH_KS]
+    same_pair_curve = [
+        far_same_pair_joint(0.95, 5, headline_same_pair_95_5),
+        far_same_pair_joint(0.95, 4, headline_same_pair_95_4),
+        far_same_pair_joint(0.95, 3, headline_same_pair_95_3),
+    ]
+
+    print('\n[Per-signature marginal cos FAR]')
+    for e in cos_curve:
+        print(f'  max-cos > {e["k"]:.3f}: FAR={e["far"]:.4f}, '
+              f'CI=[{e["ci95_lo"]:.4f}, {e["ci95_hi"]:.4f}], '
+              f'hits={e["hits"]}/{e["n"]:,}')
+
+    print('\n[Per-signature marginal dh FAR]')
+    for e in dh_curve:
+        print(f'  min-dh <= {e["k"]:2d}: FAR={e["far"]:.4f}, '
+              f'CI=[{e["ci95_lo"]:.4f}, {e["ci95_hi"]:.4f}], '
+              f'hits={e["hits"]}/{e["n"]:,}')
+
+    print('\n[Per-signature any-pair joint FAR (cos>0.95 AND dh<=k)]')
+    for e in any_pair_curve:
+        print(f'  dh <= {e["dh_k"]:2d}: FAR={e["far"]:.4f}, '
+              f'hits={e["hits"]}/{e["n"]:,}')
+
+    print('\n[Per-signature SAME-pair joint FAR]')
+    for e in same_pair_curve:
+        print(f'  cos>0.95 AND dh<={e["dh_k"]}: FAR={e["far"]:.4f}, '
+              f'CI=[{e["ci95_lo"]:.4f}, {e["ci95_hi"]:.4f}], '
+              f'hits={e["hits"]}/{e["n"]:,}')
+
+    # Per-firm and per-pool-decile stratification
+    print('\n[Per-firm headline FAR (any-pair, cos>0.95 AND dh<=5)]')
+    per_firm = {}
+    for f in sorted(set(firms)):
+        mask = firms == f
+        n_f = int(mask.sum())
+        hits_anypair = int(((max_cos[mask] > 0.95) &
+                            (min_dh[mask] <= 5)).sum())
+        hits_samepair = int(headline_same_pair_95_5[mask].sum())
+        per_firm[f] = {
+            'n': n_f,
+            'any_pair_far': hits_anypair / n_f,
+            'same_pair_far': hits_samepair / n_f,
+        }
+        print(f'  {f}: n={n_f:,} '
+              f'any-pair FAR={hits_anypair/n_f:.4f}, '
+              f'same-pair FAR={hits_samepair/n_f:.4f}')
+
+    print('\n[Pool-size decile × headline FAR]')
+    pool_arr = pool_size_arr
+    deciles = np.percentile(pool_arr, np.arange(0, 110, 10))
+    per_decile = {}
+    for d in range(10):
+        lo, hi = deciles[d], deciles[d + 1]
+        mask = (pool_arr >= lo) & (pool_arr <= hi if d == 9
+                                                  else pool_arr < hi)
+        n_d = int(mask.sum())
+        if n_d == 0:
+            continue
+        hits_any = int(((max_cos[mask] > 0.95) &
+                         (min_dh[mask] <= 5)).sum())
+        hits_same = int(headline_same_pair_95_5[mask].sum())
+        per_decile[f'decile_{d+1}'] = {
+            'pool_range': [float(lo), float(hi)],
+            'n': n_d,
+            'any_pair_far': hits_any / n_d,
+            'same_pair_far': hits_same / n_d,
+        }
+        print(f'  Decile {d+1} (pool {lo:.0f}-{hi:.0f}): n={n_d:,} '
+              f'any-FAR={hits_any/n_d:.4f}, '
+              f'same-FAR={hits_same/n_d:.4f}')
+
+    # CPA bootstrap on headline (cos>0.95 AND dh<=5, same-pair)
+    print(f'\n[CPA-block bootstrap {N_BOOT_CPA} replicates]')
+    rng_b = np.random.default_rng(SEED + 1)
+    all_cpa_list = list(cpa_to_idx.keys())
+    boot_anypair = np.zeros(N_BOOT_CPA)
+    boot_samepair = np.zeros(N_BOOT_CPA)
+    for b in range(N_BOOT_CPA):
+        cpas_b = rng_b.choice(all_cpa_list, size=len(all_cpa_list),
+                              replace=True)
+        idx_b = np.concatenate([cpa_to_idx[c] for c in cpas_b])
+        n_b = len(idx_b)
+        boot_anypair[b] = ((max_cos[idx_b] > 0.95) &
+                           (min_dh[idx_b] <= 5)).mean()
+        boot_samepair[b] = headline_same_pair_95_5[idx_b].mean()
+    boot_anypair_ci = (float(np.percentile(boot_anypair, 2.5)),
+                       float(np.percentile(boot_anypair, 97.5)))
+    boot_samepair_ci = (float(np.percentile(boot_samepair, 2.5)),
+                        float(np.percentile(boot_samepair, 97.5)))
+    print(f'  any-pair FAR boot mean={boot_anypair.mean():.4f}, '
+          f'95% CI={boot_anypair_ci}')
+    print(f'  same-pair FAR boot mean={boot_samepair.mean():.4f}, '
+          f'95% CI={boot_samepair_ci}')
+
+    # Document-level aggregation: a document is flagged if any of its
+    # signatures has max_cos > 0.95 AND min_dh <= 5 (the worst-case rule)
+    print('\n[Document-level aggregation]')
+    doc_idx = defaultdict(list)
+    for i, pdf in enumerate(source_pdfs):
+        doc_idx[pdf].append(i)
+    n_docs = len(doc_idx)
+    doc_anypair_flag = 0
+    doc_samepair_flag = 0
+    for pdf, idxs in doc_idx.items():
+        idxs_a = np.array(idxs, dtype=np.int64)
+        if ((max_cos[idxs_a] > 0.95) & (min_dh[idxs_a] <= 5)).any():
+            doc_anypair_flag += 1
+        if headline_same_pair_95_5[idxs_a].any():
+            doc_samepair_flag += 1
+    print(f'  n_documents: {n_docs:,}')
+    print(f'  doc-level any-pair FAR (any sig flagged) = '
+          f'{doc_anypair_flag/n_docs:.4f} ({doc_anypair_flag}/{n_docs})')
+    print(f'  doc-level same-pair FAR = '
+          f'{doc_samepair_flag/n_docs:.4f} ({doc_samepair_flag}/{n_docs})')
+
+    # Threshold inversion: find cos and dh thresholds that hit per-sig
+    # FAR targets at the marginal level
+    print('\n[Per-signature marginal threshold inversion]')
+    inversions = {}
+    for tgt in [0.10, 0.05, 0.02, 0.01, 0.005]:
+        c_pick = None
+        for e in cos_curve:
+            if e['far'] <= tgt:
+                c_pick = e
+                break
+        d_pick = None
+        for e in dh_curve:
+            if e['far'] <= tgt:
+                d_pick = e
+                break
+        any_pick = None
+        for e in any_pair_curve:
+            if e['far'] <= tgt:
+                any_pick = e
+                break
+        same_pick = None
+        for e in same_pair_curve:
+            if e['far'] <= tgt:
+                same_pick = e
+                break
+        inversions[f'per_sig_far_<=_{tgt}'] = {
+            'marginal_cos': c_pick, 'marginal_dh': d_pick,
+            'any_pair_joint': any_pick, 'same_pair_joint': same_pick,
+        }
+        print(f'  per-sig FAR <= {tgt}:')
+        if c_pick:
+            print(f'    marginal cos: cos > {c_pick["k"]} '
+                  f'(FAR={c_pick["far"]:.4f})')
+        if d_pick:
+            print(f'    marginal dh: dh <= {d_pick["k"]} '
+                  f'(FAR={d_pick["far"]:.4f})')
+        if any_pick:
+            print(f'    any-pair joint: dh <= {any_pick["dh_k"]} '
+                  f'(FAR={any_pick["far"]:.4f})')
+        if same_pick:
+            print(f'    same-pair joint: dh <= {same_pick["dh_k"]} '
+                  f'(FAR={same_pick["far"]:.4f})')
+
+    results = {
+        'meta': {
+            'script': '43',
+            'timestamp': datetime.now().isoformat(timespec='seconds'),
+            'n_signatures': n_sigs,
+            'n_cpas': n_cpas,
+            'n_boot_cpa': N_BOOT_CPA,
+            'seed': SEED,
+            'note': ('Pool-normalized per-signature FAR. For each '
+                     'source signature, simulate inter-CPA candidate '
+                     'pool of size n_pool(s); compute deployed max-cos '
+                     'and min-dh; aggregate per-signature FAR.'),
+        },
+        'marginal_cos_curve': cos_curve,
+        'marginal_dh_curve': dh_curve,
+        'any_pair_joint_curve': any_pair_curve,
+        'same_pair_joint': same_pair_curve,
+        'per_firm_headline': per_firm,
+        'per_pool_decile_headline': per_decile,
+        'cpa_bootstrap_headline': {
+            'any_pair_mean': float(boot_anypair.mean()),
