pdf_signature_extraction/paper/paper_a_v4_combined.md

Abstract

Regulations require Certified Public Accountants (CPAs) to attest each audit report with a signature, but digitization makes it feasible to reuse a stored signature image across reports — through administrative stamping or firm-level electronic signing — thereby undermining individualized attestation. We build an end-to-end pipeline for screening such non-hand-signed signatures at scale: a Vision-Language Model identifies signature pages, YOLOv11 localizes signatures, ResNet-50 supplies deep features, and a dual-descriptor layer combines cosine similarity with an independent-minimum perceptual hash (dHash) to separate style consistency from image reproduction. Applied to 90,282 Taiwan audit reports (2013–2023), the pipeline yields 182,328 signatures from 758 CPAs; primary analyses are scoped to the Big-4 sub-corpus (437 CPAs; 150,442 signatures). Distributional diagnostics show that the apparent multimodality of the descriptor distribution dissolves under joint firm-mean centring and integer-tie jitter (p rises to 0.35), so no within-population bimodal antimode anchors the operational thresholds. We instead adopt an anchor-based inter-CPA coincidence-rate (ICCR) calibration at three units: per-comparison (0.0006 at cos$>0.95$; 0.0013 at dHash$\leq 5$; 0.00014 jointly), pool-normalised per-signature (0.11 under the deployed any-pair high-confidence rule), and per-document (0.34 for the operational HC+MC alarm). Firm heterogeneity is decisive: Firm A's per-document HC+MC inter-CPA proxy ICCR is 0.62 versus $0.09$–0.16 at Firms B/C/D, and a per-signature logistic regression confirms the firm gap persists after controlling for pool size; under the deployed any-pair rule $77$–99\% of inter-CPA collisions concentrate within the source firm — consistent with firm-level template-like reuse. We position the system as a specificity-proxy-anchored screening framework with human-in-the-loop review, not as a validated forensic detector; no calibrated error rates are reportable without signature-level ground truth.

I. Introduction

Financial audit reports serve as a critical mechanism for ensuring corporate accountability and investor protection. In Taiwan, the Certified Public Accountant Act (會計師法 §4) and the Financial Supervisory Commission's attestation regulations (查核簽證核准準則 §6) require certifying CPAs to affix their signature or seal (簽名或蓋章) to each audit report [1]. While the law permits either a handwritten signature or a seal, the CPA's attestation on each report is intended to represent a deliberate, individual act of professional endorsement for that specific audit engagement [2].

The digitization of financial reporting has introduced a practice that complicates this intent. As audit reports are now routinely generated, transmitted, and archived as PDF documents, it is technically and operationally straightforward to reproduce a CPA's stored signature image across many reports rather than re-executing the signing act for each one. This reproduction can occur either through an administrative stamping workflow — in which scanned signature images are affixed by staff as part of the report-assembly process — or through a firm-level electronic signing system that automates the same step. We refer to signatures produced by either workflow collectively as non-hand-signed. Although this practice may fall within the literal statutory requirement of "signature or seal," it raises substantive concerns about audit quality, as an identically reproduced signature applied across hundreds of reports may not represent meaningful individual attestation for each engagement. The accounting literature has examined the audit-quality consequences of partner-level engagement transparency: studies of partner-signature mandates in the United Kingdom find measurable downstream effects [31], cross-jurisdictional evidence on individual partner signature requirements highlights similar quality channels [32], and Taiwan-specific evidence on mandatory partner rotation documents how individual-partner identification interacts with audit-quality outcomes [33]. Unlike traditional signature forgery, where a third party attempts to imitate another person's handwriting, non-hand-signing involves the legitimate signer's own stored signature being reused, and is visually invisible to report users at scale.

The distinction between non-hand-signing detection and signature forgery detection is conceptually and technically important. The extensive body of research on offline signature verification [3]–[8] focuses almost exclusively on forgery detection — determining whether a questioned signature was produced by its purported author. In our context, identity is not in question; the CPA is indeed the legitimate signer. The question is whether the physical act of signing occurred for each individual report, or whether a single signing event was reproduced as an image across many reports. This detection problem differs fundamentally from forgery detection: while it does not require modeling skilled-forger variability, it introduces the distinct challenge of separating legitimate intra-signer consistency from image-level reproduction.

A methodological concern shapes the research design. Many prior similarity-based classification studies rely on ad-hoc thresholds — declaring two images equivalent above a hand-picked cosine cutoff, for example — without principled statistical justification. Such thresholds are fragile in an archival-data setting. A defensible approach requires (i) explicit calibration of the operational thresholds against measurable negative-anchor evidence; (ii) diagnostic procedures that test whether the descriptor distribution itself supports a within-population threshold, including formal decomposition of apparent multimodality into between-group composition and integer-tie artefacts; (iii) annotation-free reporting of operational alarm rates at multiple analysis units (per-comparison, per-signature pool, per-document) with Wilson 95% confidence intervals; (iv) per-firm stratification of the reported rates to surface heterogeneity that aggregate metrics conceal; and (v) explicit disclosure of the unsupervised setting's limits — in particular, the inability to estimate true error rates without signature-level ground-truth labels.

Despite the significance of the problem for audit quality and regulatory oversight, to our knowledge no prior work has specifically addressed non-hand-signing detection in financial audit documents at scale with these methodological safeguards. Woodruff et al. [9] developed an automated pipeline for signature analysis in corporate filings for anti-money-laundering investigations, but their work focused on author clustering rather than detecting image reuse. Copy-move forgery detection methods [10], [11] address duplicated regions within or across images but are designed for natural images and do not account for the specific characteristics of scanned document signatures. Research on near-duplicate image detection using perceptual hashing combined with deep learning [12], [13] provides relevant methodological foundations but has not been applied to document forensics or signature analysis. From the statistical side, the methods we adopt for distributional characterisation — the Hartigan dip test [37] and finite mixture modelling via the EM algorithm [40], [41], complemented by a Burgstahler-Dichev / McCrary density-smoothness diagnostic [38], [39] — have been developed in statistics and accounting-econometrics but have not been combined as a joint diagnostic toolkit for document-forensics threshold characterisation.

In this paper we present a fully automated, end-to-end pipeline for screening non-hand-signed CPA signatures in audit reports at scale, together with an anchor-calibrated screening framework that characterises the pipeline's operational behaviour under explicit unsupervised assumptions. The pipeline processes raw PDF documents through (1) signature page identification with a Vision-Language Model; (2) signature region detection with a trained YOLOv11 object detector; (3) deep feature extraction via a pre-trained ResNet-50; (4) dual-descriptor similarity (cosine + independent-minimum dHash); (5) anchor-based threshold calibration at three units of analysis (per-comparison, pool-normalised per-signature, per-document) against an inter-CPA negative-anchor coincidence-rate proxy (§III-L); (6) firm-stratified per-rule reporting and a within-firm cross-CPA hit-matrix analysis (§III-L.4); (7) a composition decomposition that establishes the absence of a within-population bimodal antimode in the descriptor distributions (§III-I.4); and (8) disclosure of each diagnostic's untested assumption (§III-M).

