Apply codex round-25 final polish: §III v6 + §IV v3.2

Codex round 25 returned Minor Revision: round-24's empirical and cross-reference issues mostly CLOSED. Remaining items were all partner-facing cosmetic / internal-notes hygiene. §III v6 polish: 1. §III:11 v5 changelog reprint of real firm names removed ("real firm names 'EY' and 'KPMG'" -> "real firm names/aliases") -- this was a self-regression I introduced in v5 while documenting the v5 anonymisation fix. 2. §III:14 empirical anchor range updated: "Scripts 32-40" -> "Scripts 32-42" (includes Scripts 41 + 42). 3. New v6 changelog entry added under the draft note documenting the round-25 fixes. 4. Draft note version stamp refreshed: v5 -> v6. §IV v3.2 polish: 1. §IV draft note rewritten and version label corrected: "Draft v3" -> "Draft v3.2"; "post codex rounds 21-23" -> "post codex rounds 21-25". The v3 -> v3.1 -> v3.2 lineage is now recorded. 2. §IV close-out checklist item 2 rewritten to remove residual "Tables IV-XVIII" wording. v3.2 explicitly states: v4 table sequence is Tables V-XVIII plus Table XV-B; no v4 Table IV is printed; the inherited v3.20.0 Table IV (per-firm detection counts) remains a v3.x reference only. Verification: - Strict-case grep for KPMG / Deloitte / PwC / EY (with word boundaries) + Chinese firm names: ZERO matches in either file. Anonymisation is now complete throughout the manuscript body AND internal notes. Round 25 closure post-polish: Major: all CLOSED (round 24 Major 1 table numbering: now fully explicit V-XVIII + XV-B with v4 Table IV absent; Major 4 anonymisation: §III:11 leak removed) Minor: all CLOSED (weight drift 0.023 confirmed across 4 sites; cos <= 0.837 confirmed across 2 sites; n=686 provenance row confirmed) Editorial: 1 still PARTIAL (internal draft notes + Phase 3 close-out checklist remain in the files but explicitly marked "internal -- remove before submission"; these are author working artefacts intentionally retained until submission packaging) Phase 4 readiness: technically Yes; the §III/§IV technical content is converged across 5 codex review rounds. Internal notes will be stripped at submission packaging time. Ready to proceed to Phase 4 (Abstract/Intro/Discussion/Conclusion prose). Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
Apply codex round-24 final cleanup: §III v5 + §IV v3.1
2026-05-12 22:36:16 +08:00 · 2026-05-12 22:26:14 +08:00 · 2026-05-12 17:03:33 +08:00 · 2026-05-12 16:45:22 +08:00 · 2026-05-12 16:35:37 +08:00 · 2026-05-12 16:32:39 +08:00
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 # 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*
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 # 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
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 # 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).*
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 # STATE — Current snapshot
 **Date**: 2026-05-12
 **Active milestone**: Paper A v4.0 — Big-4 reframe
 **Active branch**: `paper-a-v4-big4` (12 commits ahead of `yolo-signature-pipeline`)
 **Active phase**: Phase 2 — Methodology rewrite, draft delivered, **awaiting user review of 5 open questions in `paper/v4/paper_a_methodology_v4_section_iii.md`** before Phase 3 begins
 ## Recently completed
 **Phase 1 (Foundation, 7 spike + foundation scripts)**:
 - Script 32 (`e1d81e3`): non-Firm-A calibration verdict C
 - Script 33 (`8ac0988`): reverse-anchor PAPER_C_STRONG (directional ρ=+0.744)
 - Script 34 (`55f9f94`): Big-4 K=2 dip-test multimodal p<0.0001, bootstrap CI [0.974, 0.977] / [3.48, 3.97]
 - Script 35 (`55f9f94`): firm × cluster — Firm A 0% C1 / 82.5% C3, PwC 23.5% C1
 - Script 36 (`ccd9f23`): K=2 LOOO **UNSTABLE** (firm-mass conflation; max Δcos=0.028)
 - Script 37 (`92f1db8`): K=3 LOOO **PARTIAL** (component shape stable, membership ±5-13pp)
 - Script 38 (`bc36dcc`): convergence **STRONG** — 3 lenses pairwise ρ ≥ 0.879
 - Script 39 (`39575ce`): per-signature convergence **MODERATE** — κ=0.87 between per-CPA and per-sig K=3 fits
 - Script 40 (`338737d`): pixel-identity FAR = **0%** on n=262 ground-truth replicated
 **Phase 2 (Methodology rewrite)**: §III-G..L draft delivered at `paper/v4/paper_a_methodology_v4_section_iii.md` (commit on the same branch). Single coherent rewrite covering 6 sub-sections (G/H/I/J/K/L); cross-references to all 9 spike scripts; 5 open questions flagged at end of draft for user decision.