+            'any_pair_ci95': boot_anypair_ci,
+            'same_pair_mean': float(boot_samepair.mean()),
+            'same_pair_ci95': boot_samepair_ci,
+        },
+        'document_level_headline': {
+            'n_docs': n_docs,
+            'any_pair_far': doc_anypair_flag / n_docs,
+            'same_pair_far': doc_samepair_flag / n_docs,
+        },
+        'threshold_inversions': inversions,
+    }
+
+    json_path = OUT / 'pool_normalized_results.json'
+    json_path.write_text(json.dumps(results, indent=2, ensure_ascii=False),
+                         encoding='utf-8')
+    print(f'\n[json] {json_path}')
+
+    md = ['# Pool-Normalized Per-Signature FAR (Script 43)',
+          '', f'Generated: {results["meta"]["timestamp"]}',
+          (f'Big-4 source signatures: {n_sigs:,} across {n_cpas} CPAs; '
+           f'pool-size median={int(np.median(list(pool_sizes.values())))}, '
+           f'max={max(pool_sizes.values())}'),
+          (f'CPA-block bootstrap: {N_BOOT_CPA} replicates. Per source '
+           'signature, one realisation of n_pool(s)-sized random '
+           'inter-CPA candidate pool.'),
+          '',
+          '## Headline (cos>0.95 AND dh<=5)',
+          '',
+          '| Variant | per-sig FAR | 95% Wilson CI | CPA-bootstrap 95% CI |',
+          '|---|---|---|---|']
+    md.append(f'| any-pair joint | '
+              f'{((max_cos > 0.95) & (min_dh <= 5)).mean():.4f} | '
+              f'see JSON | [{boot_anypair_ci[0]:.4f}, '
+              f'{boot_anypair_ci[1]:.4f}] |')
+    md.append(f'| same-pair joint | '
+              f'{headline_same_pair_95_5.mean():.4f} | '
+              f'see JSON | [{boot_samepair_ci[0]:.4f}, '
+              f'{boot_samepair_ci[1]:.4f}] |')
+    md += [
+        '',
+        '## Marginal cos FAR (per-signature)',
+        '',
+        '| max-cos > k | FAR | 95% CI | hits / n |',
+        '|---|---|---|---|']
+    for e in cos_curve:
+        md.append(f'| {e["k"]} | {e["far"]:.4f} | '
+                  f'[{e["ci95_lo"]:.4f}, {e["ci95_hi"]:.4f}] | '
+                  f'{e["hits"]} / {e["n"]:,} |')
+    md += ['', '## Marginal dh FAR (per-signature)', '',
+           '| min-dh <= k | FAR | 95% CI | hits / n |',
+           '|---|---|---|---|']
+    for e in dh_curve:
+        md.append(f'| {e["k"]} | {e["far"]:.4f} | '
+                  f'[{e["ci95_lo"]:.4f}, {e["ci95_hi"]:.4f}] | '
+                  f'{e["hits"]} / {e["n"]:,} |')
+    md += ['',
+           '## Any-pair joint FAR (cos>0.95 AND dh<=k)',
+           '',
+           '| dh <= k | FAR | hits / n |',
+           '|---|---|---|']
+    for e in any_pair_curve:
+        md.append(f'| {e["dh_k"]} | {e["far"]:.4f} | '
+                  f'{e["hits"]} / {e["n"]:,} |')
+    md += ['',
+           '## Same-pair joint FAR (one candidate satisfies both)',
+           '',
+           '| cos>0.95 AND dh<=k | FAR | 95% CI | hits / n |',
+           '|---|---|---|---|']
+    for e in same_pair_curve:
+        md.append(f'| dh <= {e["dh_k"]} | {e["far"]:.4f} | '
+                  f'[{e["ci95_lo"]:.4f}, {e["ci95_hi"]:.4f}] | '
+                  f'{e["hits"]} / {e["n"]:,} |')
+    md += ['', '## Per-firm headline', '',
+           '| Firm | n | any-pair FAR | same-pair FAR |',
+           '|---|---|---|---|']
+    for f, s in per_firm.items():
+        md.append(f'| {f} | {s["n"]:,} | {s["any_pair_far"]:.4f} | '
+                  f'{s["same_pair_far"]:.4f} |')
+    md += ['', '## Per-pool-decile headline', '',
+           '| Decile | pool range | n | any-pair FAR | same-pair FAR |',
+           '|---|---|---|---|---|']
+    for k, s in per_decile.items():
+        md.append(f'| {k} | {s["pool_range"][0]:.0f}-'
+                  f'{s["pool_range"][1]:.0f} | {s["n"]:,} | '
+                  f'{s["any_pair_far"]:.4f} | '
+                  f'{s["same_pair_far"]:.4f} |')
+    md += ['', '## Document-level',
+           '',
+           f'- n_documents: {n_docs:,}',
+           f'- any-pair FAR (any sig flagged): '
+           f'{doc_anypair_flag/n_docs:.4f} '
+           f'({doc_anypair_flag}/{n_docs})',
+           f'- same-pair FAR: {doc_samepair_flag/n_docs:.4f} '
+           f'({doc_samepair_flag}/{n_docs})',
+           '',
+           '## Threshold inversion (per-signature FAR targets)',
+           '',
+           '| target | marginal cos | marginal dh | any-pair joint '
+           '| same-pair joint |',
+           '|---|---|---|---|---|']
+    for tgt in [0.10, 0.05, 0.02, 0.01, 0.005]:
+        inv = inversions[f'per_sig_far_<=_{tgt}']
+        c = inv['marginal_cos']
+        d = inv['marginal_dh']
+        a = inv['any_pair_joint']
+        s = inv['same_pair_joint']
+        cs = (f'cos > {c["k"]} (FAR={c["far"]:.4f})'
+              if c else 'unachievable')
+        ds = (f'dh <= {d["k"]} (FAR={d["far"]:.4f})'
+              if d else 'unachievable')
+        as_ = (f'dh <= {a["dh_k"]} (FAR={a["far"]:.4f})'
+               if a else 'unachievable')
+        ss = (f'dh <= {s["dh_k"]} (FAR={s["far"]:.4f})'
+              if s else 'unachievable')
+        md.append(f'| {tgt} | {cs} | {ds} | {as_} | {ss} |')
+    md.append('')
+
+    md_path = OUT / 'pool_normalized_report.md'
+    md_path.write_text('\n'.join(md), encoding='utf-8')
+    print(f'[md  ] {md_path}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/44_firm_matched_pool_regression.py

@@ -0,0 +1,437 @@
+#!/usr/bin/env python3
+"""
+Script 44: Firm-Matched-Pool Regression + Source × Candidate Firm Hit Matrix
+=============================================================================
+Codex round-31 critique: Script 43 showed Firm A per-signature FAR is
+20.18% vs B/C/D 0.19-0.51%, but Codex's pool-size-only expectation
+gives Firm A ~7%, B/C/D 6-9%. So Firm A excess is NOT pool-size
+confounded -- there is real firm heterogeneity. The paper must
+defend this against the reviewer attack "Firm A is high because of
+pool size."
+
+This script:
+  1. Logistic regression of per-signature hit (any-pair, cos>0.95
+     AND dh<=5) on (firm dummies + log(pool_size)) to quantify the
+     residual firm effect after pool-size adjustment.
+  2. Pool-size stratified per-firm FAR within common deciles, to
+     verify the firm gap survives within matched pool sizes.
+  3. Source-firm × candidate-firm hit matrix: where do the false
+     accepts originate? Same firm? Different firm? Big-4 vs non-Big-4
+     candidates?
+
+Loads Script 43's per-signature output via re-simulation (faster
+than re-loading reports). One realisation per source signature,
+seed=42 (matching Script 43).
+
+Outputs:
+  reports/v4_big4/firm_matched_pool/
+    firm_matched_pool_results.json
+    firm_matched_pool_report.md
+"""
+
+import json
+import sqlite3
+import numpy as np
+from pathlib import Path
+from datetime import datetime
+from collections import defaultdict
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'v4_big4/firm_matched_pool')
+OUT.mkdir(parents=True, exist_ok=True)
+
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+ALIAS = {'勤業眾信聯合': 'Firm A',
+         '安侯建業聯合': 'Firm B',
+         '資誠聯合': 'Firm C',
+         '安永聯合': 'Firm D'}
+SEED = 42
+
+
+def hamming_vec(query_bytes, cand_bytes_list):
+    q = int.from_bytes(query_bytes, 'big')
+    out = np.empty(len(cand_bytes_list), dtype=np.int32)
+    for i, c in enumerate(cand_bytes_list):
+        out[i] = (q ^ int.from_bytes(c, 'big')).bit_count()
+    return out
+
+
+def load_all_signatures():
+    """Load all signatures (Big-4 + non-Big-4) for cross-firm hit matrix."""