A key empirical finding is that the descriptor distributions do not support a within-population natural threshold. The apparent multimodality in the Big-4 accountant-level distribution is explained by between-firm location-shift effects (Firm A's mean dHash of 2.73 versus Firms B/C/D's 6.46, 7.39, 7.21) and integer mass-point artefacts on the integer-valued dHash axis. After joint firm-mean centring and uniform integer-tie jitter, the pooled dHash dip-test rejection disappears (p_{\text{median}} = 0.35 across five seeds). Within-firm diagnostics in every Big-4 firm fail to reveal stable bimodal structure after accounting for integer ties; eligible non-Big-4 firms provide corroborating raw-axis evidence on the cosine dimension (§III-I.4). We therefore treat mixture fits as descriptive summaries of firm-compositional structure rather than threshold-generating mechanisms, and calibrate the deployed operating rules using inter-CPA coincidence-rate anchors.

In place of distributional anchoring, we adopt an anchor-based inter-CPA coincidence-rate (ICCR) calibration. At the per-comparison unit, the cos$>0.95$ operating point yields ICCR = 0.00060 on a $5 \times 10^5$-pair Big-4 sample; the dHash$\leq 5$ structural cutoff yields ICCR = 0.00129; the joint rule cos$>0.95$ AND dHash$\leq 5$ yields joint ICCR = 0.00014 (any-pair semantics, matching the deployed extrema rule). At the pool-normalised per-signature unit, the same rule's effective coincidence rate is materially higher because the deployed classifier takes max-cosine and min-dHash over a same-CPA pool: pooled Big-4 any-pair ICCR is 0.1102 (Wilson 95% CI [0.1086, 0.1118]; CPA-block bootstrap 95% [0.0908, 0.1330]). At the per-document unit, the operational HC$+$MC alarm fires on 33.75\% of Big-4 documents under the inter-CPA candidate-pool counterfactual.

The pooled per-signature and per-document rates conceal striking firm heterogeneity. A logistic regression of the per-signature hit indicator on firm dummies (Firm A reference) and centred log pool size yields odds ratios of 0.053 (Firm B), 0.010 (Firm C), and 0.027 (Firm D) — Firms B/C/D are an order of magnitude below Firm A even after controlling for the pool-size confound. Cross-firm hit matrix analysis under the deployed any-pair rule shows within-firm collision concentrations of 98.8\% at Firm A and $76.7$–83.7\% at Firms B/C/D (Table XXV; the stricter same-pair joint event saturates at $97.0$–99.96\% within-firm across all four firms). The pattern is consistent with firm-specific template, stamp, or document-production reuse mechanisms — though not by itself diagnostic of deliberate sharing. The deployed five-way box rule defines a reproducible screening classifier; the calibration contribution is to characterise its multi-level inter-CPA coincidence behaviour rather than to derive new thresholds. The high-confidence sub-rule (cos > 0.95 AND dHash \leq 5) and moderate-confidence sub-rule (cos > 0.95 AND 5 < \text{dHash} \leq 15) are explicit decision rules whose calibrated false-positive and false-negative error rates remain unknown in the absence of signature-level labels.

Three feature-derived scores converge on the per-CPA descriptor-position ranking with Spearman \rho \geq 0.879: the K=3 mixture posterior (a firm-compositional position score under §III-J's reading, not a mechanism cluster posterior), a reverse-anchor cosine percentile relative to a strictly-out-of-target non-Big-4 reference, and the box-rule less-replication-dominated rate. The three scores are deterministic functions of the same per-CPA descriptor pair, so the convergence is documented as internal consistency among feature-derived ranks rather than external validation. A conservative hard-positive subset for image replication is provided by 262 byte-identical signatures in the Big-4 subset (Firm A 145, Firm B 8, Firm C 107, Firm D 2), against which all three candidate checks achieve 0\% positive-anchor miss rate (Wilson 95% upper bound 1.45\%). For the box rule this result is close to tautological at byte-identity; we discuss the conservative-subset caveat in §V-G.

We apply this pipeline to 90,282 audit reports filed by publicly listed companies in Taiwan between 2013 and 2023, extracting and analyzing 182,328 individual CPA signatures from 758 unique accountants. The Big-4 sub-corpus comprises 437 CPAs and 150,442 signatures with both descriptors available.

The contributions of this paper are:

1. Problem formulation. We define non-hand-signing detection as distinct from signature forgery detection and frame it as a detection problem on intra-signer similarity distributions.

2. End-to-end pipeline. We present a pipeline that processes raw PDF audit reports through VLM-based page identification, YOLO-based signature detection, ResNet-50 feature extraction, and dual-descriptor similarity computation, with automated inference and no manual intervention after initial training.

3. Dual-descriptor similarity. We demonstrate that combining deep-feature cosine similarity with independent-minimum dHash resolves the ambiguity between style consistency and image reproduction, and we support the backbone choice through a feature-backbone ablation.

4. Composition decomposition disproves the distributional-threshold path. We show via a 2×2 factorial diagnostic (firm-mean centring × integer-tie jitter) that the apparent multimodality of the Big-4 accountant-level descriptor distribution is fully attributable to between-firm location shifts and integer mass-point artefacts. The descriptor distributions contain no within-population bimodal antimode; a distributional "natural threshold" reading of the operating points is not empirically supported.

5. Anchor-based multi-level inter-CPA coincidence-rate calibration. We characterise the deployed five-way classifier at three units of analysis: per-comparison ICCR (cos$>0.95$: 0.0006; dHash$\leq 5$: 0.0013; joint: 0.00014), pool-normalised per-signature ICCR (0.11 for the deployed any-pair high-confidence rule), and per-document ICCR (0.34 for the operational HC$+$MC alarm). We adopt "inter-CPA coincidence rate" as the metric name throughout and reserve "False Acceptance Rate" for terminology that requires ground-truth negative labels, which the corpus does not provide.

6. Firm heterogeneity quantification and within-firm cross-CPA collision concentration. Per-document D2 inter-CPA proxy ICCRs differ by an order of magnitude across firms (Firm A: 0.62 versus Firms B/C/D: $0.09$–0.16); a per-signature logistic regression of the any-pair HC hit indicator on firm dummies and centred log pool size confirms the firm gap persists after pool-size control. Cross-firm hit matrix analysis shows within-firm collision concentrations of 98.8\% at Firm A and $76.7$–83.7\% at Firms B/C/D under the deployed any-pair rule (the stricter same-pair joint event saturates at $97.0$–99.96\% within-firm across all four firms); the pattern is consistent with firm-specific template, stamp, or document-production reuse mechanisms — a descriptive finding about deployed-rule behaviour, not a claim of deliberate template sharing.

7. K=3 as descriptive firm-compositional partition; three-score convergent internal consistency. We fit a K=3 Gaussian mixture as a descriptive partition of the Big-4 accountant-level distribution (interpreted as firm-compositional structure, not as three mechanism clusters). Three feature-derived scores agree on the per-CPA descriptor-position ranking at Spearman \rho \geq 0.879; we report this as internal consistency rather than external validation, given that the scores share the underlying descriptor pair.

8. Annotation-free positive-anchor capture check and unsupervised-setting disclosure. We achieve 0\% positive-anchor miss rate (Wilson 95% upper bound 1.45\%) on 262 byte-identical Big-4 signatures, with the conservative-subset caveat that byte-identical pairs are by construction near cos$=1$ and dHash$=0$. Each supporting diagnostic in §III-M addresses one specific failure mode of an unsupervised screening classifier — composition artefacts, inter-CPA coincidence, pool-size confounding, firm heterogeneity, threshold sensitivity, or positive-anchor capture — with an explicitly disclosed untested assumption. We do not claim a validated forensic detector; we position the system as a specificity-proxy-anchored screening framework with human-in-the-loop review.