 ## Pending — Phase 2 user review (BEFORE Phase 3)
 5 decisions needed from user before Phase 3 (Results regeneration) starts:
 1. §III-G scope justification — three-point argument enough, or add a fourth?
 2. §III-H Firm A phrasing — "case study of templated end" vs an alternative framing?
 3. §III-J K=3 vs K=2 selection — lean on LOOO (current draft) or strengthen BIC argument?
 4. §III-L hybrid classifier — keep inherited 5-way box rule, or commit to K=3 hard label as primary?
 5. Section IV table numbering scheme — confirm before Phase 3 builds tables.
 Plus: any prose-level edits the user wants on the §III draft.
 ## Blockers
 None.
 ## Open questions deferred from spike
 - Bootstrap stability of cosine and dHash crossings *jointly* (not just marginally) — addressed in Phase 1 if time permits
 - K=2 vs K=3 final choice for §III-J — both reported, but operational classifier needs to commit to one (recommend K=2 for interpretability; K=3 in supplementary)
 ## Things to remember (per memory)
 - Provenance-verify all empirical claims against fresh sqlite/grep ([[feedback-provenance-fabrication]])
 - Don't mock the DB or use placeholders — every number must trace to a script + query
 - Partner Jimmy already proposed Big-4 direction (this is execution, not pitching a new direction)
 - Paper C standalone is shelved — folded into v4.0 §IV-K
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 # 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.
@@ -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.
@@ -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.
@@ -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`.
@@ -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).
@@ -0,0 +1,210 @@
 # Section III. Methodology — v4.0 Draft v6 (post codex rounds 21–25)
 > **Draft note (2026-05-12, v6; internal — remove before submission).** This file replaces the §III-G through §III-L 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.
 >
 > **v2** incorporated codex gpt-5.5 round-21 review (`paper/codex_review_gpt55_v4_round1.md`, Major Revision); key revisions were: (i) the inherited five-way per-signature box rule restored as the **primary operational classifier** (§III-L), (ii) the K=3 Gaussian mixture positioned as **accountant-level descriptive characterisation** (§III-J), (iii) "convergent validation" softened to "convergent internal-consistency checks" since the three scores share underlying features (§III-K), (iv) the pixel-identity metric renamed from FAR to positive-anchor miss rate (§III-K), (v) five empirical/wording slips corrected.
 >
 > **v3** incorporates codex gpt-5.5 round-22 review (`paper/codex_review_gpt55_v4_round2.md`, Minor Revision); five narrow fixes applied: per-firm ranking corrected (Score 2 reverse-anchor ranks Firm D fractionally above Firm C while Scores 1 and 3 rank Firm C highest), "smallest scope" language narrowed to "comparison scopes tested in Script 32", §III-L Spearman correlations explicitly tied to the K=3 *posterior* P(C1), provenance for $n = 150{,}442$ cites Script 39 directly, "max fold-to-fold deviation" wording made precise ($0.028$ = max absolute deviation from across-fold mean; pairwise range $0.0376$).
 >
 > **v4** incorporates the §III ↔ §IV cross-reference cleanup that codex round-23 review flagged: §III-G unit references now point to actual §IV locations (§IV-J for five-way per-signature counts; §IV-I for inherited inter-CPA FAR), §III-G scope statement enumerates v4-new vs inherited sub-sections explicitly, §III-K cites v3.20.0 Tables IX/XI/XII/XII-B for moderate-band capture-rate (was "§IV-F" which is now Convergent Internal-Consistency), and §III-L's "without recalibration" claim is narrowed to apply only to the binary high-confidence sub-rule.
