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Phase 6 manuscript splice (2/2): §IV / §V / §VI spliced

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Lands v4.0 §IV / §V / §VI content into v3.20.0 master sub-files.
Strips internal close-out checklists, draft notes, and open-questions
blocks at splice. Completes the Phase 6 manuscript-master file
assembly.

§IV Results (paper_a_results_v3.md):
- §IV-A..C: kept v3.20.0 inherited content (experimental setup,
  detection performance, all-pairs distribution); added v4 scope
  note (Big-4 primary) at the §IV header
- §IV-D..K: replaced v3.20.0 §IV-D..H with v4.0 §IV-D..K (Big-4
  distributional / mixture / convergence / LOOO / pixel-identity /
  inter-CPA reference / five-way classification / full-dataset
  robustness)
- §IV-L: renumbered v3.20.0 §IV-I (backbone ablation) content to
  match v4's "§IV-L inherited from v3.20.0 §IV-I" reframing
- §IV-M: appended v4.0 ICCR calibration tables (XX-XXVI):
  composition decomposition, per-comparison/per-signature/
  per-document ICCRs, firm heterogeneity + cross-firm hit matrix,
  alert-rate sensitivity
- §III-K ablation cross-ref updated to §IV-L (was §IV-I)
- Phase 3 close-out checklist (lines 365+) stripped

§V Discussion (paper_a_discussion_v3.md):
- Replaced v3.20.0 §V with v4.0 §V (8 sub-sections A-H):
  A. Distinct problem framing
  B. Continuous quality spectrum + composition-driven multimodality
  C. Firm A as templated end (case study, not anchor)
  D. K=2 / K=3 descriptive partitions
  E. Three-score convergent internal-consistency
  F. Anchor-based multi-level calibration
  G. Pixel-identity hard positive anchor + ICCR reframing
  H. Limitations (14 items: 9 v4-specific + 5 inherited from v3.x)

§VI Conclusion (paper_a_conclusion_v3.md):
- Replaced v3.20.0 §VI with v4.0 §VI (8 contribution items mirroring
  §I contributions; 4-direction future work).

Known splice-time issue (deferred to typesetting): §IV table numbering
is sequential by label (V, VI, ..., XXVI) but Table XIX (document-level
worst-case) appears physically before Tables XVI/XVII/XVIII in §IV-J
narrative flow. IEEE Access typesetters typically normalize table order
during typesetting; we accept the in-file ordering quirk to preserve
the §IV-J narrative arc (per-signature -> document-level worst-case ->
K=3 cross-tab). Renumbering to strictly-ascending physical order would
require renaming Tables XVI/XVII/XVIII -> XVII/XVIII/XIX with
downstream cross-reference updates; deferred unless partner Jimmy
review or IEEE Access submission portal flags it.

Manuscript splice complete. Working drafts in paper/v4/ retained as
archive of the round-by-round Phase 5 fix history.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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# IV. Experiments and Results
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. §IV-A through §IV-C report inherited corpus-wide v3.x material; §IV-L (feature backbone ablation) is also inherited. §IV-M consolidates the v4-new anchor-based ICCR calibration tables.
## A. Experimental Setup
Experiments used mixed hardware: YOLOv11n training and inference for signature detection, and ResNet-50 forward inference for feature extraction over all 182,328 detected signatures, were performed on an Nvidia RTX 4090 (CUDA); the downstream statistical analyses (KDE antimode, Hartigan dip test, Beta-mixture EM with logit-Gaussian robustness check, Burgstahler-Dichev/McCrary density-smoothness diagnostic, and pairwise cosine/dHash computations) were performed on an Apple Silicon workstation with Metal Performance Shaders (MPS) acceleration.
@@ -25,10 +27,12 @@ The high VLM--YOLO agreement rate (98.8%) further corroborates detection reliabi
| Processing rate | 43.1 docs/sec |
-->
The Big-4 subset of the detection output yields 150,442 signatures with both descriptors (cosine and independent dHash) successfully computed; this is the per-signature population used in all §IV v4 primary analyses (§IV-D through §IV-J).
## C. All-Pairs Intra-vs-Inter Class Distribution Analysis
Fig. 2 presents the cosine similarity distributions computed over the full set of *pairwise comparisons* under two groupings: intra-class (all signature pairs belonging to the same CPA) and inter-class (signature pairs from different CPAs).
This all-pairs analysis is a different unit from the per-signature best-match statistics used in Sections IV-D onward; we report it first because it supplies the reference point for the KDE crossover used in per-document classification (Section III-K).
This all-pairs analysis is a different unit from the per-signature best-match statistics used in Sections IV-D onward; we report it first because it supplies the reference point for the KDE crossover used in per-document classification (Section III-L).
Table IV summarizes the distributional statistics.
<!-- TABLE IV: Cosine Similarity Distribution Statistics
@@ -54,417 +58,248 @@ We emphasize that pairwise observations are not independent---the same signature
We therefore rely primarily on Cohen's $d$ as an effect-size measure that is less sensitive to sample size.
A Cohen's $d$ of 0.669 indicates a medium effect size [29], confirming that the distributional difference is practically meaningful, not merely an artifact of the large sample count.
## D. Signature-Level Distributional Characterisation
This section applies the threshold-estimator and density-smoothness diagnostic of Section III-I to the per-signature similarity distribution.
The joint reading is that per-signature similarity is a continuous quality spectrum rather than a clean two-mechanism mixture, which is why the operational classifier (Section III-K) anchors its cosine cut on the whole-sample Firm A P7.5 percentile rather than on any mixture-fit crossing.
### 1) Hartigan Dip Test: Unimodality at the Signature Level
Applying the Hartigan & Hartigan dip test [37] to the per-signature best-match distributions reveals a critical structural finding (Table V).
The $N = 168{,}740$ count used in Table V and in the downstream same-CPA per-signature best-match analyses (Tables V and XII, and the Firm-A per-signature rows of Tables XIII and XVIII) is $15$ signatures smaller than the $168{,}755$ CPA-matched count reported in Table III: these $15$ signatures belong to CPAs with exactly one signature in the entire corpus, for whom no same-CPA pairwise best-match statistic can be computed, and are therefore excluded from all same-CPA similarity analyses.
<!-- TABLE V: Hartigan Dip Test Results
| Distribution | N | dip | p-value | Verdict (α=0.05) |
|--------------|---|-----|---------|------------------|
| Firm A cosine (max-sim) | 60,448 | 0.0019 | 0.169 | Unimodal |
| Firm A min dHash (independent) | 60,448 | 0.1051 | <0.001 | Multimodal |
| All-CPA cosine (max-sim) | 168,740 | 0.0035 | <0.001 | Multimodal |
| All-CPA min dHash (independent) | 168,740 | 0.0468 | <0.001 | Multimodal |
-->
Firm A's per-signature cosine distribution *fails to reject unimodality* ($p = 0.17$), a pattern consistent with a dominant high-similarity regime plus a long left tail attributable to within-firm heterogeneity in signing outputs (Section III-G discusses the scope of partner-level claims).
The all-CPA cosine distribution, which mixes many firms with heterogeneous signing practices, is *multimodal* ($p < 0.001$).
The Firm A unimodal-long-tail finding is, in conjunction with the byte-identity, partner-ranking, and intra-report evidence reported below, consistent with the replication-dominated framing (Section III-H): a dominant high-similarity regime plus residual within-firm heterogeneity, rather than two cleanly separated mechanisms.
### 2) Burgstahler-Dichev / McCrary Density-Smoothness Diagnostic
Applying the BD/McCrary procedure (Section III-I.3) to the per-signature cosine distribution yields a nominally significant $Z^- \rightarrow Z^+$ transition at cosine 0.985 for Firm A and 0.985 for the full sample; the min-dHash distributions exhibit a transition at Hamming distance 2 for both Firm A and the full sample under the bin width ($0.005$ / $1$) used here.
Two cautions, however, prevent us from treating these signature-level transitions as thresholds.
First, the cosine transition at 0.985 lies *inside* the non-hand-signed mode rather than at the separation between two mechanisms, consistent with the dip-test finding that per-signature cosine is not cleanly bimodal.
Second, Appendix A documents that the signature-level transition locations are not bin-width-stable (Firm A cosine drifts across 0.987, 0.985, 0.980, 0.975 as the bin width is widened from 0.003 to 0.015, and full-sample dHash transitions drift across 2, 10, 9 as bin width grows from 1 to 3), which is characteristic of a histogram-resolution artifact rather than of a genuine density discontinuity between two mechanisms.
We therefore use BD/McCrary as a density-smoothness diagnostic rather than as an independent threshold estimator.
### 3) Beta Mixture at Signature Level: A Forced Fit
Fitting 2- and 3-component Beta mixtures to Firm A's per-signature cosine via EM yields a clear BIC preference for the 3-component fit ($\Delta\text{BIC} = 381$), with a parallel preference under the logit-GMM robustness check.
For the full-sample cosine the 3-component fit is likewise strongly preferred ($\Delta\text{BIC} = 10{,}175$).
Under the forced 2-component fit the Firm A Beta crossing lies at 0.977 and the logit-GMM crossing at 0.999---values sharply inconsistent with each other, indicating that the 2-component parametric structure is not supported by the data.
Under the full-sample 2-component forced fit no Beta crossing is identified; the logit-GMM crossing is at 0.980.
### 4) Joint Reading of the Three Diagnostics
The three diagnostics agree that per-signature similarity does not form a clean two-mechanism mixture:
(i) the Hartigan dip test fails to reject unimodality for Firm A and rejects it for the heterogeneous-firm pooled sample;
(ii) BIC strongly prefers a 3-component over a 2-component Beta fit, so the 2-component crossing is a *forced fit* and the Beta-vs-logit-Gaussian disagreement (0.977 vs 0.999 for Firm A) reflects parametric-form sensitivity rather than a stable two-mechanism boundary;
(iii) the BD/McCrary procedure locates its candidate transition *inside* the non-hand-signed mode rather than between modes, and the transition is not bin-width-stable.
Table VI summarises the signature-level threshold-estimator outputs for cross-method comparison.