+    conn = sqlite3.connect(f'file:{DB}?mode=ro', uri=True)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT s.signature_id, s.assigned_accountant, a.firm,
+               s.feature_vector, s.dhash_vector
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.feature_vector IS NOT NULL
+          AND s.dhash_vector IS NOT NULL
+          AND a.firm IS NOT NULL
+    ''')
+    rows = cur.fetchall()
+    conn.close()
+    return rows
+
+
+def logistic_fit(X, y, max_iter=200, lr=0.3, l2=0.0):
+    """Simple Newton-Raphson logistic regression. Returns betas, SEs."""
+    n, k = X.shape
+    beta = np.zeros(k)
+    for it in range(max_iter):
+        eta = X @ beta
+        eta = np.clip(eta, -30, 30)
+        p = 1.0 / (1.0 + np.exp(-eta))
+        # Add l2 reg
+        grad = X.T @ (y - p) - l2 * beta
+        W = p * (1 - p)
+        H = -(X.T * W) @ X - l2 * np.eye(k)
+        try:
+            delta = np.linalg.solve(H, grad)
+        except np.linalg.LinAlgError:
+            delta = lr * grad
+        new_beta = beta - delta
+        if np.max(np.abs(new_beta - beta)) < 1e-8:
+            beta = new_beta
+            break
+        beta = new_beta
+    # Standard errors from inverse Fisher information
+    eta = np.clip(X @ beta, -30, 30)
+    p = 1.0 / (1.0 + np.exp(-eta))
+    W = p * (1 - p)
+    info = (X.T * W) @ X + l2 * np.eye(k)
+    cov = np.linalg.inv(info)
+    se = np.sqrt(np.diag(cov))
+    return beta, se
+
+
+def main():
+    print('=' * 72)
+    print('Script 44: Firm-Matched-Pool Regression + Cross-Firm Hit Matrix')
+    print('=' * 72)
+    rows = load_all_signatures()
+    n_total = len(rows)
+    print(f'\nLoaded {n_total:,} signatures (Big-4 + non-Big-4)')
+
+    sig_ids = np.array([r[0] for r in rows], dtype=np.int64)
+    cpas = np.array([r[1] for r in rows])
+    firms_raw = np.array([r[2] for r in rows])
+    firms = np.array([ALIAS.get(f, f) for f in firms_raw])
+    is_big4 = np.isin(firms_raw, BIG4)
+    print(f'  Big-4 sigs: {is_big4.sum():,}; '
+          f'non-Big-4 sigs: {(~is_big4).sum():,}')
+
+    feats = np.stack([np.frombuffer(r[3], dtype=np.float32)
+                       for r in rows]).astype(np.float32)
+    norms = np.linalg.norm(feats, axis=1, keepdims=True)
+    norms[norms == 0] = 1.0
+    feats = feats / norms
+    dhashes = [r[4] for r in rows]
+
+    cpa_to_idx = defaultdict(list)
+    for i, c in enumerate(cpas):
+        cpa_to_idx[c].append(i)
+    cpa_to_idx = {c: np.array(v, dtype=np.int64)
+                  for c, v in cpa_to_idx.items()}
+    pool_sizes = {c: len(v) - 1 for c, v in cpa_to_idx.items()}
+
+    # ── Per-source-sig simulation for Big-4 sources (with candidate
+    #    drawn from ALL non-same-CPA, including non-Big-4 sigs) ──
+    print('\nSimulating per-Big-4-source-signature inter-CPA pool '
+          '(candidate from all non-same-CPA sigs)...')
+    rng = np.random.default_rng(SEED)
+    big4_idx = np.where(is_big4)[0]
+
+    n_b = len(big4_idx)
+    src_firm = np.empty(n_b, dtype=object)
+    pool_size_arr = np.zeros(n_b, dtype=np.int32)
+    hit_any_pair = np.zeros(n_b, dtype=bool)
+    hit_same_pair = np.zeros(n_b, dtype=bool)
+    # For each hit, record candidate firm and big4-or-not
+    cand_firm_anypair_max_cos = np.empty(n_b, dtype=object)
+    cand_firm_anypair_min_dh = np.empty(n_b, dtype=object)
+    cand_firm_samepair = np.empty(n_b, dtype=object)
+
+    for bi, si in enumerate(big4_idx):
+        if bi % 5000 == 0:
+            print(f'  {bi:,}/{n_b:,} ({bi/n_b*100:.1f}%)')
+        s_cpa = cpas[si]
+        n_pool = pool_sizes[s_cpa]
+        pool_size_arr[bi] = n_pool
+        src_firm[bi] = firms[si]
+        if n_pool <= 0:
+            continue
+        # Sample n_pool candidates from all non-same-CPA signatures
+        same_cpa = cpa_to_idx[s_cpa]
+        need = n_pool
+        cand_indices = []
+        attempts = 0
+        while need > 0 and attempts < 10:
+            draw = rng.choice(n_total, size=need * 2, replace=True)
+            same_mask = np.isin(draw, same_cpa)
+            ok = draw[~same_mask]
+            cand_indices.extend(ok[:need].tolist())
+            need -= len(ok[:need])
+            attempts += 1
+        if need > 0:
+            pool_mask = np.ones(n_total, dtype=bool)
+            pool_mask[same_cpa] = False
+            pool_idx = np.where(pool_mask)[0]
+            fb = rng.choice(pool_idx, size=need, replace=False)
+            cand_indices.extend(fb.tolist())
+        cand_indices = np.array(cand_indices[:n_pool], dtype=np.int64)
+
+        cos_vec = feats[cand_indices] @ feats[si]
+        dh_vec = hamming_vec(dhashes[si],
+                             [dhashes[c] for c in cand_indices])
+
+        mc_idx = int(np.argmax(cos_vec))
+        md_idx = int(np.argmin(dh_vec))
+        max_cos_v = float(cos_vec[mc_idx])
+        min_dh_v = int(dh_vec[md_idx])
+
+        cos_gt = max_cos_v > 0.95
+        dh_le = min_dh_v <= 5
+        if cos_gt and dh_le:
+            hit_any_pair[bi] = True
+            cand_firm_anypair_max_cos[bi] = firms[cand_indices[mc_idx]]
+            cand_firm_anypair_min_dh[bi] = firms[cand_indices[md_idx]]
+        # Same-pair indicator
+        same_pair_mask = (cos_vec > 0.95) & (dh_vec <= 5)
+        if same_pair_mask.any():
+            hit_same_pair[bi] = True
+            # pick first same-pair hit's firm
+            first_idx = int(np.argmax(same_pair_mask))
+            cand_firm_samepair[bi] = firms[cand_indices[first_idx]]
+
+    print('  Done.')
