Paper A v3.18: remove accountant-level + replication-dominated calibration + Gemini 2.5 Pro review minor fixes

Major changes (per partner red-pen + user decision):
- Delete entire accountant-level analysis (III.J, IV.E, Tables VI/VII/VIII,
  Fig 4) -- cross-year pooling assumption unjustified, removes the implicit
  "habitually stamps = always stamps" reading.
- Renumber sections III.J/K/L (was K/L/M) and IV.E/F/G/H/I (was F/G/H/I/J).
- Title: "Three-Method Convergent Thresholding" -> "Replication-Dominated
  Calibration" (the three diagnostics do NOT converge at signature level).
- Operational cosine cut anchored on whole-sample Firm A P7.5 (cos > 0.95).
- Three statistical diagnostics (Hartigan/Beta/BD-McCrary) reframed as
  descriptive characterisation, not threshold estimators.
- Firm A replication-dominated framing: 3 evidence strands -> 2.
- Discussion limitation list: drop accountant-level cross-year pooling and
  BD/McCrary diagnostic; add auditor-year longitudinal tracking as future work.
- Tone-shift: "we do not claim / do not derive" -> "we find / motivates".

Reference verification (independent web-search audit of all 41 refs):
- Fix [5] author hallucination: Hadjadj et al. -> Kao & Wen (real authors of
  Appl. Sci. 10:11:3716; report at paper/reference_verification_v3.md).
- Polish [16] [21] [22] [25] (year/volume/page-range/model-name).

Gemini 2.5 Pro peer review (Minor Revision verdict, A-F all positive):
- Neutralize script-path references in tables/appendix -> "supplementary
  materials".
- Move conflict-of-interest declaration from III-L to new Declarations
  section before References (paper_a_declarations_v3.md).

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

paper/paper_a_discussion_v3.md

@@ -11,41 +11,39 @@ Forgery detection systems optimize for inter-class discriminability---maximizing
 Non-hand-signing detection, by contrast, requires sensitivity to the *upper tail* of the intra-class similarity distribution, where the boundary between consistent handwriting and image reproduction becomes ambiguous.
 The dual-descriptor framework we propose---combining semantic-level features (cosine similarity) with structural-level features (dHash)---addresses this ambiguity in a way that single-descriptor approaches cannot.
 
-## B. Continuous-Quality Spectrum vs. Clustered Accountant-Level Heterogeneity
+## B. Per-Signature Similarity is a Continuous Quality Spectrum
 
-The most consequential empirical finding of this study is the asymmetry between signature level and accountant level revealed by the convergent threshold framework and the Hartigan dip test (Sections IV-D and IV-E).
+A central empirical finding of this study is that per-signature similarity does not form a clean two-mechanism mixture (Section IV-D).
+Firm A's signature-level cosine is formally unimodal (Hartigan dip test $p = 0.17$) with a long left tail.
+The all-CPA signature-level cosine rejects unimodality ($p < 0.001$), reflecting the heterogeneity of signing practices across firms, but its structure is not well approximated by a two-component Beta mixture: BIC clearly prefers a three-component fit ($\Delta\text{BIC} = 381$ for Firm A; $10{,}175$ for the full sample), and the forced 2-component Beta crossing and its logit-GMM robustness counterpart disagree sharply on the candidate threshold (0.977 vs. 0.999 for Firm A).
+The BD/McCrary discontinuity test locates its transition at cosine 0.985---*inside* the non-hand-signed mode rather than at a boundary between two mechanisms---and the transition is not bin-width-stable (Appendix A).
 
-At the per-signature level, the distribution of best-match cosine similarity is *not* cleanly bimodal.
-Firm A's signature-level cosine is formally unimodal (dip test $p = 0.17$) with a long left tail.
-The all-CPA signature-level cosine rejects unimodality ($p < 0.001$), but its structure is not well approximated by a two-component Beta mixture: BIC clearly prefers a three-component fit, and the 2-component Beta crossing and its logit-GMM counterpart disagree sharply on the candidate threshold (0.977 vs. 0.999 for Firm A).
-The BD/McCrary discontinuity test locates its transition at 0.985---*inside* the non-hand-signed mode rather than at a boundary between two mechanisms.
-Taken together, these results indicate that non-hand-signed signatures form a continuous quality spectrum rather than a discrete class.
+Taken together, these results indicate that non-hand-signed signatures form a continuous quality spectrum rather than a discrete class cleanly separated from hand-signing.
 Replication quality varies continuously with scan equipment, PDF compression, stamp pressure, and firm-level e-signing system generation, producing a heavy-tailed distribution that no two-mechanism mixture explains at the signature level.
 