The remainder of the paper is organized as follows. Section II reviews related work on signature verification, document forensics, perceptual hashing, and the statistical methods used. Section III describes the proposed methodology. Section IV presents the experimental results — distributional characterisation, mixture fits, convergent internal-consistency checks, leave-one-firm-out reproducibility, pixel-identity positive-anchor check, and full-dataset robustness. Section V discusses the implications and limitations. Section VI concludes with directions for future work.

II. Related Work

A. Offline Signature Verification

Offline signature verification---determining whether a static signature image is genuine or forged---has been studied extensively using deep learning. Bromley et al. [3] introduced the Siamese neural network architecture for signature verification, establishing the pairwise comparison paradigm that remains dominant. Hafemann et al. [14] demonstrated that deep CNN features learned from signature images provide strong discriminative representations for writer-independent verification, establishing the foundational baseline for subsequent work. Dey et al. [4] proposed SigNet, a convolutional Siamese network for writer-independent offline verification, extending this paradigm to generalize across signers without per-writer retraining. Kao and Wen [5] addressed offline verification and forgery detection using only a single known genuine signature per writer with an explainable deep-learning approach. More recently, Li et al. [6] introduced TransOSV, the first Vision Transformer-based approach, achieving state-of-the-art results. Tehsin et al. [7] evaluated distance metrics for triplet Siamese networks, finding that Manhattan distance outperformed cosine and Euclidean alternatives. Zois et al. [15] proposed similarity distance learning on SPD manifolds for writer-independent verification, achieving robust cross-dataset transfer. Hafemann et al. [16] further addressed the practical challenge of adapting to new users through meta-learning, reducing the enrollment burden for signature verification systems.

A common thread in this literature is the assumption that the primary threat is identity fraud: a forger attempting to produce a convincing imitation of another person's signature. Our work addresses a fundamentally different problem---detecting whether the legitimate signer's stored signature image has been reproduced across many documents---which requires analyzing the upper tail of the intra-signer similarity distribution rather than modeling inter-signer discriminability.

Brimoh and Olisah [8] are closest in spirit in using reference evidence to discipline threshold choice. Their setting, however, uses standard verification benchmarks with known genuine references, whereas our archival setting lacks signature-level labels and therefore characterises a fixed deployed screening rule through inter-CPA coincidence-rate anchors.

B. Document Forensics and Copy Detection

Image forensics encompasses a broad range of techniques for detecting manipulated visual content [17], with recent surveys highlighting the growing role of deep learning in forgery detection [18]. Copy-move forgery detection (CMFD) identifies duplicated regions within or across images, typically targeting manipulated photographs [11]. Abramova and Böhme [10] adapted block-based CMFD to scanned text documents, noting that standard methods perform poorly in this domain because legitimate character repetitions produce high similarity scores that confound duplicate detection.

Woodruff et al. [9] developed the work most closely related to ours: a fully automated pipeline for extracting and analyzing signatures from corporate filings in the context of anti-money-laundering investigations. Their system uses connected component analysis for signature detection, GANs for noise removal, and Siamese networks for author clustering. While their pipeline shares our goal of large-scale automated signature analysis on real regulatory documents, their objective---grouping signatures by authorship---differs fundamentally from ours, which is detecting image-level reproduction within a single author's signatures across documents.

In the domain of image copy detection, Pizzi et al. [13] proposed SSCD, a self-supervised descriptor using ResNet-50 with contrastive learning for large-scale copy detection on natural images. Their work demonstrates that pre-trained CNN features with cosine similarity provide a strong baseline for identifying near-duplicate images, a finding that supports our feature-extraction approach.

C. Perceptual Hashing

Perceptual hashing algorithms generate compact fingerprints that are robust to minor image transformations while remaining sensitive to substantive content changes [19]. Unlike cryptographic hashes, which change entirely with any pixel modification, perceptual hashes produce similar outputs for visually similar inputs, making them suitable for near-duplicate detection in scanned documents where minor variations arise from the scanning process.

Jakhar and Borah [12] demonstrated that combining perceptual hashing with deep learning features significantly outperforms either approach alone for near-duplicate image detection, achieving AUROC of 0.99 on standard benchmarks. Their two-stage architecture---pHash for fast structural comparison followed by deep features for semantic verification---provides methodological precedent for our dual-descriptor approach, though applied to natural images rather than document signatures.

Our work differs from prior perceptual-hashing studies in its application context and in the specific challenge it addresses: distinguishing legitimate high visual consistency (a careful signer producing similar-looking signatures) from image-level reproduction in scanned financial documents.

D. Deep Feature Extraction for Signature Analysis

Several studies have explored pre-trained CNN features for signature comparison without metric learning or Siamese architectures. Engin et al. [20] used ResNet-50 features with cosine similarity for offline signature verification on real-world scanned documents, incorporating CycleGAN-based stamp removal as preprocessing---a pipeline design closely paralleling our approach. Tsourounis et al. [21] demonstrated successful transfer from handwritten text recognition to signature verification, showing that CNN features trained on related but distinct handwriting tasks generalize effectively to signature comparison. Chamakh and Bounouh [22] confirmed that a simple ResNet backbone with cosine similarity achieves competitive verification accuracy across multilingual signature datasets without fine-tuning, supporting the viability of our off-the-shelf feature-extraction approach.

Babenko et al. [23] established that CNN-extracted neural codes with cosine similarity provide an effective framework for image retrieval and matching, a finding that underpins our feature-comparison approach. These findings collectively suggest that pre-trained CNN features, when L2-normalized and compared via cosine similarity, provide a robust and computationally efficient representation for signature comparison---particularly suitable for large-scale applications where the computational overhead of Siamese training or metric learning is impractical.

E. Statistical Methods for Threshold Characterisation and Calibration

Our threshold-characterisation and calibration framework combines three families of methods developed in statistics and accounting-econometrics.

Non-parametric density estimation. Kernel density estimation [28] provides a smooth estimate of a similarity distribution without parametric assumptions. In idealized two-class mixture settings with equal priors and equal misclassification costs, the local density minimum (antimode) between the two modes coincides with the Bayes-optimal decision boundary. The statistical validity of the unimodality-vs-multimodality dichotomy can be tested via the Hartigan & Hartigan dip test [37], which tests the null of unimodality; we use rejection of this null as evidence consistent with (though not a direct test for) bimodality.

Discontinuity tests on empirical distributions. Burgstahler and Dichev [38], working in the accounting-disclosure literature, proposed a test for smoothness violations in empirical frequency distributions. Under the null that the distribution is generated by a single smooth process, the expected count in any histogram bin equals the average of its two neighbours, and the standardized deviation from this expectation is approximately N(0,1). The test was placed on rigorous asymptotic footing by McCrary [39], whose density-discontinuity test provides full asymptotic distribution theory, bandwidth-selection rules, and power analysis. The BD/McCrary pairing provides a local-density-discontinuity diagnostic that is informative about distributional smoothness under minimal assumptions; we use it in that diagnostic role (rather than as a threshold estimator) because its transitions in our corpus are bin-width-sensitive at the signature level and rarely significant at the accountant level (Appendix A).