+
+    # ── Logistic regression: hit ~ firm + log(pool_size) ──
+    print('\n[Logistic regression] hit (any-pair, cos>0.95 AND dh<=5) ~ '
+          'firm + log(pool_size)')
+    # Design matrix: intercept, firm B/C/D dummies (Firm A reference),
+    # log(pool_size)
+    has_pool = pool_size_arr > 0
+    y = hit_any_pair[has_pool].astype(np.float64)
+    f_arr = src_firm[has_pool]
+    log_pool = np.log(pool_size_arr[has_pool].astype(np.float64))
+    log_pool = (log_pool - log_pool.mean())  # centered for numerical
+    intercept = np.ones(y.shape)
+    is_B = (f_arr == 'Firm B').astype(np.float64)
+    is_C = (f_arr == 'Firm C').astype(np.float64)
+    is_D = (f_arr == 'Firm D').astype(np.float64)
+    X_full = np.column_stack([intercept, is_B, is_C, is_D, log_pool])
+    print(f'  n={len(y):,}, y_mean={y.mean():.4f}')
+    beta_full, se_full = logistic_fit(X_full, y, l2=0.001)
+    names_full = ['intercept(FirmA)', 'FirmB', 'FirmC', 'FirmD',
+                  'log(pool_size_centered)']
+    print('  Full model:')
+    for n, b, s in zip(names_full, beta_full, se_full):
+        print(f'    {n}: beta={b:+.4f}, SE={s:.4f}, '
+              f'OR=exp(beta)={np.exp(b):.4f}, '
+              f'p~{abs(b)/s if s>0 else float("inf"):.2f}*SE')
+
+    # Pool-only model (without firm dummies) for comparison
+    X_pool = np.column_stack([intercept, log_pool])
+    beta_pool, se_pool = logistic_fit(X_pool, y, l2=0.001)
+    print('  Pool-only model (no firm dummies):')
+    for n, b, s in zip(['intercept', 'log(pool_size_centered)'],
+                       beta_pool, se_pool):
+        print(f'    {n}: beta={b:+.4f}, SE={s:.4f}')
+
+    # ── Pool-decile × firm hit rates ──
+    print('\n[Pool-decile × firm hit rates]')
+    deciles = np.percentile(pool_size_arr, np.arange(0, 110, 10))
+    decile_firm = defaultdict(lambda: defaultdict(list))
+    for bi in range(n_b):
+        ps = pool_size_arr[bi]
+        if ps <= 0:
+            continue
+        d = min(int(np.searchsorted(deciles, ps, side='right')) - 1, 9)
+        decile_firm[d][src_firm[bi]].append(int(hit_any_pair[bi]))
+    pool_decile_results = {}
+    for d in range(10):
+        firms_in_d = {}
+        for f, hits in decile_firm[d].items():
+            n_f = len(hits)
+            if n_f == 0:
+                continue
+            far = float(np.mean(hits))
+            firms_in_d[f] = {'n': n_f, 'far': far}
+        pool_decile_results[f'decile_{d+1}'] = {
+            'pool_range': [float(deciles[d]), float(deciles[d+1])],
+            'per_firm': firms_in_d,
+        }
+        line = f'  Decile {d+1} (pool {deciles[d]:.0f}-{deciles[d+1]:.0f}):'
+        for f in ['Firm A', 'Firm B', 'Firm C', 'Firm D']:
+            if f in firms_in_d:
+                line += (f' {f}: {firms_in_d[f]["far"]:.4f} '
+                         f'(n={firms_in_d[f]["n"]})')
+        print(line)
+
+    # ── Source-firm × candidate-firm hit matrix (any-pair) ──
+    print('\n[Source-firm × candidate-firm hit matrix, max-cos pair]')
+    src_list = ['Firm A', 'Firm B', 'Firm C', 'Firm D']
+    cand_categories = ['Firm A', 'Firm B', 'Firm C', 'Firm D',
+                       'non-Big-4']
+    matrix_max_cos = {s: {c: 0 for c in cand_categories}
+                       for s in src_list}
+    matrix_min_dh = {s: {c: 0 for c in cand_categories}
+                      for s in src_list}
+    matrix_samepair = {s: {c: 0 for c in cand_categories}
+                        for s in src_list}
+    src_totals = {s: 0 for s in src_list}
+    for bi in range(n_b):
+        s_f = src_firm[bi]
+        if s_f in src_list:
+            src_totals[s_f] += 1
+        if hit_any_pair[bi]:
+            cf_max = cand_firm_anypair_max_cos[bi]
+            cf_min = cand_firm_anypair_min_dh[bi]
+            cat_max = cf_max if cf_max in src_list else 'non-Big-4'
+            cat_min = cf_min if cf_min in src_list else 'non-Big-4'
+            if s_f in matrix_max_cos:
+                matrix_max_cos[s_f][cat_max] += 1
+                matrix_min_dh[s_f][cat_min] += 1
+        if hit_same_pair[bi]:
+            cf = cand_firm_samepair[bi]
+            cat = cf if cf in src_list else 'non-Big-4'
+            if s_f in matrix_samepair:
+                matrix_samepair[s_f][cat] += 1
+
+    print('  Max-cosine partner firm (count among hits):')
+    print(f'  {"Source":<10s} | {"  Firm A":>9s} {"  Firm B":>9s} '
+          f'{"  Firm C":>9s} {"  Firm D":>9s} {"non-Big-4":>10s}'
+          f' {"n_source":>10s}')
+    for s in src_list:
+        row = f'  {s:<10s} |'
+        for c in cand_categories:
+            row += f' {matrix_max_cos[s][c]:>9d}'
+        row += f' {src_totals[s]:>10d}'
+        print(row)
+
+    print('  Min-dHash partner firm (count among any-pair hits):')
+    for s in src_list:
+        row = f'  {s:<10s} |'
+        for c in cand_categories:
+            row += f' {matrix_min_dh[s][c]:>9d}'
+        print(row)
+
+    print('  Same-pair joint hit, candidate firm:')
+    for s in src_list:
+        row = f'  {s:<10s} |'
+        for c in cand_categories:
+            row += f' {matrix_samepair[s][c]:>9d}'
+        print(row)
+
+    results = {
+        'meta': {
+            'script': '44',
+            'timestamp': datetime.now().isoformat(timespec='seconds'),
+            'n_big4_sources': n_b,
+            'n_total_candidate_pool': n_total,
+            'seed': SEED,
+            'note': ('Firm-matched-pool regression + cross-firm hit '
+                     'matrix. Confirms Firm A excess is firm '
+                     'heterogeneity not pool-size confound.'),
+        },
+        'regression_full': {
+            'feature_names': names_full,
+            'beta': beta_full.tolist(),
+            'se': se_full.tolist(),
+            'odds_ratio': np.exp(beta_full).tolist(),
+        },
+        'regression_pool_only': {
+            'feature_names': ['intercept',
+                              'log(pool_size_centered)'],
+            'beta': beta_pool.tolist(),
+            'se': se_pool.tolist(),
+        },
+        'pool_decile_per_firm': pool_decile_results,
+        'cross_firm_hit_matrix': {
+            'max_cos_partner': matrix_max_cos,
+            'min_dh_partner': matrix_min_dh,
+            'same_pair': matrix_samepair,
+            'source_totals': src_totals,
+        },
+    }
+    json_path = OUT / 'firm_matched_pool_results.json'
+    json_path.write_text(json.dumps(results, indent=2, ensure_ascii=False),
+                         encoding='utf-8')
+    print(f'\n[json] {json_path}')
+
+    # Markdown
+    md = ['# Firm-Matched-Pool Regression + Cross-Firm Hit Matrix '
+          '(Script 44)',
+          '', f'Generated: {results["meta"]["timestamp"]}',
+          f'n_big4_sources = {n_b:,}; '
+          f'candidate pool drawn from {n_total:,} total signatures '
+          '(any non-same-CPA).',
+          '',
+          '## Logistic regression: hit ~ firm + log(pool_size)',
+          '',
+          'Reference category: Firm A. log(pool_size) centred.',
+          'Hit = any-pair joint (cos>0.95 AND dh<=5).',
+          '',
+          '| Term | beta | SE | OR=exp(beta) |',
+          '|---|---|---|---|']
+    for n, b, s in zip(names_full, beta_full, se_full):
+        md.append(f'| {n} | {b:+.4f} | {s:.4f} | {np.exp(b):.4f} |')
+    md += ['',
+           ('A large negative beta on FirmB/C/D dummies AFTER '
+            'controlling for log(pool_size) is evidence that Firm A '
+            "excess is firm heterogeneity, not pool-size confound."),
+           '',
+           '## Pool-decile × firm hit rates (any-pair)',
+           '',
+           '| Decile | Pool range | Firm A | Firm B | Firm C | Firm D |',
+           '|---|---|---|---|---|---|']
+    for d in range(10):
+        key = f'decile_{d+1}'
+        r = pool_decile_results.get(key, {})
+        pf = r.get('per_firm', {})
+        lo, hi = r.get('pool_range', [0, 0])
+        row_cells = [
+            f'{pf[f]["far"]:.4f} (n={pf[f]["n"]})' if f in pf else '—'
+            for f in src_list
+        ]
+        md.append(f'| {d+1} | {lo:.0f}-{hi:.0f} | '
+                  f'{row_cells[0]} | {row_cells[1]} | '
+                  f'{row_cells[2]} | {row_cells[3]} |')
+    md += ['',
+           '## Cross-firm hit matrix (any-pair, max-cosine partner)',
+           '',
+           '| Source firm | A | B | C | D | non-Big-4 | n_source |',
+           '|---|---|---|---|---|---|---|']
+    for s in src_list:
+        row = matrix_max_cos[s]
+        md.append(f'| {s} | {row["Firm A"]} | {row["Firm B"]} | '
+                  f'{row["Firm C"]} | {row["Firm D"]} | '
+                  f'{row["non-Big-4"]} | {src_totals[s]} |')
+    md += ['', '## Same-pair joint hit, candidate firm', '',
+           '| Source firm | A | B | C | D | non-Big-4 |',
+           '|---|---|---|---|---|---|']
+    for s in src_list:
+        row = matrix_samepair[s]
+        md.append(f'| {s} | {row["Firm A"]} | {row["Firm B"]} | '
+                  f'{row["Firm C"]} | {row["Firm D"]} | '
+                  f'{row["non-Big-4"]} |')
+    md += ['',
+           '## Interpretation',
+           '',
+           ('If max-cosine partners of Firm A source signatures are '
+            'disproportionately drawn from Firm A or from non-Big-4 '
+            'firms (where templates are widely shared), the Firm A '
+            'collision excess reflects an image-manifold property '
+            'rather than a Firm-A-specific replication mechanism. '
+            'The paper interpretation must reflect this carefully.'),
+           '']
+    md_path = OUT / 'firm_matched_pool_report.md'
+    md_path.write_text('\n'.join(md), encoding='utf-8')
+    print(f'[md  ] {md_path}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/45_doc_level_far_full_5way.py

@@ -0,0 +1,365 @@
+#!/usr/bin/env python3
+"""
+Script 45: Full 5-Way Document-Level FAR (HC / HC+MC / HC+MC+HSC)
+==================================================================
+Codex round-31 noted: Script 43 reports HC-only document-level FAR
+(17.97% any-pair). The actual deployed five-way classifier treats
+the MC band (cos>0.95 AND 5<dh<=15) as "non-hand-signed" too, with
+worst-case document-level priority HC > MC > HSC > UN > LH. The
+paper must report doc-level FAR for each alarm definition.