 >
 > **v5** incorporates codex gpt-5.5 round-24 review (`paper/codex_review_gpt55_v4_round4.md`, Minor Revision); seven narrow §III-side cleanups: (1) anonymisation leak repaired (real firm names/aliases removed from §III prose; Firm A–D used throughout); (2) K=3 LOOO weight-drift value $0.025$ corrected to $0.023$ at three §III sites (matches Script 37); (3) §III-K positive-anchor paragraph cross-ref repaired (now points to §IV-I and v3.20.0 §IV-F.1 Table X, was the meaningless "§III-J inherited; Table X"); (4) §III-L five-way rule's Likely-hand-signed band made inclusive ($\text{cos} \leq 0.837$, matches Script 42); (5) open question 1's location pointer changed from current §IV-F to v3.20.0 Tables IX/XI/XII/XII-B and §IV-J descriptive proportions; (6) provenance row added for the full-dataset $n = 686$ claim citing Script 41; (7) draft-note dates and version stamps refreshed.
 >
 > **v6** incorporates codex gpt-5.5 round-25 review (`paper/codex_review_gpt55_v4_round5.md`, Minor Revision): empirical anchor range updated to Scripts 32–42 (was 32–40, missed Scripts 41 and 42).
 >
 > Empirical anchors throughout reference Scripts 32–42 on branch `paper-a-v4-big4`; a provenance table appears at the end of this section listing every numerical claim with its 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 v3.20.0's inherited inter-CPA FAR 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.
 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, 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 FAR), 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 $n_{\text{sig}} \geq 10$ (Scripts 36, 38), totalling 150,442 Big-4 signatures with both descriptors available (Script 39 reports the explicit per-signature $n$ used in the signature-level K=3 fit). Restricting the v4-new analyses to Big-4 is a methodological choice driven by four considerations:
 1. **Within-pool homogeneity for mixture characterisation.** Pooling Big-4 with mid- and small-firm CPAs introduces a heterogeneous tail of $\sim$249 CPAs distributed across multiple firms with idiosyncratic signing practices and small per-firm samples. The full-sample and Big-4-only calibrations *differ* in their fitted marginal crossings (full-sample published $\overline{\text{cos}}^* = 0.945$, $\overline{\text{dHash}}^* = 8.10$ from v3.x; Big-4-only $\overline{\text{cos}}^* = 0.975$, $\overline{\text{dHash}}^* = 3.76$ from Script 34; bootstrap 95% CIs $[0.974, 0.977]$ / $[3.48, 3.97]$, $n_{\text{boot}} = 500$); the offset is large compared to the Big-4 bootstrap CI half-width of $0.0015$. We report this as a *scope-dependent shift* rather than asserting a causal "mid/small-firm tail distorts" claim.
 2. **Statistical multimodality at the accountant level.** Within the Big-4 sub-corpus, the Hartigan dip test rejects unimodality on both axes (cosine $p = 0.0000$, dHash $p = 0.0000$ in the bootstrap-2000 implementation, i.e., no bootstrap replicate exceeded the observed statistic; reported here as $p < 5 \times 10^{-4}$; Script 34). No such rejection holds in the comparison scopes tested by Script 32: Firm A pooled alone ($p_{\text{cos}} = 0.992$, $p_{\text{dHash}} = 0.924$, $n = 171$); Firms B + C + D pooled (Script 32, `big4_non_A` subset: $p_{\text{cos}} = 0.998$, $p_{\text{dHash}} = 0.906$); all non-Firm-A pooled (Script 32, `all_non_A` subset: other Big-4 plus mid/small firms; $p_{\text{cos}} = 0.998$, $p_{\text{dHash}} = 0.907$). Among the comparison scopes we evaluated, Big-4 is the smallest scope at which the dip test supports applying a finite-mixture model to the per-CPA distribution; we did not separately test single-firm dip statistics for Firms B, C, or D.
 3. **Reproducibility under leave-one-firm-out cross-validation.** §III-K reports leave-one-firm-out (LOOO) cross-validation of the Big-4 mixture 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.
 4. **Restricted generalizability claim.** v4.0's primary claims are scoped to the Big-4 audit-report context; we do not assert that the same mixture structure or operational thresholds extend 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 and (b) as a robustness comparison in §IV-K. Generalisation beyond Big-4 is left as future work.
 ## 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 most digitally-replicated of the Big-4. In the Big-4 K=3 mixture (§III-J; Scripts 35, 38), Firm A accounts for 0% of the C1 hand-leaning component (cos $\approx 0.946$, dHash $\approx 9.17$, weight $\approx 0.143$), 17.5% of the C2 mixed component, and 82.5% of the C3 replicated component; 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 hand-leaning component (cos $= 0.946$, dHash $= 9.17$; §III-J); compared to the Big-4 templated 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 more hand-leaning Big-4 CPA. This is a "deviation in the hand-leaning direction" measure, not a "deviation toward replication" measure; the reference is the less-replicated population.
