Paper A v3.11: reframe Section III-G unit hierarchy + propagate implications

Rewrites Section III-G (Unit of Analysis and Summary Statistics) after
self-review identified three logical issues in v3.10:

1. Ordering inversion: the three units are now ordered signature ->
   auditor-year -> accountant, with auditor-year as the principled
   middle unit under within-year assumptions and accountant as a
   deliberate cross-year pooling.

2. Oversold assumption: the old "within-auditor-year no-mixing
   identification assumption" is split into A1 (pair-detectability,
   weak statistical, cross-year scope matching the detector) and A2
   (within-year label uniformity, interpretive convention). The
   arithmetic statistics reported in the paper do not require A2; A2
   only underwrites interpretive readings (notably IV-H.1's partner-
   level "minority of hand-signers" framing).

3. Motivation-assumption mismatch: removed the "longitudinal behaviour
   of interest" framing and explicitly disclaimed across-year
   homogeneity. Accountant-level coordinates are now described as a
   pooled observed tendency rather than a time-invariant regime.

Propagated implications across Introduction, Discussion, and Results:
softened "tends to cluster into a dominant regime" and "directly
quantifying the minority of hand-signers" to "pooled observed
tendency" / "consistent with within-firm heterogeneity"; rewrote the
Limitations fifth point (was "treats all signatures from a CPA as
a single class"); added a seventh Limitation acknowledging the
source-template edge case; added a per-signature best-match cross-year
caveat to Section IV-H.2; softened IV-H.2's "direct consequence" to
"consistent with"; reframed pixel-identity anchor as pair-level proof
of image reuse (with source-template exception) rather than absolute
signature-level positive.

Process: self-review (9 findings) -> full-pass fixes -> codex
gpt-5.5 xhigh round-10 verification (8 RESOLVED, 1 PARTIAL, 4 MINOR
regression findings) -> regression fixes.

No re-computation. All tables (IV-XVIII) and Appendix A numbers
unchanged. Abstract at 248/250 words.

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

paper/paper_a_discussion_v3.md

@@ -40,7 +40,7 @@ Our evidence across multiple analyses rules out that assumption for Firm A while
 Three convergent strands of evidence support the replication-dominated framing.
 First, the visual-inspection evidence: randomly sampled Firm A reports exhibit pixel-identical signature images across different audit engagements and fiscal years for the majority of partners---a physical impossibility under independent hand-signing events.
 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---directly quantifying the within-firm minority of hand-signers.
+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 practice (spanning a minority of hand-signers, CPAs undergoing mid-sample mechanism transitions, and CPAs whose pooled coordinates reflect mixed-quality replication) rather than a pure replication population.
 Nine additional Firm A CPAs are excluded from 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.
@@ -72,7 +72,7 @@ The framing we adopt---replication-dominated rather than replication-pure---is a
 ## F. Pixel-Identity and Inter-CPA Anchors as Annotation-Free Validation
 
 A further methodological contribution is the combination of byte-level pixel identity as an annotation-free *conservative* gold positive and a large random-inter-CPA negative anchor.
-Handwriting physics makes byte-identity impossible under independent signing events, so any pair of same-CPA signatures that are byte-identical after crop and normalization is an absolute positive for non-hand-signing, requiring no human review.
+Handwriting physics makes byte-identity impossible under independent signing events, so any pair of same-CPA signatures that are byte-identical after crop and normalization is pair-level proof of image reuse and, modulo the narrow source-template edge case discussed in the seventh limitation below, a conservative positive for non-hand-signing without requiring human review.
 In our corpus 310 signatures satisfied this condition.
 We emphasize that byte-identical pairs are a *subset* of the true non-hand-signed positive class---they capture only those whose nearest same-CPA match happens to be bytewise identical, excluding replications that are pixel-near-identical but not byte-identical (for example, under different scan or compression pathways).
 Perfect recall against this subset therefore does not generalize to perfect recall against the full non-hand-signed population; it is a lower-bound calibration check on the classifier's ability to catch the clearest positives rather than a generalizable recall estimate.
@@ -99,13 +99,17 @@ This effect would bias classification toward false negatives rather than false p
 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.
 
-Fifth, the classification framework treats all signatures from a CPA as belonging to a single class, not accounting for potential changes in signing practice over time (e.g., a CPA who signed genuinely in early years but adopted non-hand-signing later).
-Extending the accountant-level analysis to auditor-year units is a natural next step.
+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.
 
 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.
+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.
+
 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.