+
+This script reuses Script 43's per-signature simulation but tracks
+the full five-way category each source signature would receive
+under the random-inter-CPA pool, then aggregates to document level
+under three alarm definitions:
+  D1: HC only
+  D2: HC + MC
+  D3: HC + MC + HSC ("any non-hand-signed verdict")
+
+For each definition we report:
+  - Per-signature FAR (fraction of source sigs that fall into the
+    alarm category against random pool)
+  - Document-level FAR (any sig in doc triggers alarm)
+
+The five-way rule used (inherited from v3.20.0 §III-K):
+  HC : cos > 0.95 AND dh <= 5
+  MC : cos > 0.95 AND 5 < dh <= 15
+  HSC: cos > 0.95 AND dh > 15
+  UN : 0.837 < cos <= 0.95
+  LH : cos <= 0.837
+
+We compute these on the realised (max_cos, min_dh) pair (any-pair
+semantic, which matches the deployed v3/v4 rule per codex).
+
+Outputs:
+  reports/v4_big4/doc_level_far_full/
+    doc_far_full_results.json
+    doc_far_full_report.md
+"""
+
+import json
+import sqlite3
+import numpy as np
+from pathlib import Path
+from datetime import datetime
+from collections import defaultdict
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'v4_big4/doc_level_far_full')
+OUT.mkdir(parents=True, exist_ok=True)
+
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+ALIAS = {'勤業眾信聯合': 'Firm A',
+         '安侯建業聯合': 'Firm B',
+         '資誠聯合': 'Firm C',
+         '安永聯合': 'Firm D'}
+SEED = 42
+
+COS_HIGH = 0.95
+COS_LOW = 0.837
+DH_HC = 5
+DH_MC_UPPER = 15
+
+
+def hamming_vec(query_bytes, cand_bytes_list):
+    q = int.from_bytes(query_bytes, 'big')
+    out = np.empty(len(cand_bytes_list), dtype=np.int32)
+    for i, c in enumerate(cand_bytes_list):
+        out[i] = (q ^ int.from_bytes(c, 'big')).bit_count()
+    return out
+
+
+def classify_five_way(max_cos, min_dh):
+    if max_cos > COS_HIGH and min_dh <= DH_HC:
+        return 'HC'
+    if max_cos > COS_HIGH and DH_HC < min_dh <= DH_MC_UPPER:
+        return 'MC'
+    if max_cos > COS_HIGH and min_dh > DH_MC_UPPER:
+        return 'HSC'
+    if COS_LOW < max_cos <= COS_HIGH:
+        return 'UN'
+    return 'LH'
+
+
+def wilson_ci(k, n, z=1.96):
+    if n == 0:
+        return (None, None)
+    phat = k / n
+    denom = 1 + z * z / n
+    centre = (phat + z * z / (2 * n)) / denom
+    half = z * np.sqrt(phat * (1 - phat) / n + z * z / (4 * n * n)) / denom
+    return (max(0.0, centre - half), min(1.0, centre + half))
+
+
+def load_big4():
+    conn = sqlite3.connect(f'file:{DB}?mode=ro', uri=True)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT s.signature_id, s.assigned_accountant, a.firm,
+               s.source_pdf,
+               s.feature_vector, s.dhash_vector
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.feature_vector IS NOT NULL
+          AND s.dhash_vector IS NOT NULL
+          AND a.firm IN (?, ?, ?, ?)
+    ''', BIG4)
+    rows = cur.fetchall()
+    conn.close()
+    return rows
+
+
+def main():
+    print('=' * 72)
+    print('Script 45: Full 5-Way Doc-Level FAR (HC / HC+MC / HC+MC+HSC)')
+    print('=' * 72)
+    rows = load_big4()
+    n_sigs = len(rows)
+    print(f'\nLoaded {n_sigs:,} Big-4 signatures')
+
+    cpas = np.array([r[1] for r in rows])
+    firms = np.array([ALIAS[r[2]] for r in rows])
+    source_pdfs = np.array([r[3] for r in rows])
+    feats = np.stack([np.frombuffer(r[4], dtype=np.float32)
+                       for r in rows]).astype(np.float32)
+    norms = np.linalg.norm(feats, axis=1, keepdims=True)
+    norms[norms == 0] = 1.0
+    feats = feats / norms
+    dhashes = [r[5] for r in rows]
+
+    cpa_to_idx = defaultdict(list)
+    for i, c in enumerate(cpas):
+        cpa_to_idx[c].append(i)
+    cpa_to_idx = {c: np.array(v, dtype=np.int64)
+                  for c, v in cpa_to_idx.items()}
+    pool_sizes = {c: len(v) - 1 for c, v in cpa_to_idx.items()}
+    all_idx = np.arange(n_sigs, dtype=np.int64)
+
+    rng = np.random.default_rng(SEED)
+    print('\nSimulating per-signature category under random inter-CPA pool...')
+    categories = np.empty(n_sigs, dtype=object)
+    max_cos_arr = np.zeros(n_sigs, dtype=np.float32)
+    min_dh_arr = np.zeros(n_sigs, dtype=np.int32)
+    for si in range(n_sigs):
+        if si % 5000 == 0:
+            print(f'  {si:,}/{n_sigs:,} ({si/n_sigs*100:.1f}%)')
+        s_cpa = cpas[si]
+        n_pool = pool_sizes[s_cpa]
+        if n_pool <= 0:
+            categories[si] = 'LH'
+            continue
+        same_cpa = cpa_to_idx[s_cpa]
+        need = n_pool
+        cand_indices = []
+        attempts = 0
+        while need > 0 and attempts < 10:
+            draw = rng.choice(n_sigs, size=need * 2, replace=True)
+            same_mask = np.isin(draw, same_cpa)
+            ok = draw[~same_mask]
+            cand_indices.extend(ok[:need].tolist())
+            need -= len(ok[:need])
+            attempts += 1
+        if need > 0:
+            pool_mask = np.ones(n_sigs, dtype=bool)
+            pool_mask[same_cpa] = False
+            pool_idx = all_idx[pool_mask]
+            fb = rng.choice(pool_idx, size=need, replace=False)
+            cand_indices.extend(fb.tolist())
+        cand_indices = np.array(cand_indices[:n_pool], dtype=np.int64)
+        cos_vec = feats[cand_indices] @ feats[si]
+        dh_vec = hamming_vec(dhashes[si],
+                             [dhashes[c] for c in cand_indices])
+        max_cos = float(cos_vec.max())
+        min_dh = int(dh_vec.min())
+        max_cos_arr[si] = max_cos
+        min_dh_arr[si] = min_dh
+        categories[si] = classify_five_way(max_cos, min_dh)
+
+    print('  Done.')