<!-- TABLE VI: Signature-Level Threshold-Estimator Summary
| Population | Method | Cosine threshold | dHash threshold | Status |
|------------|--------|------------------|-----------------|--------|
| **Threshold estimators (signature-level distributional fits)** | | | | |
| Firm A signature-level | KDE antimode + Hartigan dip (Section III-I.1) | undefined | — | unimodal at $\alpha=0.05$ ($p=0.169$); antimode not defined for unimodal data |
| Firm A signature-level | Beta-2 EM crossing (Section III-I.2) | 0.977 | — | forced fit; BIC strongly prefers $K{=}3$ ($\Delta\text{BIC} = 381$) |
| Firm A signature-level | logit-Gaussian-2 crossing (robustness check) | 0.999 | — | forced fit; sharply inconsistent with Beta-2 crossing—reflects parametric-form sensitivity |
| Full-sample signature-l. | KDE antimode + Hartigan dip | (multiple modes) | — | multimodal ($p<0.001$); KDE crossover at full-sample is dominated by between-firm heterogeneity |
| Full-sample signature-l. | Beta-2 EM crossing | no crossing | — | forced fit; component densities do not cross over $[0,1]$ under recovered parameters |
| Full-sample signature-l. | logit-Gaussian-2 crossing | 0.980 | — | forced fit; BIC strongly prefers $K{=}3$ ($\Delta\text{BIC} = 10{,}175$) |
| **Density-smoothness diagnostics (not threshold estimators)** | | | | |
| Firm A signature-level | BD/McCrary candidate transition (Section III-I.3) | 0.985 (bin 0.005)| 2.0 (bin 1) | bin-unstable across $\{0.003, 0.005, 0.010, 0.015\}$ (Appendix A); transition lies *inside* the non-hand-signed mode |
| Full-sample signature-l. | BD/McCrary candidate transition | 0.985 (bin 0.005) | 2.0 (bin 1) | bin-unstable across $\{0.003, 0.005, 0.010, 0.015\}$ (Appendix A) |
| **Reference: between-class KDE (different unit of analysis)** | | | | |
| All-pairs intra/inter (pair-level; Section IV-C) | KDE crossover | 0.837 | — | reference point for the Uncertain/Likely-hand-signed boundary in the operational classifier |
| **Operational classifier anchors and percentile cross-references** | | | | |
| Firm A whole-sample | P7.5 (operational anchor; Section III-K) | 0.95 | — | operational cosine cut for the five-way classifier |
| Firm A whole-sample | dHash$_\text{indep}$ P75 | — | 4 | informs the $\leq 5$ high-confidence band edge in the classifier |
| Firm A whole-sample | dHash$_\text{indep}$ style-consistency ceiling | — | 15 | operational $> 15$ style-consistency boundary |
| Firm A calibration fold (70%) | cosine P5 (Section IV-F.2) | 0.9407 | — | calibration-fold cross-reference; held-out fold reports rates at this cut |
| Firm A calibration fold (70%) | dHash$_\text{indep}$ P95 | — | 9 | calibration-fold cross-reference (Tables IX and XI report rates at the rounded $\leq 8$ cut for continuity) |
Read this table by *population × method*: each row reports one method applied to one population.
The first three blocks (threshold estimators; density-smoothness diagnostics; between-class KDE) are *characterisation* outputs; the bottom block is the operational anchor set used by the classifier of Section III-K.
The disagreement between Firm A Beta-2 (0.977) and Firm A logit-Gaussian-2 (0.999) is the parametric-form sensitivity referenced in the prose of Section IV-D.3; it cannot be resolved from the data because BIC rejects the underlying $K{=}2$ assumption itself.
-->
Non-hand-signed replication quality is therefore best read as a continuous spectrum produced by firm-specific reproduction technologies (administrative stamping in early years, firm-level e-signing later) acting on a common stored exemplar.
This finding has a direct methodological pay-off: it is *why* the operational cosine cut is anchored on the whole-sample Firm A P7.5 percentile (Section III-K), and it is *why* the byte-level pixel-identity anchor (Section IV-F.1) is the natural threshold-free positive reference for downstream validation.
## E. Calibration Validation with Firm A
Fig. 3 presents the per-signature cosine and dHash distributions of Firm A compared to the overall population.
Table IX reports the proportion of Firm A signatures crossing each candidate threshold; these rates play the role of calibration-validation metrics (what fraction of a known replication-dominated population does each threshold capture?).
<!-- TABLE IX: Firm A Whole-Sample Capture Rates (consistency check, NOT external validation)
| Rule | Firm A rate | k / N |
|------|-------------|-------|
| **Cosine-only marginal rates** | | |
| cosine > 0.837 (all-pairs KDE crossover) | 99.93% | 60,408 / 60,448 |
| cosine > 0.9407 (calibration-fold P5) | 95.15% | 57,518 / 60,448 |
| cosine > 0.945 (calibration-fold P5 rounded) | 94.02% | 56,836 / 60,448 |
| cosine > 0.95 (operational; whole-sample Firm A P7.5) | 92.51% | 55,922 / 60,448 |
| **dHash-only marginal rates** | | |
| dHash_indep ≤ 5 (operational high-confidence cap) | 84.20% | 50,897 / 60,448 |
| dHash_indep ≤ 8 (calibration-fold P95 rounded) | 95.17% | 57,527 / 60,448 |
| dHash_indep ≤ 15 (operational style-consistency boundary) | 99.83% | 60,348 / 60,448 |
| **Operational classifier dual rules (Section III-K)** | | |
| cosine > 0.95 AND dHash_indep ≤ 5 (high-confidence non-hand-signed) | 81.70% | 49,389 / 60,448 |
| cosine > 0.95 AND 5 < dHash_indep ≤ 15 (moderate-confidence) | 10.76% | 6,503 / 60,448 |
| cosine > 0.95 AND dHash_indep ≤ 15 (combined non-hand-signed) | 92.46% | 55,892 / 60,448 |
| **Calibration-fold-adjacent cross-reference (not the operational classifier rule)** | | |
| cosine > 0.95 AND dHash_indep ≤ 8 | 89.95% | 54,370 / 60,448 |
All rates computed exactly from the full Firm A sample (N = 60,448 signatures); per-rule counts and codes are available in the supplementary materials.
The two operational dHash cuts ($\leq 5$ for the high-confidence cap and $\leq 15$ for the style-consistency boundary) come from the classifier definition in Section III-K and are the rules used by the five-way classifier of Tables XII and XVII; the dHash $\leq 8$ row is *not* an operational classifier rule but a calibration-fold-adjacent reference (Section IV-F.2 calibration-fold dHash P95 = 9; we report the $\leq 8$ rate as the integer-valued threshold immediately below P95, included here so that Firm A capture in the calibration-fold-P95 neighbourhood can be read off the same table).
-->
Table IX is a whole-sample consistency check rather than an external validation: the cosine cut $0.95$ and the operational dHash band edges ($\leq 5$ high-confidence cap and $\leq 15$ style-consistency boundary) are themselves anchored to the whole-sample Firm A distribution described in Section III-K (the 70/30 calibration-fold thresholds of Table XI are separate and slightly different, e.g., calibration-fold cosine P5 = 0.9407 rather than the whole-sample heuristic 0.95).
The operational dual rule used by the five-way classifier of Section III-K---cosine $> 0.95$ AND $\text{dHash}_\text{indep} \leq 15$ (the union of the high-confidence and moderate-confidence non-hand-signed buckets)---captures 92.46% of Firm A; the high-confidence component alone (cosine $> 0.95$ AND $\text{dHash}_\text{indep} \leq 5$) captures 81.70%.
For continuity with prior calibration-fold reporting (Section IV-F.2 reports the calibration-fold rate at the calibration-fold-P95-adjacent cut $\text{dHash}_\text{indep} \leq 8$), Table IX also lists the cosine $> 0.95$ AND $\text{dHash}_\text{indep} \leq 8$ rate of 89.95%; this is *not* the operational classifier rule but a cross-reference value.
Both operational rates are consistent with the dip-test-confirmed unimodal-long-tail shape of Firm A's per-signature cosine distribution (Section IV-D.1) and the 92.5% / 7.5% signature-level split (Section III-H).
Section IV-F.2 reports the corresponding rates on the 30% Firm A hold-out fold, which provides the external check these whole-sample rates cannot.
## F. Pixel-Identity, Inter-CPA, and Held-Out Firm A Validation
We report three validation analyses corresponding to the anchors of Section III-J.
### 1) Pixel-Identity Positive Anchor with Inter-CPA Negative Anchor
Of the 182,328 extracted signatures, 310 have a same-CPA nearest match that is byte-identical after crop and normalization (pixel-identical-to-closest = 1); these form the byte-identity positive anchor---a pair-level proof of image reuse that serves as conservative ground truth for non-hand-signed signatures, subject to the source-template edge case discussed in Section V-G.
Within Firm A specifically, 145 of these byte-identical signatures are distributed across 50 distinct partners (of 180 registered Firm A partners), with 35 of the byte-identical pairs spanning different fiscal years; reproduction artifact for this Firm A decomposition is listed in Appendix B.
As the gold-negative anchor we sample 50,000 i.i.d. random cross-CPA signature pairs from the full 168,755-signature matched corpus (inter-CPA cosine: mean $= 0.763$, $P_{95} = 0.886$, $P_{99} = 0.915$, max $= 0.992$).
Because the positive and negative anchor populations are constructed from different sampling units (byte-identical same-CPA pairs vs random inter-CPA pairs), their relative prevalence in the combined anchor set is arbitrary, and precision / $F_1$ / recall therefore have no meaningful population interpretation.
We accordingly report FAR with Wilson 95% confidence intervals against the large inter-CPA negative anchor in Table X.
The primary quantity reported by Table X is FAR: the probability that a random pair of signatures from *different* CPAs exceeds the candidate threshold.
We do not report an Equal Error Rate: EER is meaningful only when the positive and negative error-rate curves cross in a nontrivial interior region, but byte-identical positives all sit at cosine $\approx 1$ by construction, so FRR against that subset is trivially $0$ at every threshold below $1$. An EER calculation against this anchor would be arithmetic tautology rather than biometric performance, and we therefore omit it.
<!-- TABLE X: Cosine Threshold Sweep — FAR Against 50,000 Inter-CPA Negative Pairs
| Threshold | FAR | FAR 95% Wilson CI |
|-----------|-----|-------------------|
| 0.837 (all-pairs KDE crossover) | 0.2101 | [0.2066, 0.2137] |
| 0.900 | 0.0250 | [0.0237, 0.0264] |
| 0.945 (calibration-fold P5 rounded) | 0.0008 | [0.0006, 0.0011] |
| 0.950 (whole-sample Firm A P7.5; operational cut) | 0.0005 | [0.0003, 0.0007] |
| 0.977 (Firm A Beta-2 forced-fit crossing; Section IV-D) | 0.00014 | [0.00007, 0.00029] |
| 0.985 (BD/McCrary candidate transition; Appendix A) | 0.00004 | [0.00001, 0.00015] |
Table note: We do not include FRR against the byte-identical positive anchor as a column here: the byte-identical subset has cosine $\approx 1$ by construction, so FRR against that subset is trivially $0$ at every threshold below $1$ and carries no biometric information beyond verifying that the threshold does not exceed $1$. The conservative-subset FRR role of the byte-identical anchor is instead discussed qualitatively in Section V-F.
-->
Two caveats apply.
First, the byte-identical positive anchor referenced above is a *conservative subset* of the true non-hand-signed population: it captures only those non-hand-signed signatures whose nearest match happens to be byte-identical, not those that are near-identical but not bytewise identical.
A would-be FRR computed against this subset is definitionally $0$ at every threshold below $1$ (since byte-identical pairs have cosine $\approx 1$), so such an FRR is a mathematical boundary check rather than an empirical miss-rate estimate; we discuss the generalization limits of this conservative-subset framing in Section V-F.
Second, the 0.945 / 0.95 thresholds are derived from the Firm A whole-sample and calibration-fold percentiles rather than from this anchor set, so the FAR values in Table X are post-hoc-fit-free evaluations of thresholds that were not chosen to optimize Table X.
The very low FAR at the operational cut is therefore informative about specificity against a realistic inter-CPA negative population.
### 2) Held-Out Firm A Validation (within-Firm-A sampling variance disclosure)
We split Firm A CPAs randomly 70 / 30 at the CPA level into a calibration fold (124 CPAs, 45,116 signatures) and a held-out fold (54 CPAs, 15,332 signatures).
The total of 178 Firm A CPAs differs from the 180 in the Firm A registry by two registered Firm A partners whose signatures in the corpus are singletons (only one signature each, so the per-signature best-match cosine is undefined and they do not appear in the same-CPA matched-signature table that script `24_validation_recalibration.py` reads); they are therefore not represented in either fold by construction rather than by an explicit exclusion rule.
Thresholds are re-derived from calibration-fold percentiles only.
Table XI reports both calibration-fold and held-out-fold capture rates with Wilson 95% CIs and a two-proportion $z$-test.