Finite mixture models. When the empirical distribution is viewed as a weighted sum of two (or more) latent component distributions, the Expectation-Maximization algorithm [40] provides consistent maximum-likelihood estimates of the component parameters. For observations bounded on $[0,1]$---such as cosine similarity and normalized Hamming-based dHash similarity---the Beta distribution is the natural parametric choice, with applications spanning bioinformatics and Bayesian estimation. Under mild regularity conditions, White's quasi-MLE result [41] supports interpreting maximum-likelihood estimates under a mis-specified parametric family as consistent estimators of the pseudo-true parameter that minimizes the Kullback-Leibler divergence to the data-generating distribution within that family; we use this result to justify the Beta-mixture fit as a principled approximation rather than as a guarantee that the true distribution is Beta.

The present study uses these tools diagnostically: first to test whether the descriptor distribution supports a natural operating boundary, and then, when that support fails under composition decomposition, to motivate anchor-based ICCR calibration of a fixed deployed rule.

Cross-validation in a small-cluster scope. Cross-validation methodology in the leave-one-out tradition has been developed extensively in statistics since Stone [42] and Geisser [43], and modern surveys including Vehtari et al. [44] discuss its application to mixture models. In document-forensics calibration the technique has been used selectively, typically with the individual document or signature as the hold-out unit. Our application in §III-K differs in two respects from the standard usage: (i) the hold-out unit is the firm (not the individual CPA or signature), so the analysis directly probes cross-firm reproducibility of the fitted mixture rather than within-firm sampling variance; and (ii) the held-out predictions are interpreted as a composition-sensitivity band on the candidate mixture boundary, not as a sufficiency claim for the deployed five-way operational classifier (§III-H.1; calibrated separately in §III-L). We treat LOOO drift as descriptive information about how the mixture characterisation moves when training composition changes, not as a pass/fail test for the operational classifier.

III. Methodology

A. Pipeline Overview

We propose a six-stage pipeline for large-scale non-hand-signed auditor signature detection in scanned financial documents. Fig. 1 illustrates the overall architecture. The pipeline takes as input a corpus of PDF audit reports and produces five-way operational screening labels (§III-H.1) whose behaviour is characterised by pixel-identity positive-anchor capture checks and inter-CPA coincidence-rate calibration (§III-L).

Throughout this paper we use the term non-hand-signed rather than "digitally replicated" to denote any signature produced by reproducing a previously stored image of the partner's signature---whether by administrative stamping workflows (dominant in the early years of the sample) or firm-level electronic signing systems (dominant in the later years). From the perspective of the output image the two workflows are equivalent: both can reproduce one or more stored signature images, producing same-CPA signatures that are identical or near-identical up to reproduction, scanning, compression, and template-variant noise.

B. Data Collection

The dataset comprises 90,282 annual financial audit reports filed by publicly listed companies in Taiwan, covering fiscal years 2013 to 2023. The reports were collected from the Market Observation Post System (MOPS) operated by the Taiwan Stock Exchange Corporation, the official repository for mandatory corporate filings. An automated web-scraping pipeline using Selenium WebDriver was developed to systematically download all audit reports for each listed company across the study period. Each report is a multi-page PDF document containing, among other content, the auditor's report page bearing the signatures of the certifying CPAs.

CPA names, affiliated accounting firms, and audit engagement tenure were obtained from a publicly available audit-firm tenure registry encompassing 758 unique CPAs across 15 document types, with the majority (86.4%) being standard audit reports. Table I summarizes the dataset composition.

Table I. Dataset Summary.

Attribute	Value
Total PDF documents	90,282
Date range	2013–2023
Signature-page candidates (VLM-positive)	86,084 (95.3%)
Processed for signature extraction	86,071 (95.3%)
Unique CPAs identified	758
Accounting firms	>50

C. Signature Page Identification

To identify which page of each multi-page PDF contains the auditor's signatures, we employed the Qwen2.5-VL vision-language model (32B parameters) [24], one of the multimodal generative models surveyed in [35], as an automated pre-screening mechanism. Each PDF page was rendered to JPEG at 180 DPI and submitted to the VLM with a structured prompt requesting a binary determination of whether the page contains a Chinese handwritten signature. The model was configured with temperature 0 for deterministic output.

The scanning range was restricted to the first quartile of each document's page count, reflecting the regulatory structure of Taiwanese audit reports in which the auditor's report page is consistently located in the first quarter of the document. Scanning terminated upon the first positive detection. This process identified 86,084 documents with signature pages; the remaining 4,198 documents (4.6%) were classified as having no signatures and excluded. An additional 13 PDFs that could not be rendered (corruption or read errors) were excluded, yielding a final set of 86,071 documents.

Cross-validation between the VLM and subsequent YOLO detection confirmed high agreement: YOLO successfully detected signature regions in 98.8% of VLM-positive documents. The 1.2% disagreement reflects the combined rate of (i) VLM false positives (pages incorrectly flagged as containing signatures) and (ii) YOLO false negatives (signature regions missed by the detector), and we do not attempt to attribute the residual to either source without further labeling.

D. Signature Detection

We adopted YOLOv11n (nano variant) [25], a lightweight descendant of the original YOLO single-stage detector [34], for signature region localization. A training set of 500 randomly sampled signature pages was annotated using a custom web-based interface following a two-stage protocol: primary annotation followed by independent review and correction. A region was labeled as "signature" if it contained any Chinese handwritten content attributable to a personal signature, regardless of overlap with official stamps.

The model was trained for 100 epochs on a 425/75 training/validation split with COCO pre-trained initialization, achieving strong detection performance (Table II).

Table II. YOLO Detection Performance.

Metric	Value
Precision	0.97–0.98
Recall	0.95–0.98
mAP@0.50	0.98–0.99
mAP@0.50:0.95	0.85–0.90

Batch inference on all 86,071 documents extracted 182,328 signature images at a rate of 43.1 documents per second (8 workers). A red stamp removal step was applied to each cropped signature using HSV color-space filtering, replacing detected red regions with white pixels to isolate the handwritten content.

Each signature was matched to its corresponding CPA using positional order (first or second signature on the page) against the official CPA registry, achieving a 92.6% match rate (168,755 of 182,328 signatures). The matched records assume standard two-signature ordering; residual order-mismatch risk remains for nonstandard layouts. The remaining 7.4% (13,573 signatures) could not be matched to a registered CPA name---typically because the auditor's report page format deviates from the standard two-signature layout, or because OCR of the printed CPA name on the page returns a name not present in the registry---and these signatures are excluded from all subsequent same-CPA pairwise analyses (a same-CPA best-match statistic is undefined when a signature has no assigned CPA). The 92.6% matched subset forms the candidate pool for same-CPA analyses, before the Big-4 and descriptor-completeness restrictions described in §III-G.

E. Feature Extraction

Each extracted signature was encoded into a feature vector using a pre-trained ResNet-50 convolutional neural network [26] with ImageNet-1K V2 weights, used as a fixed feature extractor without fine-tuning. The final classification layer was removed, yielding the 2048-dimensional output of the global average pooling layer.

Preprocessing consisted of resizing to 224×224 pixels with aspect-ratio preservation and white padding, followed by ImageNet channel normalization. All feature vectors were L2-normalized, ensuring that cosine similarity equals the dot product.

The choice of ResNet-50 without fine-tuning was motivated by three considerations: (1) the task is similarity comparison rather than classification, making general-purpose discriminative features sufficient; (2) ImageNet features have been shown to transfer effectively to document analysis tasks [20], [21]; and (3) avoiding domain-specific fine-tuning reduces the risk of overfitting to dataset-specific artifacts, though we note that a fine-tuned model could potentially improve discriminative performance (see Section V-H, Engineering-level caveats). This design choice is supported by an ablation study (Section IV-L) comparing ResNet-50 against VGG-16 and EfficientNet-B0.