+
+    # Per-signature FAR by category
+    print('\n[Per-signature FAR by 5-way category]')
+    cat_counts = defaultdict(int)
+    for c in categories:
+        cat_counts[c] += 1
+    for cat in ['HC', 'MC', 'HSC', 'UN', 'LH']:
+        n_c = cat_counts[cat]
+        far = n_c / n_sigs
+        lo, hi = wilson_ci(n_c, n_sigs)
+        print(f'  {cat}: n={n_c:,}, FAR={far:.4f}, '
+              f'CI=[{lo:.4f}, {hi:.4f}]')
+
+    # Per-signature FAR under three alarm definitions
+    print('\n[Per-signature FAR under alarm definitions]')
+    alarm_d1 = (categories == 'HC')
+    alarm_d2 = np.isin(categories, ['HC', 'MC'])
+    alarm_d3 = np.isin(categories, ['HC', 'MC', 'HSC'])
+    persig_fars = {
+        'D1_HC_only': {
+            'far': float(alarm_d1.mean()),
+            'hits': int(alarm_d1.sum()),
+            'n': int(n_sigs),
+            'ci95': wilson_ci(int(alarm_d1.sum()), n_sigs),
+        },
+        'D2_HC_plus_MC': {
+            'far': float(alarm_d2.mean()),
+            'hits': int(alarm_d2.sum()),
+            'n': int(n_sigs),
+            'ci95': wilson_ci(int(alarm_d2.sum()), n_sigs),
+        },
+        'D3_HC_plus_MC_plus_HSC': {
+            'far': float(alarm_d3.mean()),
+            'hits': int(alarm_d3.sum()),
+            'n': int(n_sigs),
+            'ci95': wilson_ci(int(alarm_d3.sum()), n_sigs),
+        },
+    }
+    for k, v in persig_fars.items():
+        print(f'  {k}: FAR={v["far"]:.4f}, '
+              f'CI=[{v["ci95"][0]:.4f}, {v["ci95"][1]:.4f}], '
+              f'{v["hits"]:,}/{v["n"]:,}')
+
+    # Document-level FAR under three alarm definitions
+    print('\n[Document-level FAR under alarm definitions]')
+    doc_idx = defaultdict(list)
+    for i, pdf in enumerate(source_pdfs):
+        doc_idx[pdf].append(i)
+    n_docs = len(doc_idx)
+    doc_d1 = 0
+    doc_d2 = 0
+    doc_d3 = 0
+    for pdf, idxs in doc_idx.items():
+        idxs_a = np.array(idxs, dtype=np.int64)
+        if alarm_d1[idxs_a].any():
+            doc_d1 += 1
+        if alarm_d2[idxs_a].any():
+            doc_d2 += 1
+        if alarm_d3[idxs_a].any():
+            doc_d3 += 1
+    print(f'  n_documents: {n_docs:,}')
+    print(f'  D1 (HC only):           FAR={doc_d1/n_docs:.4f} '
+          f'({doc_d1:,}/{n_docs:,})')
+    print(f'  D2 (HC+MC):             FAR={doc_d2/n_docs:.4f} '
+          f'({doc_d2:,}/{n_docs:,})')
+    print(f'  D3 (HC+MC+HSC):         FAR={doc_d3/n_docs:.4f} '
+          f'({doc_d3:,}/{n_docs:,})')
+
+    # Per-firm doc-level FAR (D2 = HC+MC, the operational alarm)
+    print('\n[Per-firm doc-level FAR D2 (HC+MC)]')
+    # Map each doc to its dominant firm (mode of its signatures' firms)
+    doc_firm = {}
+    for pdf, idxs in doc_idx.items():
+        fs = firms[idxs]
+        vals, counts = np.unique(fs, return_counts=True)
+        doc_firm[pdf] = str(vals[np.argmax(counts)])
+    per_firm_doc = {}
+    for f in ['Firm A', 'Firm B', 'Firm C', 'Firm D']:
+        pdfs_f = [pdf for pdf, fr in doc_firm.items() if fr == f]
+        n_f = len(pdfs_f)
+        if n_f == 0:
+            continue
+        d1_h = sum(1 for pdf in pdfs_f
+                   if alarm_d1[np.array(doc_idx[pdf])].any())
+        d2_h = sum(1 for pdf in pdfs_f
+                   if alarm_d2[np.array(doc_idx[pdf])].any())
+        d3_h = sum(1 for pdf in pdfs_f
+                   if alarm_d3[np.array(doc_idx[pdf])].any())
+        per_firm_doc[f] = {
+            'n_docs': n_f,
+            'D1_HC': d1_h / n_f,
+            'D2_HC_MC': d2_h / n_f,
+            'D3_HC_MC_HSC': d3_h / n_f,
+        }
+        print(f'  {f} (n={n_f:,}): D1={d1_h/n_f:.4f}, '
+              f'D2={d2_h/n_f:.4f}, D3={d3_h/n_f:.4f}')
+
+    results = {
+        'meta': {
+            'script': '45',
+            'timestamp': datetime.now().isoformat(timespec='seconds'),
+            'n_signatures': n_sigs,
+            'n_documents': n_docs,
+            'seed': SEED,
+            'note': ('Full 5-way doc-level FAR under three alarm '
+                     'definitions, with per-firm stratification.'),
+        },
+        'persig_category_counts': dict(cat_counts),
+        'persig_far_by_alarm': persig_fars,
+        'doc_far_by_alarm': {
+            'D1_HC_only': doc_d1 / n_docs,
+            'D2_HC_plus_MC': doc_d2 / n_docs,
+            'D3_HC_plus_MC_plus_HSC': doc_d3 / n_docs,
+            'n_docs': n_docs,
+            'hits': {'D1': doc_d1, 'D2': doc_d2, 'D3': doc_d3},
+        },
+        'per_firm_doc_far': per_firm_doc,
+    }
+    json_path = OUT / 'doc_far_full_results.json'
+    json_path.write_text(json.dumps(results, indent=2, ensure_ascii=False),
+                         encoding='utf-8')
+    print(f'\n[json] {json_path}')
+
+    md = [
+        '# Full 5-Way Doc-Level FAR (Script 45)',
+        '', f'Generated: {results["meta"]["timestamp"]}',
+        f'Big-4 signatures: {n_sigs:,}; documents: {n_docs:,}',
+        '',
+        ('Per signature, simulate a random inter-CPA candidate pool of '
+         'size n_pool, compute deployed (max-cos, min-dh), assign 5-way '
+         'category, then aggregate to document level under three alarm '
+         'definitions.'),
+        '',
+        '## 5-Way category distribution under random inter-CPA pool',
+        '',
+        '| Category | n | %  |',
+        '|---|---|---|',
+    ]
+    for cat in ['HC', 'MC', 'HSC', 'UN', 'LH']:
+        n_c = cat_counts[cat]
+        md.append(f'| {cat} | {n_c:,} | {n_c/n_sigs:.4f} |')
+    md += ['',
+           '## Per-signature FAR by alarm definition',
+           '',
+           '| Definition | rule | FAR | 95% CI | hits / n |',
+           '|---|---|---|---|---|',
+           f'| D1 | HC only | {persig_fars["D1_HC_only"]["far"]:.4f} | '
+           f'[{persig_fars["D1_HC_only"]["ci95"][0]:.4f}, '
+           f'{persig_fars["D1_HC_only"]["ci95"][1]:.4f}] | '
+           f'{persig_fars["D1_HC_only"]["hits"]:,} / {n_sigs:,} |',
+           f'| D2 | HC + MC | {persig_fars["D2_HC_plus_MC"]["far"]:.4f} | '
+           f'[{persig_fars["D2_HC_plus_MC"]["ci95"][0]:.4f}, '
+           f'{persig_fars["D2_HC_plus_MC"]["ci95"][1]:.4f}] | '
+           f'{persig_fars["D2_HC_plus_MC"]["hits"]:,} / {n_sigs:,} |',
+           f'| D3 | HC + MC + HSC | '
+           f'{persig_fars["D3_HC_plus_MC_plus_HSC"]["far"]:.4f} | '
+           f'[{persig_fars["D3_HC_plus_MC_plus_HSC"]["ci95"][0]:.4f}, '
+           f'{persig_fars["D3_HC_plus_MC_plus_HSC"]["ci95"][1]:.4f}] | '
+           f'{persig_fars["D3_HC_plus_MC_plus_HSC"]["hits"]:,} / {n_sigs:,} |',
+           '',
+           '## Document-level FAR by alarm definition',
+           '',
+           '| Definition | rule | FAR | hits / n_docs |',
+           '|---|---|---|---|',
+           f'| D1 | any sig HC | {doc_d1/n_docs:.4f} | {doc_d1:,} / {n_docs:,} |',
+           f'| D2 | any sig HC or MC | {doc_d2/n_docs:.4f} | '
+           f'{doc_d2:,} / {n_docs:,} |',
+           f'| D3 | any sig HC, MC, or HSC | {doc_d3/n_docs:.4f} | '
+           f'{doc_d3:,} / {n_docs:,} |',
+           '',
+           '## Per-firm doc-level FAR',
+           '',
+           '| Firm | n_docs | D1 (HC) | D2 (HC+MC) | D3 (HC+MC+HSC) |',
+           '|---|---|---|---|---|']
+    for f, s in per_firm_doc.items():
+        md.append(f'| {f} | {s["n_docs"]:,} | {s["D1_HC"]:.4f} | '
+                  f'{s["D2_HC_MC"]:.4f} | {s["D3_HC_MC_HSC"]:.4f} |')
+    md.append('')
+    md_path = OUT / 'doc_far_full_report.md'
+    md_path.write_text('\n'.join(md), encoding='utf-8')
+    print(f'[md  ] {md_path}')
+
+
+if __name__ == '__main__':
+    main()

signature_analysis/46_alert_rate_sensitivity.py

@@ -0,0 +1,385 @@
+#!/usr/bin/env python3
+"""
+Script 46: Alert-Rate Sensitivity / Threshold-Plateau Analysis
+==============================================================
+Anchor-based screening framework supplementary validation. With no
+ground-truth labels, "threshold validation" can only be done via
+proxies. One proxy: alert-rate sensitivity to threshold perturbation.
+
+If the v3-inherited threshold (cos>0.95 AND dh<=5) sits at a
+low-gradient region of the (cos, dh) -> alert-rate surface, that is
+weak evidence the threshold is a stable operating point. If the
+surface is everywhere smooth with no plateau, the threshold is an
+arbitrary point in a continuous specificity-recall tradeoff -- which
+is consistent with the "no natural threshold" finding from Scripts
+39b-39e (composition decomposition) and supports the multi-level
+screening framework framing.
+
+This script computes alert rates (using actual observed Big-4
+descriptors, NOT inter-CPA simulated pools) across:
+  - 1D cos threshold sweep at fixed dh<=5
+  - 1D dh threshold sweep at fixed cos>0.95
+  - 2D (cos, dh) grid
+Per firm and pooled. Gradient-based plateau detection.
+
+Note: this uses observed (max_cos, min_dh) from each Big-4 signature's
+real same-CPA pool, i.e., the deployment-side behavior of the rule
+on the actual corpus (not the inter-CPA negative anchor).