-At the per-accountant aggregate level the picture partly reverses.
-The distribution of per-accountant mean cosine (and mean dHash) rejects unimodality, a BIC-selected three-component Gaussian mixture cleanly separates (C1) a high-replication cluster dominated by Firm A, (C2) a middle band shared by the other Big-4 firms, and (C3) a hand-signed-tendency cluster dominated by smaller domestic firms, and the three 1D threshold methods applied at the accountant level produce mutually consistent estimates (KDE antimode $= 0.973$, Beta-2 crossing $= 0.979$, logit-GMM-2 crossing $= 0.976$).
-The BD/McCrary test is largely null at the accountant level---no significant transition at two of three cosine bin widths and two of three dHash bin widths, and the one cosine transition (at bin 0.005, location 0.980) sits on the upper edge of the convergence band described above rather than outside it (Appendix A).
-This pattern is consistent with a clustered *but smoothly mixed* accountant-level distribution rather than with a sharp density discontinuity: accountant-level means cluster into three recognizable groups, yet the test fails to reject the smoothness null at the sample size available ($N = 686$), and the GMM cluster boundaries appear gradual rather than sheer.
-We caveat this interpretation appropriately in Section V-G: the BD null alone cannot affirmatively establish smoothness---only fail to falsify it---and our substantive claim of smoothly-mixed clustering rests on the joint weight of the GMM fit, the dip test, and the BD null rather than on the BD null alone.
+The methodological implication is that the operational classifier's cosine cut should not be derived from a mixture-fit crossing.
+We accordingly anchor the operational cosine cut on the whole-sample Firm A P7.5 percentile (Section III-K), and treat the signature-level threshold-estimator outputs (KDE antimode, Beta and logit-Gaussian crossings) as descriptive characterisation of the similarity distribution rather than as the source of operational thresholds.
+The BD/McCrary procedure plays a *density-smoothness diagnostic* role in this framing rather than that of an independent threshold estimator.
 
-The substantive interpretation we take from this evidence is therefore narrower than a "discrete-behaviour" claim: *pixel-level output quality* is continuous and heavy-tailed, and *accountant-level aggregate behaviour* is clustered (three recognizable groups) but not sharply discrete.
-The accountant-level mixture is a useful classifier of firm-and-practitioner-level signing regimes; individual behaviour may still transition or mix over time within a practitioner, and our cross-sectional analysis does not rule this out.
-Methodologically, the implication is that the two threshold estimators (KDE antimode, Beta mixture with logit-Gaussian robustness) are meaningfully applied at the accountant level for threshold estimation, while the BD/McCrary non-transition at the same level is a failure-to-reject rather than a failure of the method---informative alongside the other evidence but subject to the power caveat recorded in Section V-G.
+This continuous-spectrum finding also has substantive implications for downstream interpretation.
+Because pixel-level output quality varies continuously, *signature-level rates* (such as the 92.5% / 7.5% Firm A split) reflect the share of signatures whose similarity falls above or below a chosen threshold rather than the share that came from a "non-hand-signing mechanism" versus a "hand-signing mechanism."
+We accordingly report all rates as signature-level quantities and abstain from partner-level frequency claims (Section III-G).
 
 ## C. Firm A as a Replication-Dominated, Not Pure, Population
 
 A recurring theme in prior work that treats Firm A or an analogous reference group as a calibration anchor is the implicit assumption that the anchor is a pure positive class.
 Our evidence across multiple analyses rules out that assumption for Firm A while affirming its utility as a calibration reference.
 