F. Dual-Method Similarity Descriptors

For each signature, we compute two complementary similarity measures against other signatures attributed to the same CPA:

Cosine similarity on deep embeddings captures high-level visual style:

\text{sim}(\mathbf{f}_A, \mathbf{f}_B) = \mathbf{f}_A \cdot \mathbf{f}_B

where \mathbf{f}_A and \mathbf{f}_B are L2-normalized 2048-dim feature vectors. Each feature dimension contributes to the angular alignment, so cosine similarity is sensitive to fine-grained execution differences---pen pressure, ink distribution, and subtle stroke-trajectory variations---that distinguish genuine within-writer variation from the reproduction of a stored image [14].

Perceptual hash distance (dHash) [27] captures structural-level similarity. Each signature image is resized to 9×8 pixels and converted to grayscale; horizontal gradient differences between adjacent columns produce a 64-bit binary fingerprint. The Hamming distance between two fingerprints quantifies perceptual dissimilarity: a distance of 0 indicates structurally identical images, while distances exceeding 15 indicate clearly different images. Unlike DCT-based perceptual hashes, dHash is computationally lightweight and particularly effective for detecting near-exact duplicates with minor scan-induced variations [19].

These descriptors provide partially independent evidence. Cosine similarity is sensitive to the full feature distribution and reflects fine-grained execution variation; dHash captures only coarse perceptual structure and is robust to scanner-induced noise. Non-hand-signing is expected to yield extreme similarity under both descriptors, since the underlying image is identical up to reproduction noise; scan-stage noise can in principle push a replicated pair off either extremum but rarely both. Hand-signing, by contrast, often yields high dHash similarity (the overall layout of a signature is typically preserved across writing occasions) but measurably lower cosine similarity (fine execution varies). Convergence of the two descriptors is therefore a natural robustness check; when they disagree, the case is flagged as borderline.

We do not use SSIM (Structural Similarity Index) [30] or pixel-level comparison as primary descriptors. SSIM was developed as a perceptual quality index for natural images and is by construction sensitive to the local-luminance and local-contrast perturbations routine in a print-scan cycle (JPEG block artefacts, scan-noise speckle, scanner-rule ghosts) — properties that penalise identically-reproduced signature crops at the very margins SSIM is designed to weight most heavily. Pixel-level distances (L_1, L_2, pixel-identity counting) are defined on geometrically aligned images at a common resolution and inflate under the sub-pixel offsets that scanner DPI, paper-handling alignment, and PDF-page rasterisation routinely introduce, so two scans of the same physical document cannot score near-identically. The supplementary materials contain the full design-level argument; pixel-identity counting is retained only as a threshold-free positive anchor (§III-K), because byte-identical pairs are necessarily produced by literal file reuse and so do not interact with the alignment-fragility argument.

Cosine similarity on L2-normalised deep embeddings and dHash both remain stable across the print-scan-rasterise cycle by design [14], [19], [21], [27]; together they constitute the dual descriptor used throughout the rest of this paper.

G. Unit of Analysis and Scope

We analyse signatures at two units of resolution. The signature — one signature image extracted from one report — is the operational unit of classification (§III-H.1) and of the signature-level analyses in §IV (notably §IV-J for the five-way per-signature category counts and the inter-CPA negative-anchor coincidence-rate analysis referenced in §IV-I). The accountant — one CPA aggregated over all of their signatures in the corpus — is the unit of mixture-model characterisation (§III-J), of per-CPA internal-consistency analysis (§III-K), and of the leave-one-firm-out reproducibility check (§III-K). At the accountant level we compute, for each CPA with n_{\text{sig}} \geq 10 signatures, the per-CPA mean of the per-signature best-match cosine (\overline{\text{cos}}_a) and the per-CPA mean of the independent-minimum dHash (\overline{\text{dHash}}_a). The minimum threshold of 10 signatures per CPA is required for the per-CPA mean to be a stable summary; CPAs below this threshold are excluded from the accountant-level analyses but remain in the per-signature analyses.

We make no within-year or across-year uniformity assumption about CPA signing mechanisms. Per-signature labels are signature-level quantities throughout this paper; we do not translate them to per-report or per-partner mechanism assignments, and we abstain from partner-level frequency inferences (such as "X% of CPAs hand-sign") that would require such a translation. A CPA's per-CPA mean is a summary statistic of their observed signatures, not a claim that all of their signatures share a single mechanism.

We adopt one stipulation about same-CPA pair detectability:

(A1) Pair-detectability. If a CPA uses image replication anywhere in the corpus, then at least one same-CPA signature pair is near-identical (after reproduction noise) within the cross-year same-CPA pool used by the max-cosine / min-dHash computation.

A1 is plausible for high-volume stamping or firm-level electronic signing workflows but is not guaranteed when (i) the corpus contains only one observed replicated report for a CPA, (ii) multiple template variants are used in parallel, or (iii) scan-stage noise pushes a replicated pair outside the detection regime. A1 is the only assumption the per-signature detector requires to be sensitive to replication.

Scope: the Big-4 sub-corpus. The primary analyses (§III-I, §III-J, §III-K, §III-L, and the corresponding §IV-D through §IV-J and §IV-M tables) are restricted to the four largest accounting firms in Taiwan, pseudonymously labelled Firm A through Firm D throughout the manuscript. §IV-A through §IV-C and §IV-L report the corpus-wide pipeline performance and feature-backbone ablation that support the descriptor choice of §III-F; §IV-K reports a deliberately narrow full-dataset cross-check at n = 686 CPAs. The Big-4 sub-corpus comprises 437 CPAs (171 / 112 / 102 / 52 across Firms A through D) with n_{\text{sig}} \geq 10 — the threshold for accountant-level analyses — totalling 150,442 Big-4 signatures with both pre-computed descriptors available. Restricting the primary analyses to Big-4 is a methodological choice driven by four considerations:

1. Restricted generalisability claim and Big-4 institutional comparability. The primary claims are scoped to the Big-4 audit-report context, where the four firms share comparable institutional scale, document-production infrastructure, and CPA-volume regime; we do not assert that the same descriptive mixture structure or operational alert behaviour extends to mid/small firms. The 249 non-Big-4 CPAs enter only (a) as an external reference population in §III-H.2's reverse-anchor internal-consistency check, (b) as a robustness comparison in §IV-K, and (c) as a corroborating-population check on the dHash discrete-mass-point artefact in §III-I.4. Generalisation beyond Big-4 is left as future work.

2. Within-firm cross-CPA collision structure analysis. §III-L.4 reports a Big-4 cross-firm hit-matrix analysis that quantifies the within-firm cross-CPA template-like collision pattern. The four-firm setting affords the cleanest signal for this analysis; replicating the same matrix structure on the heterogeneous mid/small-firm tail is left as future work.

3. Firm A as templated-end case study. Firm A is empirically the firm whose CPAs are most concentrated in the high-cosine, low-dHash corner of the descriptor plane (§III-J K=3 component cross-tab; byte-level pair analysis referenced in §III-H.2). We retain Firm A within the Big-4 scope as a descriptive case study of the templated end rather than as the calibration anchor for thresholds.