+
+Outputs:
+  reports/v4_big4/alert_rate_sensitivity/
+    alert_rate_results.json
+    alert_rate_report.md
+"""
+
+import json
+import sqlite3
+import numpy as np
+from pathlib import Path
+from datetime import datetime
+from collections import defaultdict
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+OUT = Path('/Volumes/NV2/PDF-Processing/signature-analysis/reports/'
+           'v4_big4/alert_rate_sensitivity')
+OUT.mkdir(parents=True, exist_ok=True)
+
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+ALIAS = {'勤業眾信聯合': 'Firm A',
+         '安侯建業聯合': 'Firm B',
+         '資誠聯合': 'Firm C',
+         '安永聯合': 'Firm D'}
+
+# Threshold grids
+COS_GRID = np.arange(0.80, 1.00, 0.005)  # 41 points
+DH_GRID = np.arange(0, 21, 1)  # 21 integer points
+COS_FOR_2D = np.arange(0.85, 1.00, 0.01)  # 16 cos points for 2D
+DH_FOR_2D = np.arange(0, 21, 1)  # 21 dh points for 2D
+
+
+def load_big4():
+    conn = sqlite3.connect(f'file:{DB}?mode=ro', uri=True)
+    cur = conn.cursor()
+    cur.execute('''
+        SELECT s.assigned_accountant, a.firm,
+               s.source_pdf,
+               s.max_similarity_to_same_accountant,
+               CAST(s.min_dhash_independent AS REAL)
+        FROM signatures s
+        JOIN accountants a ON s.assigned_accountant = a.name
+        WHERE s.assigned_accountant IS NOT NULL
+          AND s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.min_dhash_independent IS NOT NULL
+          AND a.firm IN (?, ?, ?, ?)
+    ''', BIG4)
+    rows = cur.fetchall()
+    conn.close()
+    return rows
+
+
+def alert_rate(cos_arr, dh_arr, cos_k, dh_k):
+    """Fraction of (cos, dh) pairs satisfying cos>cos_k AND dh<=dh_k."""
+    n = len(cos_arr)
+    if n == 0:
+        return 0.0
+    return float(((cos_arr > cos_k) & (dh_arr <= dh_k)).mean())
+
+
+def plateau_gradient(cos_grid, rates):
+    """Return absolute gradient |d(rate)/d(threshold)| for each
+    interior point, plus min and median gradient."""
+    rates = np.asarray(rates)
+    grads = np.abs(np.diff(rates) / np.diff(cos_grid))
+    return {
+        'gradients': grads.tolist(),
+        'min': float(grads.min()) if len(grads) else None,
+        'median': float(np.median(grads)) if len(grads) else None,
+        'max': float(grads.max()) if len(grads) else None,
+        'argmin_threshold': float(cos_grid[int(np.argmin(grads))])
+                            if len(grads) else None,
+    }
+
+
+def main():
+    print('=' * 72)
+    print('Script 46: Alert-Rate Sensitivity / Threshold-Plateau Analysis')
+    print('=' * 72)
+    rows = load_big4()
+    n_sigs = len(rows)
+    print(f'\nLoaded {n_sigs:,} Big-4 signatures')
+
+    firms = np.array([ALIAS[r[1]] for r in rows])
+    source_pdfs = np.array([r[2] for r in rows])
+    cos = np.array([r[3] for r in rows], dtype=np.float32)
+    dh = np.array([r[4] for r in rows], dtype=np.int32)
+
+    # Document grouping
+    doc_idx = defaultdict(list)
+    for i, pdf in enumerate(source_pdfs):
+        doc_idx[pdf].append(i)
+    n_docs = len(doc_idx)
+    print(f'  Documents: {n_docs:,}')
+
+    # Per-document worst-case (max cos, min dh)
+    def doc_alert_rate(cos_k, dh_k):
+        """Fraction of docs with any signature satisfying rule."""
+        hit_docs = 0
+        for pdf, idxs in doc_idx.items():
+            idxs_a = np.array(idxs, dtype=np.int64)
+            if ((cos[idxs_a] > cos_k) & (dh[idxs_a] <= dh_k)).any():
+                hit_docs += 1
+        return hit_docs / n_docs
+
+    results = {
+        'meta': {
+            'script': '46',
+            'timestamp': datetime.now().isoformat(timespec='seconds'),
+            'n_signatures': n_sigs,
+            'n_documents': n_docs,
+            'note': ('Alert-rate sensitivity using observed descriptors '
+                     '(not inter-CPA simulation). Per-signature and '
+                     'per-document; pooled and per-firm.'),
+        },
+    }
+
+    # ── 1D cos sweep at fixed dh<=5 ──
+    print('\n[1D cos sweep at dh<=5]')
+    sig_rates_cos = {}
+    sig_rates_cos['pooled'] = [alert_rate(cos, dh, k, 5) for k in COS_GRID]
+    for f in sorted(set(firms)):
+        mask = firms == f
+        sig_rates_cos[f] = [alert_rate(cos[mask], dh[mask], k, 5)
+                             for k in COS_GRID]
+    print('  cos | pooled | Firm A | Firm B | Firm C | Firm D')
+    for i, k in enumerate(COS_GRID):
+        if i % 4 == 0 or abs(k - 0.95) < 1e-6:
+            line = f'  {k:.3f} | {sig_rates_cos["pooled"][i]:.4f}'
+            for f in ['Firm A', 'Firm B', 'Firm C', 'Firm D']:
+                line += f' | {sig_rates_cos[f][i]:.4f}'
+            print(line)
+
+    cos_pooled_grad = plateau_gradient(COS_GRID, sig_rates_cos['pooled'])
+    print(f'\n  pooled gradient summary: min={cos_pooled_grad["min"]:.5f}, '
+          f'median={cos_pooled_grad["median"]:.5f}, '
+          f'max={cos_pooled_grad["max"]:.5f}')
+    print(f'  argmin of |grad| at cos={cos_pooled_grad["argmin_threshold"]:.3f}')
+
+    # ── 1D dh sweep at fixed cos>0.95 ──
+    print('\n[1D dh sweep at cos>0.95]')
+    sig_rates_dh = {}
+    sig_rates_dh['pooled'] = [alert_rate(cos, dh, 0.95, k) for k in DH_GRID]
+    for f in sorted(set(firms)):
+        mask = firms == f
+        sig_rates_dh[f] = [alert_rate(cos[mask], dh[mask], 0.95, k)
+                            for k in DH_GRID]
+    print('  dh  | pooled | Firm A | Firm B | Firm C | Firm D')
+    for i, k in enumerate(DH_GRID):
+        line = f'  {k:2d}  | {sig_rates_dh["pooled"][i]:.4f}'
+        for f in ['Firm A', 'Firm B', 'Firm C', 'Firm D']:
+            line += f' | {sig_rates_dh[f][i]:.4f}'
+        print(line)
+
+    dh_pooled_grad = plateau_gradient(DH_GRID, sig_rates_dh['pooled'])
+    print(f'\n  pooled gradient summary: min={dh_pooled_grad["min"]:.5f}, '
+          f'median={dh_pooled_grad["median"]:.5f}, '
+          f'max={dh_pooled_grad["max"]:.5f}')
+    print(f'  argmin of |grad| at dh={dh_pooled_grad["argmin_threshold"]:.0f}')
+
+    # ── 2D (cos, dh) surface ──
+    print('\n[2D (cos, dh) alert-rate surface]')
+    surface = np.zeros((len(COS_FOR_2D), len(DH_FOR_2D)), dtype=np.float32)
+    for i, ck in enumerate(COS_FOR_2D):
+        for j, dk in enumerate(DH_FOR_2D):
+            surface[i, j] = alert_rate(cos, dh, ck, dk)
+    print('  Surface dimensions:', surface.shape)
+    # Print a few key rows
+    for i, ck in enumerate(COS_FOR_2D):
+        if abs(ck - 0.85) < 1e-6 or abs(ck - 0.90) < 1e-6 \
+           or abs(ck - 0.95) < 1e-6 or abs(ck - 0.98) < 1e-6:
+            line = f'  cos>{ck:.2f}:'
+            for j, dk in enumerate(DH_FOR_2D):
+                if dk in [0, 3, 5, 8, 10, 15, 20]:
+                    line += f' dh<={dk}: {surface[i, j]:.4f},'
+            print(line)
+
+    # Compute 2D gradient magnitude at key threshold (cos=0.95, dh=5)
+    # Find indices
+    i95 = int(np.argmin(np.abs(COS_FOR_2D - 0.95)))
+    j5 = int(np.argmin(np.abs(DH_FOR_2D - 5)))
+    if 0 < i95 < len(COS_FOR_2D) - 1 and 0 < j5 < len(DH_FOR_2D) - 1:
+        dcos = (surface[i95 + 1, j5] - surface[i95 - 1, j5]) / \
+               (COS_FOR_2D[i95 + 1] - COS_FOR_2D[i95 - 1])
+        ddh = (surface[i95, j5 + 1] - surface[i95, j5 - 1]) / \
+              (DH_FOR_2D[j5 + 1] - DH_FOR_2D[j5 - 1])
+        grad_mag = float(np.sqrt(dcos ** 2 + ddh ** 2))
+    else:
+        dcos = ddh = grad_mag = None
+    print(f'\n  At (cos=0.95, dh=5): rate={surface[i95, j5]:.4f}')