<!-- TABLE XI: Held-Out vs Calibration Firm A Capture Rates and Generalization Test
| Rule | Calibration 70% fold (CI) | Held-out 30% fold (CI) | 2-prop z | p | k/n calib | k/n held |
|------|---------------------------|-------------------------|----------|---|-----------|----------|
| cosine > 0.837 | 99.94% [99.91%, 99.96%] | 99.93% [99.87%, 99.96%] | +0.31 | 0.756 n.s. | 45,087/45,116 | 15,321/15,332 |
| cosine > 0.9407 (calib-fold P5) | 94.99% [94.79%, 95.19%] | 95.63% [95.29%, 95.94%] | -3.19 | 0.001 | 42,856/45,116 | 14,662/15,332 |
| cosine > 0.945 (calib-fold P5 rounded) | 93.77% [93.54%, 93.99%] | 94.78% [94.41%, 95.12%] | -4.54 | <0.001 | 42,305/45,116 | 14,531/15,332 |
| cosine > 0.950 (whole-sample P7.5; operational cut) | 92.14% [91.89%, 92.38%] | 93.61% [93.21%, 93.98%] | -5.97 | <0.001 | 41,570/45,116 | 14,352/15,332 |
| dHash_indep ≤ 5 | 82.96% [82.61%, 83.31%] | 87.84% [87.31%, 88.34%] | -14.29 | <0.001 | 37,430/45,116 | 13,467/15,332 |
| dHash_indep ≤ 8 | 94.84% [94.63%, 95.04%] | 96.13% [95.82%, 96.43%] | -6.45 | <0.001 | 42,788/45,116 | 14,739/15,332 |
| dHash_indep ≤ 9 (calib-fold P95) | 96.65% [96.48%, 96.81%] | 97.48% [97.22%, 97.71%] | -5.07 | <0.001 | 43,604/45,116 | 14,945/15,332 |
| dHash_indep ≤ 15 | 99.83% [99.79%, 99.87%] | 99.84% [99.77%, 99.89%] | -0.31 | 0.754 n.s. | 45,040/45,116 | 15,308/15,332 |
| cosine > 0.95 AND dHash_indep ≤ 8 (calibration-fold P95-adjacent reference; P95 = 9) | 89.40% [89.12%, 89.68%] | 91.54% [91.09%, 91.97%] | -7.60 | <0.001 | 40,335/45,116 | 14,035/15,332 |
| cosine > 0.95 AND dHash_indep ≤ 15 (operational classifier rule, Section III-K) | 92.09% [91.84%, 92.34%] | 93.56% [93.16%, 93.93%] | -5.93 | <0.001 | 41,548/45,116 | 14,344/15,332 |
Calibration-fold thresholds: Firm A cosine median = 0.9862, P1 = 0.9067, P5 = 0.9407; dHash_indep median = 2, P95 = 9. Counts and z/p values are reproducible from the supplementary materials (fixed random seed).
-->
We report fold-versus-fold comparisons rather than fold-versus-whole-sample comparisons, because the whole-sample rate is a weighted average of the two folds and therefore cannot, in general, fall inside the Wilson CI of either fold when the folds differ in rate; the correct generalization reference is the calibration fold, which produced the thresholds.
Under this proper test the two extreme rules agree across folds (cosine $> 0.837$ and $\text{dHash}_\text{indep} \leq 15$; both $p > 0.7$).
The operationally relevant rules in the 85–95% capture band differ between folds by 1–5 percentage points ($p < 0.001$ given the $n \approx 45\text{k}/15\text{k}$ fold sizes).
Both folds nevertheless sit in the same replication-dominated regime: every calibration-fold rate in the 85–99% range has a held-out counterpart in the 87–99% range, and the calibration-fold-adjacent reference rule cosine $> 0.95$ AND $\text{dHash}_\text{indep} \leq 8$ (the integer cut immediately below the calibration-fold dHash P95 of 9) captures 89.40% of the calibration fold and 91.54% of the held-out fold; the operational classifier rule cosine $> 0.95$ AND $\text{dHash}_\text{indep} \leq 15$ used by the five-way classifier of Section III-K captures still higher rates in both folds (calibration 92.09%, 41,548 / 45,116; held-out 93.56%, 14,344 / 15,332).
The modest fold gap is consistent with within-Firm-A heterogeneity in replication intensity: the random 30% CPA sample evidently contained proportionally more high-replication CPAs.
We therefore interpret the held-out fold as confirming the qualitative finding (Firm A is strongly replication-dominated across both folds) while cautioning that exact rates carry fold-level sampling noise that a single 30% split cannot eliminate; the threshold-independent partner-ranking analysis (Section IV-G.2) is the cross-check that is robust to this fold variance.
### 3) Operational-Threshold Sensitivity: cos $> 0.95$ vs cos $> 0.945$
The per-signature classifier (Section III-K) uses cos $> 0.95$ as its operational cosine cut, anchored on the whole-sample Firm A P7.5 heuristic (i.e., 7.5% of whole-sample Firm A signatures lie at or below 0.95; see Section III-H).
We report a sensitivity check in which this round-number cut is replaced by the slightly stricter calibration-fold P5 rounded value cos $> 0.945$ (calibration-fold P5 = 0.9407, see Table XI).
Table XII reports the five-way classifier output under each cut.
<!-- TABLE XII: Classifier Sensitivity to the Operational Cosine Cut (All-Sample Five-Way Output, N = 168,740 signatures)
| Cosine cut | High-confidence | Moderate-confidence | High style consistency | Uncertain | Likely hand-signed |
|------------|-----------------|---------------------|------------------------|-----------|--------------------|
| cos > 0.940 | 81,069 (48.04%) | 55,308 (32.78%) | 801 (0.47%) | 31,026 (18.39%) | 536 (0.32%) |
| cos > 0.945 | 79,278 (46.98%) | 50,001 (29.63%) | 665 (0.39%) | 38,260 (22.67%) | 536 (0.32%) |
| cos > 0.950 (operational) | 76,984 (45.62%) | 43,906 (26.02%) | 546 (0.32%) | 46,768 (27.72%) | 536 (0.32%) |
| cos > 0.960 | 70,250 (41.63%) | 29,450 (17.45%) | 288 (0.17%) | 68,216 (40.43%) | 536 (0.32%) |
| cos > 0.970 | 60,247 (35.70%) | 14,865 ( 8.81%) | 117 (0.07%) | 92,975 (55.10%) | 536 (0.32%) |
| cos > 0.985 | 37,368 (22.15%) | 2,231 ( 1.32%) | 10 (0.01%) | 128,595 (76.21%) | 536 (0.32%) |
The dHash band edges ($\leq 5$ for high-confidence, $5 < \text{dHash}_\text{indep} \leq 15$ for moderate-confidence, $> 15$ for style) are held fixed across the grid; only the cosine cut varies. The Likely-hand-signed count is invariant across the grid because it depends only on the all-pairs KDE crossover cosine $= 0.837$.
-->
At the aggregate firm-level, the calibration-fold-adjacent reference dual rule cos $> 0.95$ AND $\text{dHash}_\text{indep} \leq 8$ captures 89.95% of whole Firm A under the 0.95 cut and 91.14% under the 0.945 cut---a shift of 1.19 percentage points.
The operational classifier rule cos $> 0.95$ AND $\text{dHash}_\text{indep} \leq 15$ used by the five-way classifier of Section III-K captures 92.46% under the 0.95 cut and 93.97% under the 0.945 cut---a shift of 1.51 percentage points.
Reading the wider grid in Table XII: the High-confidence and Moderate-confidence shares shift by less than 5 percentage points across the 0.940-0.950 neighbourhood, while pushing the cosine cut to 0.970 or 0.985 produces qualitatively different classifier behaviour (Moderate-confidence collapses from 26.02% at $0.95$ to 8.81% at $0.97$ and 1.32% at $0.985$, with the displaced mass landing in Uncertain rather than reclassifying out of the corpus).
The classifier output is therefore robust to small (~0.005-cosine) perturbations of the operational cut but not to wholesale reanchoring at the threshold-estimator outputs of Section IV-D, which is consistent with our reading that those outputs are not classifier thresholds.
At the per-signature categorization level, replacing 0.95 by 0.945 reclassifies 8,508 signatures (5.04% of the corpus) out of the Uncertain band; 6,095 of them migrate to Moderate-confidence non-hand-signed, 2,294 to High-confidence non-hand-signed, and 119 to High style consistency.
The Likely-hand-signed category is unaffected because it depends only on the fixed all-pairs KDE crossover cosine $= 0.837$.
The High-confidence non-hand-signed share grows from 45.62% to 46.98%.
We interpret this sensitivity pattern as indicating that the classifier's aggregate and high-confidence output is robust to the choice of operational cut within a 0.005-cosine neighbourhood of the Firm A P7.5 anchor, and that the movement is concentrated at the Uncertain/Moderate-confidence boundary.
To make the operating-point selection (Section III-K) auditable rather than presented as a single fixed value, Table XII-B reports the capture-vs-FAR tradeoff over the candidate threshold grid spanning the calibration-fold P5 (0.9407), its rounded value (0.945), the operational anchor (0.95), the Firm A Beta-2 forced-fit crossing from Section IV-D.3 (0.977), and the BD/McCrary candidate transition from Section IV-D.2 (0.985).
For each grid point we report Firm A capture (under both the cosine-only marginal and the operational dual rule cos $> t$ AND $\text{dHash}_\text{indep} \leq 15$ used by the five-way classifier of Section III-K), non-Firm-A capture (the cosine-only marginal in the 108,292 non-Firm-A matched signatures), and inter-CPA FAR with Wilson 95% CI against the 50,000-pair anchor of Section IV-F.1.
<!-- TABLE XII-B: Cosine-Threshold Tradeoff: Capture vs Inter-CPA FAR
| Cosine cut t | Firm A capture (cos > t) | Firm A capture (cos > t AND dHash_indep ≤ 15) | Non-Firm-A capture (cos > t) | Inter-CPA FAR | Inter-CPA FAR Wilson 95% CI |
|--------------|--------------------------|------------------------------------------------|------------------------------|---------------|------------------------------|
| 0.9407 (calibration-fold P5) | 95.15% (57,518/60,448) | 95.09% (57,482/60,448) | 72.68% (78,710/108,292) | 0.00126 | [0.00099, 0.00161] |
| 0.945 (calibration-fold P5 rounded) | 94.02% (56,836/60,448) | 93.97% (56,804/60,448) | 67.51% (73,108/108,292) | 0.00082 | [0.00061, 0.00111] |
| 0.95 (whole-sample Firm A P7.5; **operational cut**) | **92.51%** (55,922/60,448) | **92.46%** (55,892/60,448) | 60.50% (65,514/108,292) | **0.00050** | [0.00034, 0.00074] |
| 0.977 (Firm A Beta-2 forced-fit crossing) | 74.53% (45,050/60,448) | 74.51% (45,038/60,448) | 13.14% (14,233/108,292) | 0.00014 | [0.00007, 0.00029] |
| 0.985 (BD/McCrary candidate transition) | 55.27% (33,409/60,448) | 55.26% (33,406/60,448) | 5.73% (6,200/108,292) | 0.00004 | [0.00001, 0.00015] |
Inter-CPA FAR computed against 50,000 i.i.d. inter-CPA pairs (random seed 42, reproducing the anchor of Section IV-F.1 / Table X). Capture and FAR percentages are exact ratios of the displayed integer counts; gap arithmetic in the surrounding prose is computed from those exact counts and rounded to two decimal places. The dual-rule column is the operational classifier rule of Section III-K; for cuts above the dHash-15 saturation point (Firm A dHash$_\text{indep}$ $> 15$ rate is only 0.17%, Table IX), the dual-rule and cosine-only columns coincide to within the dHash$_\text{indep}$ $> 15$ residual.