4. Leave-one-firm-out fold feasibility. §III-K reports leave-one-firm-out (LOOO) cross-validation of the Big-4 K=3 fit. The Big-4 sub-corpus permits a four-fold LOOO at the firm level (one fold per Big-4 firm). No analogous firm-level fold is available outside Big-4 because mid/small firms have CPA counts of $O(1)$–O(30) per firm.

Sample-size reconciliation. Two Big-4 signature counts appear in this section and §IV: n = 150{,}442 for analyses using the pre-computed per-signature descriptors \text{cos}_s (max_similarity_to_same_accountant) and \text{dHash}_s (min_dhash_independent), and n = 150{,}453 for analyses recomputing pair-level metrics directly from the stored feature and dHash byte vectors (Scripts 40b, 43, 44). The $11$-signature difference reflects descriptor-completion status: 11 signatures have feature vectors and dHash byte vectors stored but lack the pre-computed extrema. The 11 signatures are negligible at population scale and do not affect any reported coincidence rate within 0.01 percentage point. The CPA counts 468 (all Big-4 CPAs with both vectors stored) and 437 (Big-4 CPAs with n_{\text{sig}} \geq 10 for accountant-level stability) likewise reflect a single uniform exclusion rule rather than analysis-specific subsetting.

H. Operational Classifier and Reference Populations

H.1. Deployed Operational Rule

Each Big-4 signature is assigned to one of five categories using the per-signature descriptor pair (\text{cos}_s, \text{dHash}_s) where \text{cos}_s is the maximum cosine similarity to another signature by the same CPA and \text{dHash}_s is the minimum independent dHash to another signature by the same CPA:

1. High-confidence non-hand-signed (HC): Cosine > 0.95 AND \text{dHash}_{\text{indep}} \leq 5. Both descriptors converge on strong replication evidence.
2. Moderate-confidence non-hand-signed (MC): Cosine > 0.95 AND 5 < \text{dHash}_{\text{indep}} \leq 15. Feature-level evidence is strong; structural similarity is present but below the high-confidence cutoff.
3. High style consistency (HSC): Cosine > 0.95 AND \text{dHash}_{\text{indep}} > 15. High feature-level similarity without structural corroboration — consistent with a CPA who signs very consistently but not via image reproduction.
4. Uncertain (UN): Cosine between the all-pairs intra/inter KDE crossover (0.837) and 0.95.
5. Likely hand-signed (LH): Cosine \leq 0.837.

Document-level labels are aggregated via the worst-case rule: each audit report inherits the most-replication-consistent category among its certifying-CPA signatures (rank order HC > MC > HSC > UN > LH). The thresholds (\text{cos} = 0.95 as the cosine operating point, \text{cos} = 0.837 as the all-pairs KDE crossover, \text{dHash} = 5 and 15 as structural-similarity sub-band cutoffs) retain their prior calibration provenance (see supplementary materials). These thresholds define the deployed screening rule; the present analysis does not re-derive them as optimal cutoffs but characterises their behaviour under inter-CPA coincidence anchors (developed in §III-L).

The remainder of this section (§III-H.2) describes the reference populations used to calibrate and cross-check this rule. §III-I demonstrates that the descriptor distributions do not provide a within-population natural threshold; §III-J–§III-K develop the descriptive partition and internal-consistency cross-checks; §III-L develops the anchor-based threshold calibration; §III-M discloses the unsupervised-setting limits.

H.2. Reference Populations

The calibration distinguishes two reference populations: Firm A as a within-Big-4 templated-end case study, and the 249 non-Big-4 CPAs as an out-of-target reference for internal-consistency checking.

Internal reference: Firm A as the templated-end case study. Firm A is empirically the firm whose CPAs are most concentrated in the high-cosine, low-dHash corner of the Big-4 descriptor plane. In the Big-4 K=3 descriptive partition (§III-J; Scripts 35, 38), Firm A accounts for 0% of the C1 component (low-cos / high-dHash corner; cos \approx 0.946, dHash \approx 9.17, weight \approx 0.143), 17.5% of the C2 component (central region), and 82.5% of the C3 component (high-cos / low-dHash corner); the opposite pattern holds at Firm C (Script 35: 23.5% C1, 75.5% C2, 1.0% C3, hereafter referred to as "the Firm whose CPAs are most concentrated in C1"). Byte-level decomposition of these signatures (see supplementary materials) identifies 145 Firm A pixel-identical signatures, spanning 50 distinct Firm A partners of the 180 registered, with 35 byte-identical matches occurring across different fiscal years; the 145 are the Firm A portion of the 262 byte-identical Big-4 signatures.

Firm A is not the calibration anchor for the operational threshold. Firm A enters the Big-4 mixture on equal footing with Firms B through D; the K=3 components are derived from the joint Big-4 distribution (§III-J), not from Firm A alone. Firm A's role in the methodology is descriptive: it is the Big-4 firm whose CPAs are most concentrated in the high-cosine, low-dHash corner of the descriptor plane, and the byte-level pair evidence above provides the firm-level signature-reuse evidence that anchors §III-K's pixel-identity positive-anchor miss rate.

External reference: non-Big-4 as the reverse-anchor reference for internal-consistency checking. The 249 non-Big-4 CPAs (n_{\text{sig}} \geq 10, drawn from $\sim$30 mid- and small-firms) constitute a population strictly outside the Big-4 target. Their per-CPA (\overline{\text{cos}}_a, \overline{\text{dHash}}_a) distribution defines a 2D Gaussian reference (fit by Minimum Covariance Determinant with support fraction 0.85 for robustness; Script 38). This reference is used in §III-K's reverse-anchor internal-consistency check: each Big-4 CPA's location relative to the reference centre, measured as the marginal cosine cumulative-distribution-function value under the reference, is one of three feature-derived scores used as a cross-check on the per-signature classifier. The reverse-anchor reference is not a positive or negative anchor for threshold derivation — its role is to provide a strictly out-of-target benchmark against which the within-Big-4 mixture-derived ranking can be internally cross-checked.

The reverse-anchor reference centre is at \overline{\text{cos}} = 0.935, \overline{\text{dHash}} = 9.77 (Script 38). The reference sits at a lower cosine and higher dHash than the Big-4 K=3 low-cos / high-dHash component (cos = 0.946, dHash = 9.17; §III-J); compared to the Big-4 high-cos / low-dHash component (cos = 0.983, dHash = 2.41; §III-J) the reference is markedly less replication-dominated. The reverse-anchor metric for a given Big-4 CPA is the percentile of \overline{\text{cos}}_a within the reference marginal cosine distribution, sign-flipped so that lower percentile (further into the left tail of the reference) corresponds to a Big-4 CPA whose mean cosine sits further from the templated end of the descriptor plane. This is a "deviation in the less-replication-dominated descriptor-position direction" measure, not a "deviation toward the templated descriptor-position" measure; the reference is the less-replication-dominated population.

I. Distributional Diagnostics: Why the Composition Path Does Not Yield a Natural Threshold

This section characterises the joint distribution of accountant-level descriptor means (\overline{\text{cos}}_a, \overline{\text{dHash}}_a) across the 437 Big-4 CPAs of §III-G and tests whether the distribution provides distributional support — in the form of within-population bimodality — for the deployed operational thresholds. We apply four diagnostic procedures in turn: a univariate unimodality test on each accountant-level marginal; a 2D Gaussian mixture fit (developed in §III-J); a density-smoothness diagnostic; and a composition decomposition that distinguishes within-population multimodality from between-firm location-shift artefacts. The four diagnostics jointly imply that the operational thresholds are not anchored by distributional bimodality: §III-L develops an anchor-based calibration framework that does not require this assumption.