 ## I. Distributional Characterisation at the Accountant Level
 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. Three diagnostic procedures are applied: a univariate unimodality test on each marginal axis, a 2D Gaussian mixture fit (developed in §III-J), and a density-smoothness diagnostic.
 **1. Hartigan dip test on each 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 rejected unimodality. The dip-test multimodality at the Big-4 level is the empirical justification for fitting a finite-mixture model in §III-J; without it, the mixture would be a forced fit on an essentially unimodal distribution.
 **2. Mixture-model evidence.** 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, and the operational role of each fit is developed in §III-J and §III-K.
 **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 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 the mixture-model evidence: the K=3 components overlap rather than separate sharply, so a local-discontinuity test does not flag a transition. We retain BD/McCrary in v4.0 as a non-parametric robustness diagnostic; the dHash transitions outside Big-4 are not used as operational thresholds because they are scope-dependent and lie within rather than between modes of the corresponding density.
 ## J. Mixture Model and Accountant-Level Characterisation
 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 characterisations of the joint Big-4 distribution; the operational per-signature classifier remains the inherited five-way box rule of §III-L.** Neither mixture is used to assign signature-level or document-level labels in the v4.0 primary analysis.
 **K=2 fit.** Two components at $(\overline{\text{cos}}, \overline{\text{dHash}}) = (0.954, 7.14)$ ("hand-leaning"), weight $0.689$, and $(0.983, 2.41)$ ("replicated"), weight $0.311$ (Script 34). $\text{BIC}(K{=}2) = -1108.45$. Marginal crossings: $\overline{\text{cos}}^* = 0.9755$, $\overline{\text{dHash}}^* = 3.755$.
 **K=3 fit.** Three components, sorted by ascending cosine mean (Script 35; Script 38 reproduces):
 | Component | $\overline{\text{cos}}$ | $\overline{\text{dHash}}$ | weight | descriptive label |
 |---|---|---|---|---|
 | C1 | 0.9457 | 9.17 | 0.143 | hand-leaning |
 | C2 | 0.9558 | 6.66 | 0.536 | mixed |
 | C3 | 0.9826 | 2.41 | 0.321 | replicated |
 $\text{BIC}(K{=}3) = -1111.93$, lower than $K{=}2$ by $3.48$ (mild support for K=3 under standard BIC interpretation but not by itself decisive).
 **Why we report both K=2 and K=3.** Leave-one-firm-out cross-validation (§III-K) 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* — not the bootstrap CI; the full-Big-4 bootstrap cosine half-width is the much smaller $0.0015$. 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 more 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`, with the explicit interpretation that "the C1 cluster exists but membership is not well-predicted by the held-out fit" and "K=3 is not predictively useful as an operational classifier" (Script 37 verdict legend).
 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 a 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 shape across LOOO folds; a hand-leaning component (C1) exists at $\overline{\text{cos}} \approx 0.946$, $\overline{\text{dHash}} \approx 9.17$ with weight $\approx 0.14$.
 - Hard-posterior membership in C1 is composition-sensitive (max absolute deviation $12.8$ pp across LOOO folds, exceeding the report's $5$ pp viability bar); K=3 is therefore not used to assign operational hand-leaning labels to CPAs in v4.0.
 The operational signature-level classifier remains the inherited five-way box rule of §III-L, calibrated as in v3.x. Cross-checks between the inherited rule and the K=3 mixture appear in §III-K.
 ## K. Convergent Internal-Consistency Checks
 The mixture characterisation 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. 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 (mixture posterior):** $P(\text{C1}_{\text{hand-leaning}})$ from the K=3 fit of §III-J — 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 more hand-leaning Big-4 CPA. This is a function of $\overline{\text{cos}}_a$ alone.