-->
Reading Table XII-B, three patterns motivate the choice of $0.95$ as the operating point.
First, *Firm A capture* on the operational dual rule decays smoothly from 95.09% at $t = 0.9407$ to 55.26% at $t = 0.985$.
Relaxing the cut from $0.95$ to $0.945$ buys 1.51 percentage points of additional Firm A capture, and to $0.9407$ buys 2.63 percentage points; tightening from $0.95$ to $0.977$ costs 17.96 percentage points and to $0.985$ costs 37.20 percentage points.
The selected cut at $0.95$ is the strictest cut on this grid at which Firm A capture remains above $90\%$ on the operational dual rule.
Second, *inter-CPA FAR* is small in absolute terms across the entire candidate grid ($0.00126$ at $0.9407$, falling to $0.00004$ at $0.985$): under any of these operating points the classifier's specificity against random cross-CPA pairs is in the per-mille range or better, so FAR alone does not determine the choice.
The marginal FAR cost of relaxing from $0.95$ to $0.945$ is $+0.00032$ ($25 \to 41$ false positives per 50,000 pairs) and to $0.9407$ is $+0.00076$ ($25 \to 63$); the marginal FAR savings from tightening to $0.977$ and $0.985$ are $-0.00036$ and $-0.00046$ respectively.
The FAR savings from going stricter are small in absolute terms compared with the corresponding Firm A capture loss, which makes $0.95$ a balanced operating point on this grid rather than a uniquely optimal one.
Third, *non-Firm-A capture* (the cosine-only marginal in the 108,292 non-Firm-A signatures) decays from 67.51% at $0.945$ to 60.50% at $0.95$, 13.14% at $0.977$, and 5.73% at $0.985$.
The Firm-A-minus-non-Firm-A gap widens with strictness through $0.977$ and then contracts (22.41 percentage points at $0.9407$; 26.46 at $0.945$; 31.97 at $0.95$; 61.36 at $0.977$; 49.54 at $0.985$): on the $0.95 \to 0.977$ segment non-Firm-A capture falls faster than Firm A capture in absolute terms ($-47.35$ vs $-17.96$ percentage points), so the widening is dominated by non-Firm-A removal rather than by an intrinsic property of Firm A; on the $0.977 \to 0.985$ segment Firm A capture falls faster than non-Firm-A's already-low residual, so the gap contracts.
We do *not* read the gap pattern as evidence for a particular cut; it is reported here as cross-firm replication heterogeneity rather than as a selection criterion.
The operating point at $0.95$ is therefore a defensible---not unique---selection in this neighbourhood, motivated by (i) keeping Firm A capture above $90\%$ on the operational dual rule, (ii) achieving an FAR of $0.0005$ at which marginal further savings from tightening are small relative to the corresponding capture loss, and (iii) preserving the interpretive transparency of the whole-sample Firm A P7.5 reading.
It is *not* derived from the threshold-estimator outputs of Section IV-D, which the data do not support as classifier thresholds.
The paper therefore retains cos $> 0.95$ as the primary operational cut and reports the 0.945 result of Table XII as a sensitivity check rather than as a deployed alternative; downstream document-level rates (Table XVII) and intra-report agreement (Table XVI) are robust to moderate cutoff shifts within the 0.945--0.95 neighbourhood as long as the same cutoff is applied uniformly across firms.
## G. Additional Firm A Benchmark Validation
Before presenting the three threshold-robust analyses, Fig. 4 summarises the per-firm yearly per-signature best-match cosine distribution that motivates them.
The left panel reports the mean per-signature best-match cosine within each firm bucket and fiscal year (a threshold-free statistic); the right panel reports the share of each firm-bucket-year with per-signature best-match cosine $\geq 0.95$ (the operational cut of Section III-K).
Both panels show Firm A above the other Big-4 firms in every year of the 2013-2023 sample, with non-Big-4 firms below all four Big-4 firms throughout, and the cross-firm ordering is stable across the sample period.
The mean-cosine separation between Firm A and the other Big-4 firms is on the order of 0.02-0.04 throughout the sample (e.g., 2013: Firm A $0.9733$ vs Firm B $0.9498$, Firm C $0.9464$, Firm D $0.9395$, Non-Big-4 $0.9227$; 2023: $0.9860$ vs $0.9668$, $0.9662$, $0.9525$, $0.9346$); the share-above-0.95 separation is wider (2013: Firm A $87.2\%$ vs $61.8\%$, $56.2\%$, $38.5\%$, $27.5\%$).
This visual is the most direct cross-firm evidence in the paper that Firm A's high-similarity behaviour is firm-specific rather than corpus-wide; the three subsections below decompose this gap along three threshold-free or threshold-robust dimensions.
<!-- FIGURE 4: Per-firm yearly per-signature best-match cosine
File: reports/figures/fig_yearly_big4_comparison.png (and .pdf)
Generated by: signature_analysis/30_yearly_big4_comparison.py
Caption: Per-firm yearly per-signature best-match cosine, 2013-2023.
(a) Mean per-signature best-match cosine by firm bucket and fiscal year
(threshold-free). (b) Share of per-signature best-match cosine $\geq 0.95$
(operational cut of Section III-K). Five lines: Firm A, B, C, D, Non-Big-4.
Firm A is above the other Big-4 firms in every year; Non-Big-4 is below all
four Big-4 firms in every year. Per-firm signature counts and exact values
are in `reports/firm_yearly_comparison/firm_yearly_comparison.{json,md}`.
-->
The capture rates of Section IV-E are an *internal* consistency check: they ask "how much of Firm A does our threshold capture?", but the threshold was itself derived from Firm A's percentiles, so a high capture rate is not surprising.
To go beyond this circular check, we report three further analyses, each chosen so that the *informative quantity* does not depend on the threshold's absolute value:
- **§IV-G.1 (year-by-year stability).** Holds the cosine cutoff fixed at 0.95 and asks whether the share of Firm A below the cutoff is *stable across years*. The information is in the temporal trend, not in the absolute rate; under a noise-only explanation of the left tail, the share should shrink as scan/PDF technology matured.
- **§IV-G.2 (partner-level similarity ranking).** Uses *no threshold at all*: every auditor-year is ranked by mean similarity, and we measure Firm A's share of the top decile against its baseline share. The information is in the concentration ratio, which is invariant to the choice of cutoff.
- **§IV-G.3 (intra-report agreement).** Applies the calibrated classifier and measures whether the *two co-signing CPAs on the same Firm A report* receive the same classifier label, then compares Firm A's intra-report agreement rate to the other firms'. The information is in the *cross-firm gap*; the absolute agreement rate at any one firm depends on the cutoff, but the gap is robust to moderate cutoff shifts as long as the same cutoff is applied uniformly across firms.
Together these three analyses provide threshold-free or threshold-robust evidence that complements the within-sample capture rates of Section IV-E.
### 1) Year-by-Year Stability of the Firm A Left Tail
Table XIII reports the proportion of Firm A signatures with per-signature best-match cosine below 0.95, disaggregated by fiscal year.
Under the replication-dominated interpretation (Section III-H), this signature-level left-tail rate reflects within-firm heterogeneity in signing outputs at Firm A.
Consistent with the scope-of-claims framing in Section III-G, we report the rate as a signature-level quantity without disaggregating the underlying mechanism (which may span a minority of hand-signing partners, multi-template replication workflows within the firm, or a combination); partner-level mechanism attribution is not attempted.
Under the alternative hypothesis that the left tail is an artifact of scan or compression noise, the share should shrink as scanning and PDF-compression technology improved over 2013-2023.
<!-- TABLE XIII: Firm A Per-Year Cosine Distribution
| Year | N sigs | mean best-match cosine | % below 0.95 |
|------|--------|-------------|--------------|
| 2013 | 2,167 | 0.9733 | 12.78% |
| 2014 | 5,256 | 0.9781 | 8.69% |
| 2015 | 5,484 | 0.9793 | 7.46% |
| 2016 | 5,739 | 0.9811 | 6.92% |
| 2017 | 5,796 | 0.9814 | 6.69% |
| 2018 | 5,986 | 0.9808 | 6.58% |
| 2019 | 6,122 | 0.9780 | 8.71% |
| 2020 | 6,122 | 0.9770 | 9.46% |
| 2021 | 5,996 | 0.9792 | 8.37% |
| 2022 | 5,918 | 0.9819 | 6.25% |
| 2023 | 5,862 | 0.9860 | 3.75% |
-->
The left tail is stable at 6-13% throughout the sample period and shows no pre/post-2020 level shift: the 2013-2019 mean left-tail share is 8.26% and the 2020-2023 mean is 6.96%.
The lowest observed share is in 2023 (3.75%), consistent with firm-level electronic signing systems producing more uniform output than earlier manual scanning-and-stamping, not less.
This stability supports the replication-dominated framing: a persistent within-firm heterogeneity component is consistent with a Beta left tail that is stable across production technologies, whereas a noise-only explanation would predict a shrinking share as technology improved.
### 2) Partner-Level Similarity Ranking
If Firm A applies firm-wide stamping while the other Big-4 firms use stamping only for a subset of partners, Firm A auditor-years should disproportionately occupy the top of the similarity distribution among all auditor-years (across all firms).
We test this prediction directly.
For each auditor-year (CPA $\times$ fiscal year) with at least 5 signatures we compute the mean best-match cosine similarity across the year's signatures, yielding 4,629 auditor-years across 2013-2023.
Firm A accounts for 1,287 of these (27.8% baseline share).
Table XIV reports per-firm occupancy of the top $K\%$ of the ranked distribution.
The per-signature best-match cosine underlying each auditor-year mean is taken over the full same-CPA pool (Section III-G), consistent with the unit-of-analysis framing in Section III-G.
<!-- TABLE XIV: Top-K Similarity Rank Occupancy by Firm (pooled 2013-2023)
| Top-K | k in bucket | Firm A | Firm B | Firm C | Firm D | Non-Big-4 | Firm A share |
|-------|-------------|--------|--------|--------|--------|-----------|--------------|
| 10% | 462 | 443 | 2 | 3 | 0 | 14 | 95.9% |
| 20% | 925 | 877 | 9 | 14 | 2 | 23 | 94.8% |
| 25% | 1,157 | 1,043 | 32 | 23 | 9 | 50 | 90.1% |
| 30% | 1,388 | 1,129 | 105 | 52 | 25 | 77 | 81.3% |
| 50% | 2,314 | 1,220 | 473 | 273 | 102 | 246 | 52.7% |
-->
Firm A occupies 95.9% of the top 10%, 94.8% of the top 20%, 90.1% of the top 25%, and 81.3% of the top 30% of auditor-years by similarity, against its baseline share of 27.8%---a concentration ratio of $3.5\times$ at the top decile, $3.4\times$ at the top quintile, and $2.9\times$ at the top tercile.
Firm A's share decays monotonically as the bracket widens (95.9% $\to$ 94.8% $\to$ 90.1% $\to$ 81.3% $\to$ 52.7% across top-10/20/25/30/50%), and only at the top 50% does its share approach its baseline; the over-representation is therefore concentrated in the very top of the distribution rather than spread uniformly through the upper half.
Year-by-year (Table XV), the top-10% Firm A share ranges from 88.4% (2020) to 100% (2013, 2014, 2017, 2018, 2019), showing that the concentration is stable across the sample period.