+    print(f'    d(rate)/d(cos) ~ {dcos:.4f} (per unit cos)')
+    print(f'    d(rate)/d(dh) ~ {ddh:.4f} (per unit dh)')
+    print(f'    gradient magnitude ~ {grad_mag:.4f}')
+
+    # ── Document-level 1D cos sweep ──
+    print('\n[Document-level 1D cos sweep at dh<=5]')
+    doc_rates_cos = [doc_alert_rate(k, 5) for k in COS_GRID]
+    for i, k in enumerate(COS_GRID):
+        if i % 4 == 0 or abs(k - 0.95) < 1e-6:
+            print(f'  cos > {k:.3f}: doc-FAR (HC) = {doc_rates_cos[i]:.4f}')
+
+    doc_cos_grad = plateau_gradient(COS_GRID, doc_rates_cos)
+    print(f'\n  doc gradient summary: min={doc_cos_grad["min"]:.5f}, '
+          f'median={doc_cos_grad["median"]:.5f}, '
+          f'max={doc_cos_grad["max"]:.5f}')
+
+    # ── Plateau detection summary ──
+    print('\n[Plateau detection summary]')
+    cos095_idx = int(np.argmin(np.abs(COS_GRID - 0.95)))
+    dh5_idx = int(np.argmin(np.abs(DH_GRID - 5)))
+    if 0 < cos095_idx < len(sig_rates_cos['pooled']) - 1:
+        local_grad_cos = abs(
+            sig_rates_cos['pooled'][cos095_idx + 1] -
+            sig_rates_cos['pooled'][cos095_idx - 1]) / \
+            (COS_GRID[cos095_idx + 1] - COS_GRID[cos095_idx - 1])
+    else:
+        local_grad_cos = None
+    if 0 < dh5_idx < len(sig_rates_dh['pooled']) - 1:
+        local_grad_dh = abs(
+            sig_rates_dh['pooled'][dh5_idx + 1] -
+            sig_rates_dh['pooled'][dh5_idx - 1]) / \
+            (DH_GRID[dh5_idx + 1] - DH_GRID[dh5_idx - 1])
+    else:
+        local_grad_dh = None
+    median_grad_cos = cos_pooled_grad['median']
+    median_grad_dh = dh_pooled_grad['median']
+    ratio_cos = (local_grad_cos / median_grad_cos
+                 if median_grad_cos and median_grad_cos > 0 else None)
+    ratio_dh = (local_grad_dh / median_grad_dh
+                if median_grad_dh and median_grad_dh > 0 else None)
+    print(f'  v3 inherited cos=0.95 local |grad|={local_grad_cos:.5f}, '
+          f'median |grad|={median_grad_cos:.5f}, '
+          f'ratio={ratio_cos:.2f}')
+    print(f'  v3 inherited dh=5    local |grad|={local_grad_dh:.5f}, '
+          f'median |grad|={median_grad_dh:.5f}, '
+          f'ratio={ratio_dh:.2f}')
+    if ratio_cos is not None and ratio_cos < 0.5:
+        print('  -> cos=0.95 IS at a low-gradient region (plateau-like).')
+    elif ratio_cos is not None and ratio_cos > 1.5:
+        print('  -> cos=0.95 IS at a high-gradient region (steep slope).')
+    else:
+        print('  -> cos=0.95 is at a moderate-gradient region '
+              '(no clear plateau or cliff).')
+    if ratio_dh is not None and ratio_dh < 0.5:
+        print('  -> dh=5 IS at a low-gradient region (plateau-like).')
+    elif ratio_dh is not None and ratio_dh > 1.5:
+        print('  -> dh=5 IS at a high-gradient region.')
+    else:
+        print('  -> dh=5 is at a moderate-gradient region.')
+
+    results['cos_sweep_at_dh_5'] = {
+        'cos_grid': COS_GRID.tolist(),
+        'sig_rates': {k: v for k, v in sig_rates_cos.items()},
+        'pooled_gradient_summary': cos_pooled_grad,
+    }
+    results['dh_sweep_at_cos_0_95'] = {
+        'dh_grid': DH_GRID.tolist(),
+        'sig_rates': {k: v for k, v in sig_rates_dh.items()},
+        'pooled_gradient_summary': dh_pooled_grad,
+    }
+    results['surface_2d'] = {
+        'cos_axis': COS_FOR_2D.tolist(),
+        'dh_axis': DH_FOR_2D.tolist(),
+        'rates': surface.tolist(),
+        'at_v3_threshold': {
+            'cos_0.95_dh_5_rate': float(surface[i95, j5]),
+            'd_rate_d_cos': dcos,
+            'd_rate_d_dh': ddh,
+            'gradient_magnitude': grad_mag,
+        },
+    }
+    results['doc_level_cos_sweep_at_dh_5'] = {
+        'cos_grid': COS_GRID.tolist(),
+        'doc_rates': doc_rates_cos,
+        'doc_gradient_summary': doc_cos_grad,
+    }
+    results['plateau_detection'] = {
+        'v3_cos_0_95': {
+            'local_gradient': local_grad_cos,
+            'median_gradient': median_grad_cos,
+            'ratio_local_to_median': ratio_cos,
+        },
+        'v3_dh_5': {
+            'local_gradient': local_grad_dh,
+            'median_gradient': median_grad_dh,
+            'ratio_local_to_median': ratio_dh,
+        },
+    }
+    json_path = OUT / 'alert_rate_results.json'
+    json_path.write_text(json.dumps(results, indent=2, ensure_ascii=False),
+                         encoding='utf-8')
+    print(f'\n[json] {json_path}')
+
+    md = [
+        '# Alert-Rate Sensitivity / Threshold-Plateau Analysis '
+        '(Script 46)',
+        '', f'Generated: {results["meta"]["timestamp"]}',
+        f'Big-4 signatures: {n_sigs:,}; documents: {n_docs:,}',
+        '',
+        ('Alert-rate sensitivity to threshold perturbation. If the '
+         'v3-inherited threshold cos>0.95 AND dh<=5 sits at a '
+         'low-gradient region, that is weak evidence the threshold is '
+         'a stable operating point. If the alert-rate surface is '
+         'everywhere smooth without a plateau, the threshold is one '
+         'point on a continuous specificity-recall tradeoff -- '
+         'consistent with the no-natural-threshold finding from '
+         'Scripts 39b-39e.'),
+        '',
+        '## Plateau detection at v3 inherited thresholds',
+        '',
+        '| Threshold | local |grad| | median |grad| | ratio | interpretation |',
+        '|---|---|---|---|---|',
+        f'| cos=0.95 | {local_grad_cos:.5f} | '
+        f'{median_grad_cos:.5f} | {ratio_cos:.2f} | '
+        f'{"plateau" if ratio_cos < 0.5 else ("cliff" if ratio_cos > 1.5 else "moderate")} |',
+        f'| dh=5 | {local_grad_dh:.5f} | {median_grad_dh:.5f} | '
+        f'{ratio_dh:.2f} | '
+        f'{"plateau" if ratio_dh < 0.5 else ("cliff" if ratio_dh > 1.5 else "moderate")} |',
+        '',
+        '## 1D cos sweep at dh<=5 (per-signature alert rate)',
+        '',
+        '| cos > k | pooled | Firm A | Firm B | Firm C | Firm D |',
+        '|---|---|---|---|---|---|',
+    ]
+    for i, k in enumerate(COS_GRID):
+        if i % 2 == 0:
+            md.append(f'| {k:.3f} | {sig_rates_cos["pooled"][i]:.4f} | '
+                      f'{sig_rates_cos["Firm A"][i]:.4f} | '
+                      f'{sig_rates_cos["Firm B"][i]:.4f} | '
+                      f'{sig_rates_cos["Firm C"][i]:.4f} | '
+                      f'{sig_rates_cos["Firm D"][i]:.4f} |')
+    md += ['',
+           '## 1D dh sweep at cos>0.95 (per-signature alert rate)',
+           '',
+           '| dh <= k | pooled | Firm A | Firm B | Firm C | Firm D |',
+           '|---|---|---|---|---|---|']
+    for i, k in enumerate(DH_GRID):
+        md.append(f'| {int(k):2d} | {sig_rates_dh["pooled"][i]:.4f} | '
+                  f'{sig_rates_dh["Firm A"][i]:.4f} | '
+                  f'{sig_rates_dh["Firm B"][i]:.4f} | '
+                  f'{sig_rates_dh["Firm C"][i]:.4f} | '
+                  f'{sig_rates_dh["Firm D"][i]:.4f} |')
+    md += ['',
+           '## Document-level cos sweep at dh<=5',
+           '',
+           '| cos > k | doc alert rate (HC) |',
+           '|---|---|']
+    for i, k in enumerate(COS_GRID):
+        if i % 2 == 0:
+            md.append(f'| {k:.3f} | {doc_rates_cos[i]:.4f} |')
+    md.append('')
+    md_path = OUT / 'alert_rate_report.md'
+    md_path.write_text('\n'.join(md), encoding='utf-8')
+    print(f'[md  ] {md_path}')
+
+
+if __name__ == '__main__':
+    main()