-Three convergent strands of evidence support the replication-dominated framing.
-First, the byte-level pair evidence: 145 Firm A signatures (from 50 distinct partners of 180 registered) have a byte-identical same-CPA match in a different audit report, with 35 of these matches spanning different fiscal years. Independent hand-signing cannot produce byte-identical images across distinct reports, so these pairs directly establish image reuse within Firm A.
-Second, the signature-level statistical evidence: Firm A's per-signature cosine distribution is unimodal long-tail rather than a tight single peak; 92.5% of Firm A signatures exceed cosine 0.95, with the remaining 7.5% forming the left tail.
-Third, the accountant-level evidence: of the 171 Firm A CPAs with enough signatures ($\geq 10$) to enter the accountant-level GMM, 32 (19%) fall into the middle-band C2 cluster rather than the high-replication C1 cluster---consistent with within-firm heterogeneity in signing output (potentially spanning hand-signing partners, multi-template replication workflows, CPAs undergoing mid-sample mechanism transitions, and CPAs whose pooled coordinates reflect mixed-quality replication; we do not disaggregate these mechanisms---see Section III-G for the scope of claims) rather than a pure replication population.
-Of the 178 valid Firm A CPAs (the 180 registered CPAs minus two excluded for disambiguation ties in the registry; Section IV-G.2), seven are outside the GMM for having fewer than 10 signatures, so we cannot place them in a cluster from the cross-sectional analysis alone.
-The held-out Firm A 70/30 validation (Section IV-G.2) gives capture rates on a non-calibration Firm A subset that sit in the same replication-dominated regime as the calibration fold across the full range of operating rules (extreme rules are statistically indistinguishable; operational rules in the 85–95% band differ between folds by 1–5 percentage points, reflecting within-Firm-A heterogeneity in replication intensity rather than a generalization failure).
-The accountant-level GMM (Section IV-E) and the threshold-independent partner-ranking analysis (Section IV-H.2) are the cross-checks that are robust to fold-level sampling variance.
+Two convergent strands of evidence support the replication-dominated framing.
+First, the byte-level pair evidence: 145 Firm A signatures (from 50 distinct partners of 180 registered) have a byte-identical same-CPA match in a different audit report, with 35 of these matches spanning different fiscal years.
+Independent hand-signing cannot produce byte-identical images across distinct reports, so these pairs directly establish image reuse within Firm A as a concrete, threshold-free phenomenon, and the 50/180 partner spread shows that replication is widespread rather than confined to a handful of CPAs.
+Second, the signature-level distributional evidence: Firm A's per-signature cosine distribution is unimodal long-tail (Hartigan dip test $p = 0.17$) rather than a tight single peak; 92.5% of Firm A signatures exceed cosine 0.95, with the remaining 7.5% forming the left tail.
+The unimodal-long-tail *shape*, not the precise 92.5 / 7.5 split, is the structural evidence: it is consistent with a single dominant mechanism plus residual within-firm heterogeneity, and a noise-only explanation of the left tail would predict a shrinking share as scan/PDF technology matured over 2013--2023, which is not what we observe (Section IV-G.1).
 
-The replication-dominated framing is internally coherent with all three pieces of evidence, and it predicts and explains the residuals that a "near-universal" framing would be forced to treat as noise.
+Two additional checks, reported in Section IV-G, are robust to threshold choice and complement the two primary strands:
+the held-out Firm A 70/30 validation (Section IV-F.2) gives capture rates on a non-calibration Firm A subset that sit in the same replication-dominated regime as the calibration fold across the full range of operating rules (extreme rules are statistically indistinguishable; operational rules in the 85--95% band differ between folds by 1--5 percentage points, reflecting within-Firm-A heterogeneity in replication intensity rather than a generalization failure), and the threshold-independent partner-ranking analysis (Section IV-G.2) shows that Firm A auditor-years occupy 95.9% of the top decile of similarity-ranked auditor-years against a 27.8% baseline share---a 3.5$\times$ concentration ratio that uses only ordinal ranking and is independent of any absolute cutoff.
+
+The replication-dominated framing is internally coherent with both pieces of evidence, and it predicts and explains the residuals that a "near-universal" framing would be forced to treat as noise.
 We therefore recommend that future work building on this calibration strategy should explicitly distinguish replication-dominated from replication-pure calibration anchors.
 
 ## D. The Style-Replication Gap
@@ -67,7 +65,7 @@ Our approach leverages domain knowledge---the established prevalence of non-hand
 
 This calibration strategy has broader applicability beyond signature analysis.
 Any forensic detection system operating on real-world corpora can benefit from identifying subpopulations with known dominant characteristics (positive or negative) to anchor threshold selection, particularly when the distributions of interest are non-normal and non-parametric or mixture-based thresholds are preferred over parametric alternatives.
-The framing we adopt---replication-dominated rather than replication-pure---is an important refinement of this strategy: it prevents overclaim, accommodates the within-firm heterogeneity quantified by the accountant-level mixture, and yields classification rates that are internally consistent with the data.
+The framing we adopt---replication-dominated rather than replication-pure---is an important refinement of this strategy: it prevents overclaim, accommodates the within-firm heterogeneity visible in the unimodal-long-tail shape of Firm A's per-signature cosine distribution, and yields classification rates that are internally consistent with the data.
 