<!-- TABLE XV: Firm A Share of Top-K Similarity by Year (K = 10%, 20%, 30%)
| Year | N auditor-years | Top-10% share | Top-20% share | Top-30% share | Firm A baseline |
|------|-----------------|---------------|---------------|---------------|-----------------|
| 2013 | 324 | 100.0% (32/32) | 98.4% (63/64) | 89.7% (87/97) | 32.4% |
| 2014 | 399 | 100.0% (39/39) | 98.7% (78/79) | 82.4% (98/119) | 27.8% |
| 2015 | 394 | 97.4% (38/39) | 96.2% (75/78) | 84.7% (100/118) | 27.7% |
| 2016 | 413 | 95.1% (39/41) | 96.3% (79/82) | 81.3% (100/123) | 26.2% |
| 2017 | 415 | 100.0% (41/41) | 97.6% (81/83) | 83.9% (104/124) | 27.2% |
| 2018 | 434 | 100.0% (43/43) | 97.7% (84/86) | 80.0% (104/130) | 26.5% |
| 2019 | 429 | 100.0% (42/42) | 97.6% (83/85) | 78.9% (101/128) | 27.0% |
| 2020 | 430 | 88.4% (38/43) | 91.9% (79/86) | 76.0% (98/129) | 27.7% |
| 2021 | 450 | 97.8% (44/45) | 96.7% (87/90) | 81.5% (110/135) | 28.7% |
| 2022 | 467 | 93.5% (43/46) | 95.7% (89/93) | 84.3% (118/140) | 28.3% |
| 2023 | 474 | 97.9% (46/47) | 94.7% (89/94) | 83.8% (119/142) | 27.4% |
Per-cell entries are "share (k_FirmA / k_total)". Top-25% and top-50% pooled values are reported in Table XIV; per-year top-25/50 columns are omitted from this table to reduce visual width but are reproducible from the supplementary materials.
-->
This over-representation is consistent with firm-wide non-hand-signing practice at Firm A and is not derived from any threshold we subsequently calibrate.
It therefore constitutes genuine cross-firm evidence for Firm A's benchmark status.
### 3) Intra-Report Consistency
Taiwanese statutory audit reports are co-signed by two engagement partners (a primary and a secondary signer).
Under firm-wide stamping practice at a given firm, both signers on the same report should receive the same signature-level classification.
Disagreement between the two signers on a report is informative about whether the stamping practice is firm-wide or partner-specific.
For each report with exactly two signatures and complete per-signature data (84,354 reports total: 83,970 single-firm reports, in which both signers are at the same firm, and 384 mixed-firm reports, in which the two signers are at different firms), we classify each signature using the dual-descriptor rules of Section III-K and record whether the two classifications agree.
Table XVI reports per-firm intra-report agreement for the 83,970 single-firm reports only (firm-assignment defined by the common firm identity of both signers); the 384 mixed-firm reports (0.46% of the 2-signature corpus) are excluded from the intra-report analysis because firm-level agreement is not well defined when the two signers are at different firms.
<!-- TABLE XVI: Intra-Report Classification Agreement by Firm
| Firm | Total 2-signer reports | Both non-hand-signed | Both uncertain | Both style | Both hand-signed | Mixed | Agreement rate |
|------|-----------------------|----------------------|----------------|------------|------------------|-------|----------------|
| Firm A | 30,222 | 26,435 | 734 | 0 | 4 | 3,049 | **89.91%** |
| Firm B | 17,121 | 9,260 | 2,159| 5 | 6 | 5,691 | 66.76% |
| Firm C | 19,112 | 8,983 | 3,035| 3 | 5 | 7,086 | 62.92% |
| Firm D | 8,375 | 3,028 | 2,376| 0 | 3 | 2,968 | 64.56% |
| Non-Big-4 | 9,140 | 1,671 | 3,945| 18| 27| 3,479 | 61.94% |
A report is "in agreement" if both signature labels fall in the same coarse bucket
(non-hand-signed = high+moderate; uncertain; style consistency; or likely hand-signed).
-->
Firm A achieves 89.9% intra-report agreement, with 87.5% of Firm A reports having *both* signers classified as non-hand-signed and only 4 reports (0.01%) having both classified as likely hand-signed.
The other Big-4 firms (B, C, D) and non-Big-4 firms cluster at 62-67% agreement, a 23-28 percentage-point gap.
This 23-28 percentage-point gap in intra-report agreement between Firm A and the other firms is consistent with firm-wide (rather than partner-specific) non-hand-signing practice; we do not claim a sharp discontinuity in the formal sense, since classifier calibration, firm-specific document-production pipelines, and signer-mix differences could each contribute to gap magnitude.
We note that this test uses the calibrated classifier of Section III-K rather than a threshold-free statistic; the substantive evidence lies in the *cross-firm gap* between Firm A and the other firms rather than in the absolute agreement rate at any single firm, and that gap is robust to moderate shifts in the absolute cutoff so long as the cutoff is applied uniformly across firms.
## H. Classification Results
Table XVII presents the final classification results under the dual-descriptor framework with Firm A-calibrated thresholds for 84,386 documents (656 documents excluded from the 85,042-document YOLO-detection cohort because no signature on the document could be matched to a registered CPA; see Table XVII note).
We emphasize that the document-level proportions below reflect the *worst-case aggregation rule* of Section III-K: a report carrying one stamped signature and one hand-signed signature is labeled with the most-replication-consistent of the two signature-level verdicts.
Document-level rates therefore represent the share of reports in which *at least one* signature is non-hand-signed rather than the share in which *both* are; the intra-report agreement analysis of Section IV-G.3 (Table XVI) reports how frequently the two co-signers share the same signature-level label within each firm, so that readers can judge what fraction of the non-hand-signed document-level share corresponds to fully non-hand-signed reports versus mixed reports.
<!-- TABLE XVII: Document-Level Classification (Dual-Descriptor: Cosine + dHash)
| Verdict | N (PDFs) | % | Firm A | Firm A % |
|---------|----------|---|--------|----------|
| High-confidence non-hand-signed | 29,529 | 35.0% | 22,970 | 76.0% |
| Moderate-confidence non-hand-signed | 36,994 | 43.8% | 6,311 | 20.9% |
| High style consistency | 5,133 | 6.1% | 183 | 0.6% |
| Uncertain | 12,683 | 15.0% | 758 | 2.5% |
| Likely hand-signed | 47 | 0.1% | 4 | 0.0% |
Per the worst-case aggregation rule of Section III-K, a document with two signatures inherits the most-replication-consistent of the two signature-level labels.
The 84,386-document cohort excludes 656 documents (relative to the 85,042 YOLO-detected cohort of Table III) for which no signature could be matched to a registered CPA: the per-document classifier requires at least one CPA-matched signature so that a same-CPA best-match similarity is defined. The exclusion is definitional rather than discretionary; typical causes are auditor's-report-page formats deviating from the standard two-signature layout, or OCR returning a printed CPA name not present in the registry.
-->
Within the 71,656 documents exceeding cosine $0.95$, the dHash dimension stratifies them into three distinct populations:
29,529 (41.2%) show converging structural evidence of non-hand-signing (dHash $\leq 5$);
36,994 (51.7%) show partial structural similarity (dHash in $[6, 15]$) consistent with replication degraded by scan variations;
and 5,133 (7.2%) show no structural corroboration (dHash $> 15$), suggesting high signing consistency rather than image reproduction.
A cosine-only classifier would treat all 71,656 identically; the dual-descriptor framework separates them into populations with fundamentally different interpretations.
### 1) Firm A Capture Profile (Consistency Check)
96.9% of Firm A's documents fall into the high- or moderate-confidence non-hand-signed categories, 0.6% into high-style-consistency, and 2.5% into uncertain.
This pattern is consistent with the replication-dominated framing: the large majority is captured by non-hand-signed rules, while the small residual is consistent with the within-firm heterogeneity implied by the dip-test-confirmed unimodal-long-tail shape of Firm A's per-signature cosine distribution (Section IV-D.1) and the 7.5% signature-level left tail (Section III-H).
The near-zero "likely hand-signed" rate (4 of 30,226 Firm A documents, 0.013%; the 30,226 denominator is documents with at least one Firm A signer under the 84,386-document classification cohort, which differs from the 30,222 single-firm two-signer subset of Table XVI by 4 mixed-firm reports excluded from the firm-level intra-report comparison) indicates that the within-firm heterogeneity implied by the 7.5% signature-level left tail (Section IV-D) does not project into the lowest-cosine document-level category under the dual-descriptor rules; it is absorbed instead into the uncertain or high-style-consistency categories at this threshold set.
We note that because the non-hand-signed thresholds are themselves calibrated to Firm A's empirical percentiles (Section III-H), these rates are an internal consistency check rather than an external validation; the held-out Firm A validation of Section IV-F.2 is the corresponding external check.
### 2) Cross-Firm Comparison of Dual-Descriptor Convergence
Among the 65,514 non-Firm-A signatures with per-signature best-match cosine $> 0.95$, 42.12% have $\text{dHash}_\text{indep} \leq 5$, compared to 88.32% of the 55,922 Firm A signatures meeting the same cosine condition---a $\sim 2.1\times$ difference that the structural-verification layer makes visible.
The Firm A denominator (55,922) matches Table IX exactly: both Table IX and the cross-firm decomposition define Firm A membership via the CPA registry (`accountants.firm`), and the cross-firm analysis additionally requires a non-null independent-min dHash record, which all 55,922 Firm A cosine-eligible signatures have in the current database.
This cross-firm gap is consistent with firm-wide non-hand-signing practice at Firm A versus partner-specific or per-engagement replication at other firms; it complements the partner-level ranking (Section IV-G.2) and intra-report consistency (Section IV-G.3) findings.
Reproduction artifact for these counts is listed in Appendix B.
## I. Ablation Study: Feature Backbone Comparison
## D. Big-4 Accountant-Level Distributional Characterisation
This section reports the empirical evidence for §III-I's distributional diagnostics at the Big-4 accountant level. All numbers below are direct re-statements from Scripts 32 / 34. The accountant-level dip-test rejection reported in Table V is, per §III-I.4 (Scripts 39b–39e), fully attributable to between-firm location shifts and integer mass-point artefacts rather than to within-population bimodality; the v4-new composition-decomposition diagnostics that establish this finding are tabulated in §IV-M below alongside the anchor-based ICCR calibration.
**Table V.** Hartigan dip-test results, accountant-level marginals (Big-4 primary; comparison scopes from Script 32).
| Population | $n$ CPAs | $p_{\text{cos}}$ | $p_{\text{dHash}}$ | Interpretation |
|---|---|---|---|---|
| **Big-4 pooled (primary)** | 437 | $< 5 \times 10^{-4}$ | $< 5 \times 10^{-4}$ | reject unimodality on both axes |
| Firm A pooled alone | 171 | 0.992 | 0.924 | unimodal |
| Firms B + C + D pooled | 266 | 0.998 | 0.906 | unimodal |
| All non-Firm-A pooled | 515 | 0.998 | 0.907 | unimodal |
Bootstrap implementation: $n_{\text{boot}} = 2000$; for the Big-4 cells, no bootstrap replicate exceeded the observed dip statistic, so the empirical $p$-value is bounded above by the bootstrap resolution $1 / 2000 = 5 \times 10^{-4}$ (Script 34 reports this as $p = 0.0000$; we report $p < 5 \times 10^{-4}$ to reflect the resolution). Single-firm dip statistics for Firms B, C, and D were not separately computed.