 - **Score 3 (Paper A box-rule hand-leaning 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 | $+0.963$ | $< 10^{-248}$ |
 | Score 2 vs Score 3 | $+0.889$ | $< 10^{-149}$ |
 | Score 1 vs Score 2 | $+0.879$ | $< 10^{-142}$ |
 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 hand-leaning ranking with $\rho > 0.87$. The three scores agree on placing Firm A as the most replication-dominated and the three non-Firm-A Big-4 firms as more hand-leaning, but they do not all rank the non-Firm-A firms identically: the K=3 posterior P(C1) and the box-rule hand-leaning rate (Scores 1 and 3) place Firm C at the most-hand-leaning end of Big-4 (mean P(C1) $= 0.311$; mean box-rule hand-leaning 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 most-hand-leaning 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 (replicated vs not-replicated):
 | 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 hand-leaning 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 classifier's *positive-anchor miss rate* — the fraction of byte-identical signatures misclassified as hand-leaning. This is a one-sided check against a conservative positive subset, **not a false-alarm rate in the usual two-class sense**; we do not report a paired false-alarm rate because no signature-level hand-signed ground truth exists. The corresponding signature-level false-alarm-rate evidence comes from the v3.x inter-CPA negative anchor (§IV-I, inheriting v3.20.0 §IV-F.1 Table X), which retains its v3.x interpretation:
 | Candidate classifier | 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 = replicated; 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. Signature- and Document-Level Classification
 The v4.0 operational classifier is the inherited v3.x five-way per-signature box rule, retained unchanged for two reasons: (a) it preserves continuity with the v3.x literature and its established interpretation; (b) the convergent internal-consistency checks of §III-K show that the box rule's *binary high-confidence* output (cos $> 0.95$ AND dHash $\leq 5$) agrees at $\rho \geq 0.96$ per-CPA with a K=3-posterior score and at $\rho \geq 0.89$ with a reverse-anchor score. The §III-K checks cover only the binary high-confidence rule; the five-way classifier's moderate-confidence band ($5 < \text{dHash} \leq 15$), the style-consistency band ($\text{dHash} > 15$), and the document-level worst-case aggregation are not separately validated by Scripts 38–42. We retain those rule components by reference to v3.20.0's calibration (v3.20.0 §III-K and Tables IX, XI, XII, XII-B); we do not claim that v4.0's convergent-checks evidence supports the inherited rule as a whole, only its binary high-confidence sub-rule.
 **Per-signature five-way classifier.** Operational thresholds are anchored on whole-sample Firm A percentile heuristics as in v3.x: cos $> 0.95$ (Firm A P7.5) for the cosine dimension and dHash$_{\text{indep}} \leq 5$ / $> 15$ for the structural dimension. All dHash references refer to the *independent-minimum* dHash defined in §III-G. We assign each signature to one of five signature-level categories using convergent evidence from both descriptors:
 1. **High-confidence non-hand-signed:** Cosine $> 0.95$ AND $\text{dHash}_{\text{indep}} \leq 5$. Both descriptors converge on strong replication evidence.
 2. **Moderate-confidence non-hand-signed:** 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:** 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:** Cosine between the all-pairs intra/inter KDE crossover (0.837) and 0.95 without sufficient convergent evidence in either direction.
 5. **Likely hand-signed:** Cosine at or below the all-pairs KDE crossover threshold (cos $\leq 0.837$).
 The conventions about these thresholds (cosine 0.95 as an operating point chosen for capture-vs-FAR tradeoff against the inter-CPA negative anchor, with Wilson 95% inter-CPA FAR of $0.0005$ at the operating point; cosine 0.837 as the all-pairs KDE crossover; dHash 5 and 15 as the upper tail of the high-similarity mode and the structural-similarity ceiling respectively) are inherited from v3.x §III-K and retain their v3.x calibration and capture-rate evidence (v3.x Tables IX, XI, XII, XII-B).
 **Document-level aggregation.** Each audit report typically carries two certifying-CPA signatures. We aggregate signature-level outcomes to document-level labels using the v3.x worst-case rule: the document inherits the *most-replication-consistent* signature label among the two signatures (rank order: High-confidence $>$ Moderate-confidence $>$ Style-consistency $>$ Uncertain $>$ Likely-hand-signed). The aggregation rule reflects the detection goal of flagging any potentially non-hand-signed report.
 **K=3 as accountant-level characterisation, not 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 for the accountant-level cluster cross-tabulation (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.
 ---
 ## Provenance table for numerical claims in §III-G through §III-L
 | 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 | $150{,}442$ | Script 39 (per-signature K=3 fit explicitly cites this $n$) | direct |
 | 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 FAR $0.0005$ at cos $> 0.95$ (Wilson 95% $[0.0003, 0.0007]$) | direct | v3 §IV-F.1 / Table X (inherited) | inherited from v3 |
 | 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 |
 ---
 ## Cross-reference index (author working checklist; remove before submission)
 - **Big-4 sub-corpus definition** (§III-G) — 437 CPAs, 150,442 signatures.