 ## F. Pixel-Identity and Inter-CPA Anchors as Annotation-Free Validation
 
@@ -97,23 +95,18 @@ In these overlap regions, blended pixels are replaced with white, potentially cr
 This effect would bias classification toward false negatives rather than false positives, but the magnitude has not been quantified.
 
 Fourth, scanning equipment, PDF generation software, and compression algorithms may have changed over the 10-year study period (2013--2023), potentially affecting similarity measurements.
-While cosine similarity and dHash are designed to be robust to such variations, longitudinal confounds cannot be entirely excluded, and we note that our accountant-level aggregates could mask within-accountant temporal transitions.
+While cosine similarity and dHash are designed to be robust to such variations, longitudinal confounds cannot be entirely excluded.
 
-Fifth, the accountant-level summary (Section III-J) is a cross-year pooled statistic by construction, so a CPA whose signing mechanism changed mid-sample is placed at a weighted mix of component means rather than at a single regime centroid.
-Extending the accountant-level analysis to auditor-year units---using the same convergent threshold framework at finer temporal resolution---is the natural next step for resolving such within-accountant transitions.
+Fifth, our cross-sectional analysis does not track individual CPAs longitudinally and therefore cannot confirm or rule out within-CPA mechanism transitions over the sample period (e.g., a CPA who hand-signed early in the sample and switched to firm-level e-signing later, or vice versa).
+Extending the analysis to *auditor-year* units---computing per-signature statistics within each fiscal year and observing how individual CPAs move across years---is the natural next step for resolving such within-CPA transitions and is left to future work.
 
-Sixth, the BD/McCrary transition estimates fall inside rather than between modes for the per-signature cosine distribution, and the test produces no significant transition at all at the accountant level.
-In our application, therefore, BD/McCrary contributes diagnostic information about local density-smoothness rather than an independent accountant-level threshold estimate; that role is played by the KDE antimode and the two mixture-based estimators.
-We emphasize that the accountant-level BD/McCrary null is *consistent with*---not affirmative proof of---smoothly mixed cluster boundaries: the BD/McCrary test is known to have limited statistical power at modest sample sizes, and with $N = 686$ accountants in our analysis the test cannot reliably detect anything less than a sharp cliff-type density discontinuity.
-Failure to reject the smoothness null at this sample size therefore reinforces BD/McCrary's role as a diagnostic rather than a definitive estimator; the substantive claim of smoothly-mixed accountant-level clustering rests on the joint weight of the dip-test and Beta-mixture evidence together with the BD null, not on the BD null alone.
-
-Seventh, the max/min detection logic treats both ends of a near-identical same-CPA pair as non-hand-signed.
+Sixth, the max/min detection logic treats both ends of a near-identical same-CPA pair as non-hand-signed.
 In the rare case that one of the two documents contains a genuinely hand-signed exemplar that was subsequently reused as the stamping or e-signature template, the pair correctly identifies image reuse but misattributes the non-hand-signed status to the source exemplar.
 This misattribution affects at most one source document per template variant per CPA (the exemplar from which the template was produced), is not expected to be common given that stored signature templates are typically generated in a separate acquisition step rather than extracted from submitted audit reports, and does not materially affect aggregate capture rates at the firm level.
 
-Eighth, our analyses remain at the signature level and the accountant (cross-year pooled) level; we abstain from partner-level frequency inferences such as "X% of CPAs hand-sign in a given year."
+Seventh, our analyses remain at the signature level; we abstain from partner-level frequency inferences such as "X% of CPAs hand-sign in a given year."
 Per-signature labels in this paper are not translated to per-report or per-partner mechanism assignments, because making such a translation would require an assumption of within-year uniformity of signing mechanisms that we do not adopt: a CPA's signatures within a single fiscal year may reflect a single replication template, multiple templates used in parallel (e.g., for different engagement positions or reporting pipelines), within-year mechanism mixing, or a combination, and the data at hand do not disambiguate these possibilities (Section III-G).
-The signature-level rates we report, including the 92.5% / 7.5% Firm A split and the year-by-year left-tail share of Section IV-H.1, should accordingly be read as signature-level quantities rather than partner-level frequencies.
+The signature-level rates we report, including the 92.5% / 7.5% Firm A split and the year-by-year left-tail share of Section IV-G.1, should accordingly be read as signature-level quantities rather than partner-level frequencies.
 
 Finally, the legal and regulatory implications of our findings depend on jurisdictional definitions of "signature" and "signing."
 Whether non-hand-signing of a CPA's own stored signature constitutes a violation of signing requirements is a legal question that our technical analysis can inform but cannot resolve.