**Table VI.** Burgstahler-Dichev / McCrary density-smoothness diagnostic on accountant-level marginals (cosine in 0.002 bins; dHash in integer bins; $\alpha = 0.05$, two-sided).
| Population | Cosine: significant transition? | dHash: significant transition? |
|---|---|---|
| **Big-4 pooled (primary)** | none ($p > 0.05$) | none ($p > 0.05$) |
| Firm A pooled alone | none | none |
| Firms B + C + D pooled | none | one transition at $\overline{\text{dHash}} = 10.8$ |
| All non-Firm-A pooled | none | one transition at $\overline{\text{dHash}} = 6.6$ |
The Big-4-scope null on both axes is consistent with the §IV-E mixture evidence: the K=3 components overlap in their tails rather than separating sharply, so a local-discontinuity test does not flag a transition. Outside Big-4, dHash transitions appear in some subsets but no cosine transition is identified in any tested subset (Script 32 sweeps; pre-2018 and post-2020 stratified variants exhibit dHash transitions at varying locations). These off-Big-4 dHash transitions are scope-dependent and are not used as v4.0 operational thresholds; we do not claim a specific structural interpretation for them without an explicit bin-width sensitivity sweep at those scopes.
## E. Big-4 K=2 / K=3 Mixture Fits
This section reports the K=2 and K=3 2D Gaussian mixture fits to the Big-4 accountant-level distribution and the bootstrap stability of their marginal crossings.
**Table VII.** Big-4 K=2 mixture components (descriptive partition; not mechanism clusters per §III-J) and marginal-crossing bootstrap 95% confidence intervals.
| K=2 component | $\overline{\text{cos}}$ | $\overline{\text{dHash}}$ | weight |
|---|---|---|---|
| K=2-a (low-cos / high-dHash position) | 0.954 | 7.14 | 0.689 |
| K=2-b (high-cos / low-dHash position) | 0.983 | 2.41 | 0.311 |
Marginal crossings (point + bootstrap 95% CI, $n_{\text{boot}} = 500$):
| Axis | Point | Bootstrap median | 95% CI | CI half-width |
|---|---|---|---|---|
| cos | 0.9755 | 0.9754 | $[0.9742, 0.9772]$ | 0.0015 |
| dHash | 3.755 | 3.763 | $[3.476, 3.969]$ | 0.246 |
$\text{BIC}(K{=}2) = -1108.45$ (Script 34).
**Table VIII.** Big-4 K=3 mixture components (descriptive firm-compositional partition per §III-J; not mechanism clusters).
| K=3 component | $\overline{\text{cos}}$ | $\overline{\text{dHash}}$ | weight | descriptive position |
|---|---|---|---|---|
| C1 | 0.9457 | 9.17 | 0.143 | low-cos / high-dHash corner |
| C2 | 0.9558 | 6.66 | 0.536 | central region |
| C3 | 0.9826 | 2.41 | 0.321 | high-cos / low-dHash corner |
$\text{BIC}(K{=}3) = -1111.93$, lower than $K{=}2$ by $3.48$ (mild support; not by itself decisive). The full-fit K=3 baseline above is reproduced in Scripts 35, 37, and 38 with identical hyperparameters; Script 37 additionally fits K=3 on each leave-one-firm-out training set (those fold-specific components differ from the full-fit baseline by design and are reported separately in §IV-G Table XIII). Operational use of the K=2 / K=3 fits is governed by §III-J and §III-L; §IV-G reports the LOOO reproducibility evidence that motivates reporting both fits descriptively.
## F. Convergent Internal-Consistency Checks
This section reports the empirical evidence for §III-K's three-score internal-consistency analysis. We re-emphasise the §III-K caveat: the three scores are deterministic functions of the same per-CPA descriptor pair $(\overline{\text{cos}}_a, \overline{\text{dHash}}_a)$ and are *not statistically independent measurements*. The pairwise correlations document internal consistency among feature-derived ranks rather than external validation against an independent ground truth.
**Table IX.** Per-CPA Spearman rank correlations among three feature-derived scores, Big-4, $n = 437$.
| Score pair | Spearman $\rho$ | $p$-value |
|---|---|---|
| K=3 P(C1) vs Paper A box-rule less-replication-dominated rate | $+0.9627$ | $< 10^{-248}$ |
| Reverse-anchor cosine percentile vs Paper A box-rule less-replication-dominated rate | $+0.8890$ | $< 10^{-149}$ |
| K=3 P(C1) vs Reverse-anchor cosine percentile | $+0.8794$ | $< 10^{-142}$ |
(Source: Script 38.) Reverse-anchor reference: 2D Gaussian fit by MCD (support fraction 0.85) on $n = 249$ non-Big-4 CPAs; reference centre $\overline{\text{cos}} = 0.935$, $\overline{\text{dHash}} = 9.77$.
**Table X.** Per-firm summary across the three feature-derived scores, Big-4.
| Firm | $n$ CPAs | mean $P(\text{C1})$ | mean reverse-anchor score | mean Paper A less-replication-dominated rate |
|---|---|---|---|---|
| Firm A | 171 | 0.0072 | $-0.9726$ | 0.1935 |
| Firm B | 112 | 0.1410 | $-0.8201$ | 0.6962 |
| Firm C | 102 | 0.3110 | $-0.7672$ | 0.7896 |
| Firm D | 52 | 0.2406 | $-0.7125$ | 0.7608 |
(Source: Script 38 per-firm summary; reverse-anchor score is sign-flipped so that *higher* values indicate deeper into the reference left tail = less replication-dominated relative to the non-Big-4 reference.)
The three scores agree on placing Firm A as the most replication-dominated and the three non-Firm-A firms as less replication-dominated. The K=3 posterior P(C1) and the box-rule less-replication-dominated rate (Score 1 and Score 3) place Firm C at the least-replication-dominated end of Big-4; the reverse-anchor cosine percentile (Score 2) ranks Firm D fractionally above Firm C. This residual within-Big-4-non-A disagreement is a design feature of the reverse-anchor metric: Score 2 measures only the marginal cosine percentile under the non-Big-4 reference, so a firm with a slightly higher cosine but a markedly different dHash distribution (Firm D vs Firm C) can score higher on Score 2 while scoring lower on Scores 1 and 3, both of which use both descriptors.
**Table XI.** Per-signature Cohen $\kappa$ (binary collapse, replication-dominated vs less-replication-dominated), $n = 150{,}442$ Big-4 signatures.
| Pair | Cohen $\kappa$ |
|---|---|
| Paper A binary high-confidence box rule (cos $> 0.95$ AND dHash $\leq 5$) vs per-CPA K=3 hard label | 0.662 |
| Paper A binary high-confidence box rule vs per-signature K=3 hard label | 0.559 |
| Per-CPA K=3 hard label vs per-signature K=3 hard label | 0.870 |
(Source: Script 39; verdict label `SIG_CONVERGENCE_MODERATE`.) Per-signature K=3 components ($n = 150{,}442$) sorted by ascending cosine: $(0.928, 9.75, 0.146)$ / $(0.963, 6.04, 0.582)$ / $(0.989, 1.27, 0.272)$, an absolute cosine drift of $0.018$ in C1 and $0.006$ in C3 relative to the per-CPA fit. These convergence checks cover only the binary high-confidence rule (cos $> 0.95$ AND dHash $\leq 5$); the five-way classifier's moderate-confidence band ($5 < \text{dHash} \leq 15$) inherits its v3.x calibration and capture-rate evaluation (§IV-J).
## G. Leave-One-Firm-Out Reproducibility
This section reports the firm-level cross-validation evidence motivating §III-J's "K=3 descriptive, not operational" framing.
**Table XII.** K=2 leave-one-firm-out across the four Big-4 folds.
| Held-out firm | $n_{\text{train}}$ | $n_{\text{held}}$ | Fold rule (cos cut, dHash cut) | Held-out classified as templated by fold rule |
|---|---|---|---|---|
| Firm A | 266 | 171 | cos $> 0.9380$ AND dHash $\leq 8.79$ | $171 / 171 = 100.00\%$ ($95\%$ Wilson $[97.80\%, 100.00\%]$) |
| Firm B | 325 | 112 | cos $> 0.9744$ AND dHash $\leq 3.98$ | $0 / 112 = 0\%$ ($95\%$ Wilson $[0\%, 3.32\%]$) |
| Firm C | 335 | 102 | cos $> 0.9752$ AND dHash $\leq 3.75$ | $0 / 102 = 0\%$ ($95\%$ Wilson $[0\%, 3.63\%]$) |
| Firm D | 385 | 52 | cos $> 0.9756$ AND dHash $\leq 3.74$ | $0 / 52 = 0\%$ ($95\%$ Wilson $[0\%, 6.88\%]$) |
(Source: Script 36.) Across-fold cosine crossing: pairwise range $[0.9380, 0.9756]$, range = $0.0376$; max absolute deviation from the across-fold mean is $0.028$. This exceeds the report's $0.005$ across-fold stability tolerance by $5.6\times$ and is much larger than the full-Big-4 bootstrap CI half-width of $0.0015$. Together with the all-or-nothing held-out classification pattern (Firm A held out $\Rightarrow$ all held-out CPAs templated; any non-Firm-A firm held out $\Rightarrow$ none templated), this indicates the K=2 boundary is essentially a Firm-A-vs-others separator rather than a within-Big-4 mechanism boundary.
**Table XIII.** K=3 leave-one-firm-out: C1 component shape and held-out membership.
| Held-out firm | C1 cos (fit) | C1 dHash (fit) | C1 weight (fit) | Held-out C1 hard-label rate | Full-Big-4 baseline C1% | Absolute difference |
|---|---|---|---|---|---|---|
| Full-Big-4 baseline | 0.9457 | 9.17 | 0.143 | — | — | — |
| Firm A held out | 0.9425 | 10.13 | 0.145 | $4.68\%$ | $0.00\%$ | $4.68$ pp |
| Firm B held out | 0.9441 | 9.16 | 0.127 | $7.14\%$ | $8.93\%$ | $1.76$ pp |
| Firm C held out | 0.9504 | 8.41 | 0.126 | $36.27\%$ | $23.53\%$ | $12.77$ pp |
| Firm D held out | 0.9439 | 9.29 | 0.120 | $17.31\%$ | $11.54\%$ | $5.81$ pp |
(Source: Script 37; verdict label `P2_PARTIAL`.) Component shape is reproducible across folds: max deviation of C1 cosine = $0.005$, C1 dHash = $0.96$, C1 weight = $0.023$. Hard-posterior membership for the held-out firm varies: max absolute difference from the full-Big-4 baseline is $12.77$ pp at the Firm C held-out fold, exceeding the report's $5$ pp viability bar. We accordingly do not use K=3 hard-posterior membership as an operational classifier label (§III-J, §III-L).
## H. Pixel-Identity Positive-Anchor Miss Rate
This section reports the only hard-ground-truth subset analysis available in the corpus: the positive-anchor miss rate against $n = 262$ Big-4 signatures whose nearest same-CPA match is byte-identical after crop and normalisation. Independent hand-signing cannot produce pixel-identical images, so byte-identical signatures are conservative-subset ground truth for the *replicated* class. The analysis is one-sided (positive-anchor only); a paired false-alarm rate against a hand-signed negative anchor is not available because no signature-level hand-signed ground truth exists in the corpus (§III-K item 4).
**Table XIV.** Positive-anchor miss rate, $n = 262$ Big-4 byte-identical signatures.
| Classifier | Misclassified as less-replication-dominated | Miss rate | Wilson 95% CI |
|---|---|---|---|
| Paper A binary high-confidence box rule (cos $> 0.95$ AND dHash $\leq 5$) | $0 / 262$ | $0\%$ | $[0\%, 1.45\%]$ |
| K=3 per-CPA hard label (C3 = high-cos / low-dHash; descriptive) | $0 / 262$ | $0\%$ | $[0\%, 1.45\%]$ |
| Reverse-anchor (prevalence-calibrated cut) | $0 / 262$ | $0\%$ | $[0\%, 1.45\%]$ |
(Source: Script 40.) Per-firm breakdown of the byte-identical subset: Firm A 145; Firm B 8; Firm C 107; Firm D 2. All three candidate scores correctly assign every byte-identical signature to the replicated class.