1. Hartigan dip test on each accountant-level marginal. We apply the Hartigan & Hartigan dip test [37] to each of the two marginal distributions \{\overline{\text{cos}}_a\}_{a=1}^{437} and \{\overline{\text{dHash}}_a\}_{a=1}^{437}, with bootstrap-based $p$-value estimation (n_{\text{boot}} = 2000). In both cases no bootstrap replicate exceeded the observed dip statistic, so the empirical $p$-value is bounded above by 5 \times 10^{-4}; we report this in tables as p < 5 \times 10^{-4} rather than p = 0 to reflect the bootstrap resolution (Script 34). For comparison, no rejection of unimodality holds in the comparison scopes tested in Script 32: Firm A pooled alone (p_{\text{cos}} = 0.992, p_{\text{dHash}} = 0.924, n = 171); Firms B + C + D pooled (p_{\text{cos}} = 0.998, p_{\text{dHash}} = 0.906, n = 266); all non-Firm-A CPAs pooled (p_{\text{cos}} = 0.998, p_{\text{dHash}} = 0.907, n = 515). Single-firm dip tests for Firms B, C, and D were not separately computed; the comparison scopes above sufficed to establish that no narrower-than-Big-4 tested scope at the accountant level rejected unimodality. The accountant-level Big-4 rejection is a descriptive observation; §III-I.4 below shows that the rejection is fully explained by between-firm location-shift effects rather than within-population bimodality.

2. K=2 / K=3 Gaussian mixture fits (descriptive partition). A 2-component 2D Gaussian Mixture Model (full covariance, n_{\text{init}} = 15, fixed seed 42; Script 34) recovers components at (\overline{\text{cos}}, \overline{\text{dHash}}) = (0.954, 7.14), weight 0.689, and (0.983, 2.41), weight 0.311. The marginal crossings of the K=2 fit are \overline{\text{cos}}^* = 0.9755 and \overline{\text{dHash}}^* = 3.755, with bootstrap 95% confidence intervals [0.9742, 0.9772] and [3.48, 3.97] over n_{\text{boot}} = 500 resamples. The 3-component fit (§III-J) is BIC-preferred — using the convention that lower BIC is preferred, \text{BIC}(K{=}3) - \text{BIC}(K{=}2) = -3.48 (Script 36). The $\Delta$BIC magnitude is small in absolute terms; we do not treat \Delta\text{BIC} = 3.5 alone as decisive evidence for K=3 as a population mixture. Following §III-I.4 we treat both K=2 and K=3 fits as descriptive partitions of the joint Big-4 distribution that reflect firm-composition structure (Firm A vs others; §III-J) rather than as inferential evidence for two or three latent population modes.

3. Burgstahler-Dichev / McCrary density-smoothness diagnostic. We apply the discontinuity test of [38, 39] as a density-smoothness diagnostic (rather than as a threshold estimator) on each accountant-level marginal axis (cosine in bins of 0.002, dHash in integer bins). At the Big-4 scope, the diagnostic identifies no significant transition on either marginal at \alpha = 0.05 (Script 34). Outside Big-4, the diagnostic does flag dHash transitions in some subsets (Script 32: big4_non_A dHash transition at 10.8; all_non_A dHash transition at 6.6; pre-2018 and post-2020 time-stratified variants also exhibit one or more dHash transitions), but no cosine transition is identified in any subset. The Big-4-scope null on both axes is consistent with §III-I.4 below: under the composition decomposition the Big-4 marginals are unimodal once between-firm and integer-tie confounds are removed, so a local-discontinuity test correctly fails to flag a within-population transition.

4. Composition decomposition (Scripts 39b–39e). §III-I.1 establishes that the accountant-level marginals reject unimodality at the Big-4 sub-corpus. The remaining question is whether the rejection reflects (a) genuine within-population bimodality at the signature or accountant level, (b) between-firm location-shift artefacts (firms with different mean descriptor positions pool to a multi-peaked distribution), or (c) integer mass-point artefacts on the integer-valued dHash axis (the dHash dip statistic is sensitive to spikes at integer values). We apply four diagnostics that decompose the rejection into these candidate sources:

Within-firm signature-level dip (Scripts 39b, 39c). Repeating the dip test at the signature level inside each individual Big-4 firm (Script 39b) and inside each individual non-Big-4 firm with \geq 500 signatures (Script 39c) yields a consistent picture. The cosine marginal fails to reject unimodality in every single firm tested — all four Big-4 firms (p_{\text{cos}} \in \{0.176, 0.991, 0.551, 0.976\} for Firms A through D; Script 39b) and ten non-Big-4 firms with \geq 500 signatures (p_{\text{cos}} \in [0.59, 0.99]; Script 39c). The raw dHash marginal does reject unimodality in every firm tested (p < 5 \times 10^{-4} in all 14 firms), but the raw dHash values are integer-valued in \{0, 1, \ldots, 64\}, leaving open the possibility of an integer-tie artefact.

Integer-jitter robustness (Scripts 39d, 39e). Adding independent uniform jitter \sim \mathrm{U}[-0.5, +0.5] to break exact dHash ties and re-running the dip test on the perturbed signature cloud (5 seeds, n_{\text{boot}} = 2000; Script 39d) eliminates the dHash within-firm rejection in every Big-4 firm tested (Firm A jittered p_{\text{median}} = 0.999; B 0.996; C 0.999; D 0.9995; $0$/5 seeds reject at \alpha = 0.05 in any firm). The pooled-Big-4 dHash dip does survive jitter alone (p_{\text{median}} = 0, $5$/5 seeds reject), but Firm A's mean dHash (2.73) is substantially below Firms B/C/D's (6.46, 7.39, 7.21) — a between-firm location shift. Script 39e applies a 2 \times 2 factorial correction (firm-mean centring \times integer jitter) on the Big-4 pooled dHash:

Condition	Firm-mean centred	Integer jitter	Median dip p	Reject at \alpha = 0.05
1 raw	—	—	< 5 \times 10^{-4}	5/5
2 centred only	\checkmark	—	< 5 \times 10^{-4}	5/5
3 jittered only	—	\checkmark	< 5 \times 10^{-4}	5/5
4 centred and jittered	\checkmark	\checkmark	\mathbf{0.35}	\mathbf{0/5}

Removing both the between-firm location shift and the integer mass points eliminates the Big-4 dHash rejection. The Big-4 pooled dHash multimodality is therefore fully attributable to firm-composition contrast (primarily Firm A's mean \text{dHash} = 2.73 versus Firms B/C/D $\approx 6.5$–7.4) and integer-density artefacts, with no residual continuous within-firm bimodality.

Cosine analogue. The cosine axis follows the same pattern by construction: the within-firm signature-level cosine dip tests above (Scripts 39b, 39c) fail to reject in every Big-4 firm and in every eligible non-Big-4 firm, so any pooled cosine multimodality must arise from between-firm composition rather than from within-population bimodality.

Integer-histogram valleys (Script 39d). A genuine within-firm dHash antimode would appear as a strict local minimum in the count histogram with deep relative depth. Within each of the four Big-4 firms, the dHash histogram on bins $0$–20 exhibits no strict local minimum; the Big-4 pooled histogram exhibits one shallow valley at \text{dHash} = 4 with relative depth 0.021 (a 2.1\% count drop). No valley near the deployed \text{dHash} = 5 operational boundary appears within any individual firm. The hypothesised dHash antimode near \text{dHash} \approx 5 is not empirically supported by the histogram analysis.