 - **Reference populations** (§III-H) — Firm A as templated-end case study; non-Big-4 ($n = 249$) as reverse-anchor reference (less-replicated population).
 - **Distributional characterisation** (§III-I) — Big-4 dip-test multimodality ($p < 5 \times 10^{-4}$); BD/McCrary null at Big-4 scope; mixture support.
 - **K=3 components, descriptive only** (§III-J) — C1 hand-leaning, C2 mixed, C3 replicated; LOOO supports descriptive use, not operational classification.
 - **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 $n = 262$.
 - **Per-document classifier** (§III-L) — inherited five-way rule retained as primary; K=3 demoted to characterisation only.
 ## 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).
@@ -0,0 +1,269 @@
 # Section IV. Results — v4.0 Draft v3.2 (post codex rounds 21–25)
 > **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 three-diagnostic distributional characterisation at the Big-4 accountant level. All numbers below are direct re-statements from Scripts 32 / 34; cross-citations to the v3.x (signature-level) analysis are noted where the v4.0 result differs structurally from the v3.x result.
 **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 and marginal-crossing bootstrap 95% confidence intervals.
 | K=2 component | $\overline{\text{cos}}$ | $\overline{\text{dHash}}$ | weight |
 |---|---|---|---|
 | Hand-leaning | 0.954 | 7.14 | 0.689 |
 | Replicated | 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.
 | K=3 component | $\overline{\text{cos}}$ | $\overline{\text{dHash}}$ | weight | descriptive label |
 |---|---|---|---|---|
 | C1 | 0.9457 | 9.17 | 0.143 | hand-leaning |
 | C2 | 0.9558 | 6.66 | 0.536 | mixed |
 | C3 | 0.9826 | 2.41 | 0.321 | replicated |
 $\text{BIC}(K{=}3) = -1111.93$, 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 hand-leaning rate | $+0.9627$ | $< 10^{-248}$ |
 | Reverse-anchor cosine percentile vs Paper A box-rule hand-leaning 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 hand-leaning 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 = more hand-leaning 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 more hand-leaning. The K=3 posterior P(C1) and the box-rule hand-leaning rate (Score 1 and Score 3) place Firm C at the most-hand-leaning 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, replicated vs not-replicated), $n = 150{,}442$ 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 hand-leaning | 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 = replicated; 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 Negative-Anchor FAR (inherited from v3.x)
 The signature-level inter-CPA negative-anchor FAR analysis (~50,000 random pairs from different CPAs; v3.20.0 §IV-F.1, Table X) is inherited unchanged. The v3.x result, reproduced here for reference: at the operational cosine cut of $0.95$, the inter-CPA FAR is $0.0005$ (Wilson 95% CI $[0.0003, 0.0007]$). v4.0 does not regenerate this analysis on the Big-4 subset; the inter-CPA negative-anchor logic is corpus-wide and the v3.x FAR remains the operational specificity reference for the §III-L operational rule.
 ## 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.)
 **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-replicated 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 XV-B.** 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 hand-leaning ranking (e.g., Firm D's MC fraction is lower than Firm B's while Firm D's reverse-anchor score ranks it as more hand-leaning than Firm B).
 **Table XVI.** Firm × K=3 cluster cross-tabulation, Big-4 sub-corpus.
 | Firm | $n$ | C1 (hand-leaning) | C2 (mixed) | C3 (replicated) | 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 replicated component (no Firm A CPAs in C1); Firm C has the highest hand-leaning 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 XV-B 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 hand-leaning 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 hand-leaning | 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 mixed | 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 replicated | 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 hand-leaning rate, Big-4 sub-corpus vs full dataset.
 | Scope | $n$ CPAs | Spearman $\rho$ (P(C1) vs Paper A hand-leaning 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 hand-leaning rate are preserved at the full scope. Component centres shift modestly: C3 (replicated) is essentially unchanged in centre but loses weight $0.117$ as the full population includes more non-templated CPAs (mid/small firms); C1 (hand-leaning) 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 hand-leaning CPAs 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.
 ---
 ## 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.
@@ -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()
@@ -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()
@@ -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()
@@ -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()
@@ -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()
@@ -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()
@@ -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()
@@ -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()
@@ -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()
@@ -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()
@@ -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()
@@ -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()