We caution that for the Paper A box rule this result is close to tautological (byte-identical nearest-neighbour signatures have cosine $\approx 1$ and dHash $\approx 0$, well inside the rule's high-confidence region); v3.20.0 §V-F discusses this conservative-subset caveat at length and we retain that discussion. The reverse-anchor cut is chosen by *prevalence calibration* against the inherited box rule's overall replicated rate of $49.58\%$ across Big-4 signatures; this is a documented v4.0 limitation since no signature-level hand-signed ground truth exists to permit direct ROC optimisation.
## I. Inter-CPA Pair-Level Coincidence Rate (Big-4 spike + inherited corpus-wide)
The signature-level inter-CPA pair-level coincidence-rate analysis (reported in v3.20.0 §IV-F.1, Table X as "FAR") is inherited and extended in v4.0. v4.0 retroactively reframes the metric as **inter-CPA pair-level coincidence rate (ICCR)** rather than "False Acceptance Rate" because the corpus does not provide signature-level ground-truth negative labels; the inter-CPA negative-anchor assumption underpinning the metric is itself partially violated by within-firm cross-CPA template-like collision structures (§III-L.4). The v3.20.0 corpus-wide spike on $\sim 50{,}000$ inter-CPA pairs reported a per-comparison rate of $0.0005$ (Wilson 95% CI $[0.0003, 0.0007]$) at the cosine cut $0.95$.
v4.0 additionally reports the §III-L.1 Big-4-scope spike at higher sample size ($5 \times 10^5$ inter-CPA pairs; Script 40b), which replicates and extends the v3 result and adds the structural dimension (dHash) and joint-rule rates. The §III-L.1 numbers are referenced rather than duplicated here; the consolidated v4-new ICCR calibration appears in §IV-M Tables XXI–XXVI.
## J. Five-Way Per-Signature + Document-Level Classification Output
This section reports the §III-L five-way per-signature + document-level worst-case classifier output on the Big-4 sub-corpus. The five-way category definitions are inherited unchanged from v3.20.0 §III-K (now §III-L); see §III-L for the cosine and dHash cuts.
**Table XV.** Five-way per-signature category counts, Big-4 sub-corpus, $n = 150{,}442$ classified.
| Category | Long name | $n$ signatures | % of classified |
|---|---|---|---|
| HC | High-confidence non-hand-signed | 74,593 | 49.58% |
| MC | Moderate-confidence non-hand-signed | 39,817 | 26.47% |
| HSC | High style consistency | 314 | 0.21% |
| UN | Uncertain | 35,480 | 23.58% |
| LH | Likely hand-signed | 238 | 0.16% |
(Source: Script 42; 11 of 150,453 loaded Big-4 signatures lacked one or both descriptors and were excluded. The $150{,}442$ vs $150{,}453$ distinction — descriptor-complete vs vector-complete — recurs across §IV: descriptor-complete analyses (§IV-D through §IV-J, all using accountant-level aggregates or per-signature category counts derived from the same 150,442-signature substrate) use $n = 150{,}442$; vector- or pair-recomputed analyses (§IV-M.2 Table XXI, §IV-M.3 Table XXII, §IV-M.5 Tables XXIV–XXV; Scripts 40b, 43, 44) use $n = 150{,}453$ because their pair- or pool-level computations load all vector-complete signatures including those failing the descriptor-complete filter. See §III-G for the sample-size reconciliation.)
**Per-firm five-way breakdown (% within firm).**
| Firm | HC | MC | HSC | UN | LH | total signatures |
|---|---|---|---|---|---|---|
| Firm A | 81.70% | 10.76% | 0.05% | 7.42% | 0.07% | 60,448 |
| Firm B | 34.56% | 35.88% | 0.29% | 29.09% | 0.18% | 34,248 |
| Firm C | 23.75% | 41.44% | 0.38% | 34.21% | 0.22% | 38,613 |
| Firm D | 24.51% | 29.33% | 0.22% | 45.65% | 0.29% | 17,133 |
(Source: Script 42 per-firm cross-tab.) The per-firm pattern qualitatively aligns with the K=3 cluster cross-tab of Table XVI: Firm A's signatures concentrate in the HC band (81.70%) while its CPAs concentrate at the accountant level in the K=3 C3 (high-cos / low-dHash) component (82.46%; Table XVI). These two figures address different units (per-signature classification vs per-CPA hard cluster assignment) and are not directly comparable as a like-for-like consistency check; we report the qualitative alignment but do not infer a numerical equivalence. The three non-Firm-A Big-4 firms have markedly lower HC rates than Firm A and substantially higher Uncertain rates, with Firm D having the highest Uncertain rate (45.65%).
**Document-level worst-case aggregation.** Each audit report typically carries two certifying-CPA signatures. We aggregate signature-level outcomes to document-level labels using the v3.20.0 worst-case rule (HC > MC > HSC > UN > LH; §III-L). v4.0 does not change this aggregation rule; only the population over which it is computed changes (Big-4 subset).
**Table XIX.** Document-level worst-case category counts, Big-4 sub-corpus, $n = 75{,}233$ unique PDFs.
| Category | Long name | $n$ documents | % |
|---|---|---|---|
| HC | High-confidence non-hand-signed | 46,857 | 62.28% |
| MC | Moderate-confidence non-hand-signed | 19,667 | 26.14% |
| HSC | High style consistency | 167 | 0.22% |
| UN | Uncertain | 8,524 | 11.33% |
| LH | Likely hand-signed | 18 | 0.02% |
(Source: Script 42 document-level table; 379 of 75,233 PDFs carried signatures from more than one Big-4 firm and are reported in the single-firm-PDF per-firm breakdown of the script CSV but pooled into the overall counts here.)
**Per-firm document-level breakdown (single-firm PDFs only).**
| Firm | HC | MC | HSC | UN | LH | total docs |
|---|---|---|---|---|---|---|
| Firm A | 27,600 | 1,857 | 7 | 758 | 4 | 30,226 |
| Firm B | 8,783 | 6,079 | 57 | 2,202 | 6 | 17,127 |
| Firm C | 7,281 | 8,660 | 77 | 3,099 | 5 | 19,122 |
| Firm D | 3,100 | 2,838 | 22 | 2,416 | 3 | 8,379 |
(Source: Script 42; mixed-firm PDFs $n = 379$ excluded from the per-firm rows but included in the overall counts above.)
The five-way **moderate-confidence non-hand-signed** band (cos $> 0.95$ AND $5 < \text{dHash} \leq 15$) inherits its v3.x calibration; it is **not separately validated by Scripts 38–40**, which evaluated only the binary high-confidence rule (cos $> 0.95$ AND dHash $\leq 5$). v4.0 does not re-derive the moderate-band cuts on the Big-4 subset; we report the Table XV per-firm MC proportions (10.76% / 35.88% / 41.44% / 29.33% across Firms A through D) descriptively. The v3.20.0 capture-rate calibration evidence for the moderate band (v3.20.0 Tables IX, XI, XII, XII-B) is carried into v4.0 by reference and not regenerated on the Big-4 subset. We do not claim that the MC-band per-firm ordering above is a separate validation of the §III-K Spearman convergence, since MC occupancy is not a monotone function of the per-CPA less-replication-dominated ranking (e.g., Firm D's MC fraction is lower than Firm B's while Firm D's reverse-anchor score ranks it as less replication-dominated than Firm B).
**Table XVI.** Firm × K=3 cluster cross-tabulation, Big-4 sub-corpus.
| Firm | $n$ | C1 (low-cos / high-dHash) | C2 (central) | C3 (high-cos / low-dHash) | C1 % | C3 % |
|---|---|---|---|---|---|---|
| Firm A | 171 | 0 | 30 | 141 | $0.00\%$ | $82.46\%$ |
| Firm B | 112 | 10 | 102 | 0 | $8.93\%$ | $0.00\%$ |
| Firm C | 102 | 24 | 77 | 1 | $23.53\%$ | $0.98\%$ |
| Firm D | 52 | 6 | 45 | 1 | $11.54\%$ | $1.92\%$ |
(Source: Script 35.) The cross-tab is the accountant-level descriptive output of the K=3 mixture (§III-J / §IV-E). It is reported here as a complement to the five-way per-signature classifier (Table XV), not as an operational classifier output. Reading: Firm A's CPAs are concentrated in the C3 (high-cos / low-dHash) component (no Firm A CPAs in C1); Firm C has the highest C1 (low-cos / high-dHash) concentration of the Big-4 (C1 fraction $23.5\%$); Firms B and D sit between A and C on the K=3 hard-label ordering, broadly consistent with the per-firm Spearman ordering of Table X (with the within-Big-4-non-A reverse-anchor disagreement noted there).
**Document-level worst-case aggregation outputs are reported in Table XIX above.**
## K. Full-Dataset Robustness (light scope)
This section reports the v4.0 reproducibility cross-check at the full accountant scope ($n = 686$ CPAs, Big-4 plus mid/small firms). The scope of §IV-K is deliberately narrow: we re-run only the K=3 mixture + Paper A operational-rule per-CPA less-replication-dominated rate analysis, sufficient to demonstrate that the v4.0 K=3 + Paper A convergence reproduces at the wider scope. The §III-L five-way classifier and the §IV-G LOOO analyses are not re-run at the full scope. The five-way moderate-confidence band is documented as inherited from v3.x calibration in §IV-J.
**Table XVII.** K=3 component comparison, Big-4 sub-corpus vs full dataset.
| K=3 component | Big-4 (n=437) cos / dHash / weight | Full (n=686) cos / dHash / weight | Drift Big-4 → Full |
|---|---|---|---|
| C1 (low-cos / high-dHash) | 0.9457 / 9.17 / 0.143 | 0.9278 / 11.17 / 0.284 | $\lvert\Delta\rvert$ cos 0.018, dHash 1.99, wt 0.141 |
| C2 (central) | 0.9558 / 6.66 / 0.536 | 0.9535 / 6.99 / 0.512 | $\lvert\Delta\rvert$ cos 0.002, dHash 0.33, wt 0.024 |
| C3 (high-cos / low-dHash) | 0.9826 / 2.41 / 0.321 | 0.9826 / 2.40 / 0.205 | $\lvert\Delta\rvert$ cos 0.000, dHash 0.01, wt 0.117 |
(Source: Script 41; full-dataset $\text{BIC}(K{=}3) = -792.31$ vs Big-4 $\text{BIC}(K{=}3) = -1111.93$; BIC values are not directly comparable across different $n$ and are reported only for completeness.)
**Table XVIII.** Spearman rank correlation between K=3 P(C1) and Paper A operational less-replication-dominated rate, Big-4 sub-corpus vs full dataset.
| Scope | $n$ CPAs | Spearman $\rho$ (P(C1) vs Paper A less-replication-dominated rate) | $p$-value |
|---|---|---|---|
| Big-4 (primary) | 437 | $+0.9627$ | $< 10^{-248}$ |
| Full dataset | 686 | $+0.9558$ | $< 10^{-300}$ |
| $\lvert\rho_{\text{full}} - \rho_{\text{Big-4}}\rvert$ | — | $0.0069$ | — |
(Source: Script 41.)
**Reading.** The K=3 component ordering and the strong Spearman convergence between K=3 P(C1) and the Paper A box-rule less-replication-dominated rate are preserved at the full scope. Component centres shift modestly: C3 (high-cos / low-dHash) is essentially unchanged in centre but loses weight $0.117$ as the full population includes more non-templated CPAs (mid/small firms); C1 (low-cos / high-dHash) gains weight $0.141$ and shifts to lower cosine and higher dHash (centre $(0.928, 11.17)$ vs Big-4 $(0.946, 9.17)$) as the broader population includes mid/small-firm CPAs landing toward the low-cos / high-dHash region that the Big-4-primary scope deliberately excludes. We read this as evidence that the Big-4-primary K=3 + Paper A convergence is not a Big-4-specific artefact; we do **not** read it as an endorsement of using full-dataset K=3 component centres or operational thresholds in place of the Big-4-primary analysis. Mid/small-firm composition shifts the component centres meaningfully and the v4.0 primary methodology is restricted to Big-4 by design (§III-G item 4).
## L. Ablation Study: Feature Backbone Comparison
To validate the choice of ResNet-50 as the feature extraction backbone, we conducted an ablation study comparing three pre-trained architectures: ResNet-50 (2048-dim), VGG-16 (4096-dim), and EfficientNet-B0 (1280-dim).
All models used ImageNet pre-trained weights without fine-tuning, with identical preprocessing and L2 normalization.
Table XVIII presents the comparison.
The comparison summary is inherited unchanged from the v3.20.0 backbone-ablation table (v3.20.0 Table XVIII; not the same table as v4 Table XVIII which reports Big-4 vs full-dataset Spearman drift in §IV-K).
<!-- TABLE XVIII: Backbone Comparison
<!-- v3.20.0 TABLE XVIII (inherited unchanged; reproduced here in commented form for v4 splice provenance — actual rendered table is the v3.20.0 backbone-ablation table):
| Metric | ResNet-50 | VGG-16 | EfficientNet-B0 |
|--------|-----------|--------|-----------------|
| Feature dim | 2048 | 4096 | 1280 |
@@ -492,3 +327,106 @@ ResNet-50 provides the best overall balance:
(2) its tighter distributions yield more reliable individual classifications;
(3) the highest Firm A all-pairs 1st percentile (0.543) indicates that known-replication signatures are least likely to produce low-similarity outlier pairs under this backbone; and
(4) its 2048-dimensional features offer a practical compromise between discriminative capacity and computational/storage efficiency for processing 182K+ signatures.
## M. v4-New Anchor-Based ICCR Calibration Results
This section consolidates the v4-new empirical results that support the §III-L anchor-based threshold calibration framework. Numbers below are direct re-statements from the spike scripts cited per row; the corresponding §III provenance table entries appear in §III's provenance table.
### M.1 Composition decomposition (Scripts 39b–39e)
**Table XX.** Within-firm and between-firm decomposition of the Big-4 accountant-level dip-test rejection.
| Diagnostic | Scope | Statistic | Implication |
|---|---|---|---|
| Within-firm signature-level cosine dip | Big-4 (4 firms) | $p_{\text{cos}} \in \{0.176, 0.991, 0.551, 0.976\}$ | 0/4 firms reject; cosine within-firm unimodal |
| Within-firm signature-level cosine dip | non-Big-4 (10 firms $\geq 500$ sigs) | $p_{\text{cos}} \in [0.59, 0.99]$ | 0/10 firms reject; cosine within-firm unimodal |
| Within-firm jittered-dHash dip (5 seeds, median) | Big-4 (4 firms) | $p_{\text{med}} \in \{0.999, 0.996, 0.999, 0.9995\}$ | 0/4 firms reject after integer-jitter; raw rejection was integer-tie artefact |
| Big-4 pooled dHash: 2×2 factorial | firm-centred + jittered (5 seeds) | $p_{\text{med}} = 0.35$, 0/5 seeds reject | combined corrections eliminate rejection; multimodality is composition + integer artefact |
| Integer-histogram valley near $\text{dHash} \approx 5$ | within each Big-4 firm | none (0/4 firms) | no within-firm dHash antimode at the inherited HC cutoff |
(Source: Scripts 39b, 39c, 39d, 39e; bootstrap $n_{\text{boot}} = 2000$; jitter $\sim \mathrm{U}[-0.5, +0.5]$.)
### M.2 Anchor-based inter-CPA pair-level ICCR (Script 40b)
**Table XXI.** Big-4 inter-CPA per-comparison ICCR sweep, $n = 5 \times 10^5$ pairs (Big-4 scope; v4 new).
| Threshold | Per-comparison ICCR | 95% Wilson CI |
|---|---|---|
| cos $> 0.945$ (v3.x published "natural threshold") | $0.00081$ | $[0.00073, 0.00089]$ |
| cos $> 0.95$ (inherited operating point) | $0.00060$ | $[0.00053, 0.00067]$ |
| cos $> 0.97$ | $0.00024$ | $[0.00020, 0.00029]$ |
| cos $> 0.98$ | $0.00009$ | $[0.00007, 0.00012]$ |
| dHash $\leq 5$ (inherited operating point) | $0.00129$ | $[0.00120, 0.00140]$ |
| dHash $\leq 4$ | $0.00050$ | $[0.00044, 0.00057]$ |
| dHash $\leq 3$ | $0.00019$ | $[0.00015, 0.00023]$ |
| Joint: cos $> 0.95$ AND dHash $\leq 5$ (any-pair semantics) | $0.00014$ | — |
| Joint: cos $> 0.95$ AND dHash $\leq 4$ (any-pair) | $0.00011$ | — |
Conditional ICCR(dHash $\leq 5$ | cos $> 0.95$) $= 0.234$ (Wilson 95% $[0.190, 0.285]$; $70$ of $299$ pairs).
The cos $> 0.95$ row replicates v3.20.0 §IV-F.1 Table X (v3 reported $0.0005$ under prior "FAR" terminology). The dHash row and joint row are v4 new.
### M.3 Pool-normalised per-signature ICCR (Script 43)
**Table XXII.** Pool-normalised per-signature ICCR under the deployed any-pair HC rule (cos $> 0.95$ AND dHash $\leq 5$); $n_{\text{sig}} = 150{,}453$ (vector-complete Big-4); CPA-block bootstrap $n_{\text{boot}} = 1000$.
| Scope | Per-signature ICCR | Wilson 95% CI | CPA-bootstrap 95% CI |
|---|---|---|---|
| Big-4 pooled (any-pair, deployed) | $0.1102$ | $[0.1086, 0.1118]$ | $[0.0908, 0.1330]$ |
| Big-4 pooled (same-pair, stricter alternative) | $0.0827$ | $[0.0813, 0.0841]$ | $[0.0668, 0.1021]$ |
| Firm A (any-pair) | $0.2594$ | — | — |
| Firm B (any-pair) | $0.0147$ | — | — |
| Firm C (any-pair) | $0.0053$ | — | — |
| Firm D (any-pair) | $0.0110$ | — | — |
| Pool-size decile 1 (smallest pools) any-pair | $0.0249$ | — | — |
| Pool-size decile 10 (largest pools) any-pair | $0.1905$ | — | — |
Decile trend is broadly monotone in pool size with two minor reversals (decile 5 and decile 9 dip below their predecessors). Stricter operating point cos $> 0.95$ AND dHash $\leq 3$ (same-pair) gives per-signature ICCR $0.0449$.
### M.4 Document-level ICCR under three alarm definitions (Script 45)
**Table XXIII.** Document-level inter-CPA ICCR by alarm definition; $n_{\text{docs}} = 75{,}233$.
| Alarm definition | Alarm set | Document-level ICCR | Wilson 95% CI |
|---|---|---|---|
| D1 | HC only | $0.1797$ | $[0.1770, 0.1825]$ |
| D2 (operational) | HC + MC | $0.3375$ | $[0.3342, 0.3409]$ |
| D3 | HC + MC + HSC | $0.3384$ | $[0.3351, 0.3418]$ |
Per-firm D2 document-level ICCR: Firm A $0.6201$ ($n = 30{,}226$); Firm B $0.1600$ ($n = 17{,}127$); Firm C $0.1635$ ($n = 19{,}501$); Firm D $0.0863$ ($n = 8{,}379$). The Firm C denominator $n = 19{,}501$ exceeds Table XIX's single-firm Firm C count of $19{,}122$ by exactly the $379$ mixed-firm PDFs: all $379$ are $1{:}1$ Firm C / Firm D mixed-firm documents, and Script 45's mode-of-firms implementation (`np.argmax` over `np.unique`'s alphabetically-sorted firm counts) returns the first-sorted firm on ties, which assigns these tied documents to Firm C rather than to Firm D. The four per-firm denominators here therefore sum to the full $75{,}233$, whereas Table XIX's per-firm rows sum to $74{,}854 = 75{,}233 - 379$.
### M.5 Firm heterogeneity logistic regression and cross-firm hit matrix (Script 44)
**Table XXIV.** Logistic regression of per-signature any-pair HC hit indicator on firm dummies and centred log pool size (Firm A reference).
| Term | Odds ratio (vs Firm A) | Direction |
|---|---|---|
| Firm B | $0.053$ | $\sim 19\times$ lower odds than Firm A |
| Firm C | $0.010$ | $\sim 100\times$ lower odds than Firm A |
| Firm D | $0.027$ | $\sim 37\times$ lower odds than Firm A |
| log(pool size, centred) | $4.01$ | $\sim 4\times$ higher odds per log unit pool size |
Per-decile per-firm rates (Table not duplicated here; Script 44 decile table available in the supplementary report): within every pool-size decile, Firms B/C/D show rates of $0.0006$–$0.0358$ while Firm A ranges $0.0541$–$0.5958$. The firm gap survives within matched pool sizes.
**Table XXV.** Cross-firm hit matrix among Big-4 source signatures with any-pair HC hit; max-cosine partner firm (counts).
| Source firm | Firm A cand. | Firm B | Firm C | Firm D | non-Big-4 | n hits |
|---|---|---|---|---|---|---|
| Firm A | $14{,}447$ | $95$ | $44$ | $19$ | $17$ | $14{,}622$ |
| Firm B | $92$ | $371$ | $8$ | $4$ | $9$ | $484$ |
| Firm C | $16$ | $7$ | $149$ | $5$ | $1$ | $178$ |
| Firm D | $22$ | $2$ | $6$ | $106$ | $1$ | $137$ |
Same-pair joint hits (single candidate satisfying both cos $> 0.95$ AND dHash $\leq 5$) are within-firm at rates $99.96\%$ / $97.7\%$ / $98.2\%$ / $97.0\%$ for Firms A/B/C/D respectively.
### M.6 Alert-rate sensitivity around inherited HC threshold (Script 46)
**Table XXVI.** Local-gradient / median-gradient ratio at inherited thresholds (descriptive plateau diagnostic).
| Threshold | Local / median gradient ratio | Interpretation |
|---|---|---|
| cos $= 0.95$ (HC) | $\approx 25\times$ | locally sensitive (not plateau-stable) |
| dHash $= 5$ (HC) | $\approx 3.8\times$ | locally sensitive (not plateau-stable) |
| dHash $= 15$ (MC/HSC boundary) | $\approx 0.08$ | plateau-like (saturating tail) |
Big-4 observed deployed alert rate on actual same-CPA pools: per-signature HC $= 0.4958$; per-document HC $= 0.6228$. The deployed-rate excess over the inter-CPA proxy is $0.3856$ pp per-signature and $0.4431$ pp per-document; this excess is interpreted as a same-CPA repeatability signal under the §III-M caveats, not as a presumed true-positive rate.