5. Conclusion: no natural threshold from the descriptor distribution. §III-I.4 jointly establishes that (a) the Big-4 accountant-level dip rejection is fully attributable to between-firm composition and integer mass-point artefacts; (b) within the Big-4 firms, the descriptor marginals at the signature level are unimodal once integer ties are broken (Scripts 39b, 39d); (c) eligible non-Big-4 checks provide corroborating raw-axis evidence on the cosine dimension (Script 39c) and corroborate the integer-mass-point reading of raw dHash, but are not used as calibration evidence for the deployed thresholds; and (d) no integer-histogram valley near the deployed \text{dHash} = 5 operational boundary exists within any Big-4 firm. The descriptor distributions therefore do not contain a within-population bimodal antimode that could anchor an operational threshold. The K=2 / K=3 mixture fits of §III-I.2 and §III-J are retained as descriptive partitions that reflect firm-composition contrast, not as inferential evidence for two or three population modes. §III-L develops the anchor-based threshold calibration framework, which derives operational rates from inter-CPA pair-level negative-anchor coincidences rather than from a distributional antimode.

J. K=3 as a Descriptive Partition of Firm-Composition Contrast

This section develops the K=2 and K=3 Gaussian mixture fits to the Big-4 accountant-level distribution and clarifies their role. Both fits are descriptive partitions of the joint Big-4 distribution; they reflect firm-composition contrast — primarily Firm A versus Firms B, C, D — rather than within-population mechanism modes. §III-I.4 demonstrates that the apparent multimodality of the accountant-level marginals is fully explained by between-firm location shifts and integer mass-point artefacts, leaving no residual evidence for two or three latent within-population mechanism classes. Neither mixture is used to assign signature-level or document-level labels in the primary analysis. The operational classifier of §III-H.1 is calibrated in §III-L via inter-CPA negative-anchor coincidence rates, not via mixture-derived antimodes.

K=2 fit. Two components at (\overline{\text{cos}}, \overline{\text{dHash}}) = (0.954, 7.14) (weight 0.689) and (0.983, 2.41) (weight 0.311) (Script 34). \text{BIC}(K{=}2) = -1108.45. Marginal crossings: \overline{\text{cos}}^* = 0.9755, \overline{\text{dHash}}^* = 3.755. We refer to the components by index rather than by mechanism labels, since §III-I.4 establishes that the K=2 separation is firm-compositional rather than mechanistic.

K=3 fit. Three components, sorted by ascending cosine mean (Script 35; Script 38 reproduces):

Component	\overline{\text{cos}}	\overline{\text{dHash}}	weight	descriptive position
C1	0.9457	9.17	0.143	low-cos / high-dHash corner
C2	0.9558	6.66	0.536	central region
C3	0.9826	2.41	0.321	high-cos / low-dHash corner

\text{BIC}(K{=}3) = -1111.93, lower than K{=}2 by 3.48 (mild numerical preference for K=3 under standard BIC interpretation, but not by itself decisive). The "descriptive position" column refrains from any mechanism interpretation: §III-I.4 establishes that the cosine and dHash axes both lack within-population bimodality, so component centres are best interpreted as locations in a continuous descriptor space rather than as latent mechanism modes.

Per-firm component composition (Script 35 firm × cluster cross-tab). The K=3 partition is dominated by firm membership:

• Firm A: 0\% C1, 17.5\% C2, 82.5\% C3
• Firm B: 8.9\% C1, \sim 78\% C2, \sim 13\% C3
• Firm C: 23.5\% C1, 75.5\% C2, 1.0\% C3
• Firm D: 11.5\% C1, \sim 84\% C2, \sim 4.5\% C3

Firm A accounts for 141 of the 143 C3-assigned CPAs; Firm C accounts for 24 of the 40 C1-assigned CPAs. The K=3 partition is therefore well-described as a firm-compositional decomposition: C3 is essentially "Firm A and any non-Firm-A CPA whose mean descriptors happen to land in the high-cos / low-dHash corner"; C1 is essentially "non-Firm-A CPAs whose mean descriptors land in the low-cos / high-dHash corner." The composition contrast that K=3 captures at the accountant level reappears at the deployment level in the cross-firm hit matrix of §III-L.4 (Script 44): under the deployed any-pair rule, within-firm collision concentration is 98.8\% at Firm A and $76.7$–83.7\% at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–99.96\% within-firm across all four firms). The K=3 partition and the cross-firm hit matrix therefore describe the same underlying firm-compositional structure at two different units of analysis.

Leave-one-firm-out stability (Scripts 36, 37). Leave-one-firm-out cross-validation shows that K=2 is unstable across folds: holding Firm A out gives a fold rule cos > 0.938 AND dHash \leq 8.79, while holding any single non-Firm-A Big-4 firm out gives a fold rule near cos > 0.975 AND dHash \leq 3.76 (Script 36). The maximum absolute deviation of the four fold cosine crossings from their across-fold mean is 0.028 (the corresponding pairwise across-fold range is 0.0376, from 0.9380 for the held-out-Firm-A fold to 0.9756 for the held-out-Firm-D fold; Script 36 stability summary). The 0.028 value is 5.6\times the report's 0.005 across-fold stability tolerance. K=3 in contrast has a reproducible component shape: across the four folds the C1 cosine mean varies by at most 0.005, the C1 dHash mean by at most 0.96, and the C1 weight by at most 0.023 (Script 37). K=3 hard-posterior membership for the held-out firm is composition-sensitive — for Firm C the held-out C1 rate is 36.3\% vs the full-Big-4 baseline of 23.5\%, an absolute difference of 12.8 pp; for Firm A the held-out C1 rate is 4.7\% vs baseline 0.0\%; the report's own legend classifies this pattern as P2_PARTIAL ("the C1 cluster exists but membership is not well-predicted by the held-out fit"). We accordingly do not use K=3 hard-posterior membership as an operational label.

We take the joint K=2 / K=3 LOOO evidence as supporting the following descriptive claims, all of which are used in §III-K and §V but none of which underwrites the operational classifier:

• The Big-4 K=2 marginal crossing (0.975, 3.76) is essentially a firm-mass separator between Firm A and Firms B + C + D, not a within-Big-4 mechanism boundary.
• The Big-4 K=3 mixture exhibits a reproducible three-component component shape across LOOO folds at the descriptor-position level, with C1 reproducibly located at \overline{\text{cos}} \approx 0.946, \overline{\text{dHash}} \approx 9.17.
• Hard-posterior K=3 membership is composition-sensitive across folds (max absolute deviation 12.8 pp); K=3 is therefore not used to assign operational labels to CPAs.

The operational signature-level classifier of §III-H.1 is calibrated in §III-L against inter-CPA pair-level negative-anchor coincidence rates, not against mixture-derived antimodes. Cross-checks between the deployed five-way box rule and the K=3 partition appear in §III-K.

K. Convergent Internal-Consistency Checks

The descriptive partition of §III-J is supported by three feature-derived per-CPA scores and a conservative hard-positive subset analysis. We caution at the outset that the three scores are not statistically independent measurements — all three are deterministic functions of the same per-CPA descriptor means (\overline{\text{cos}}_a, \overline{\text{dHash}}_a) — so their high pairwise rank correlations are partly a mechanical consequence of shared inputs. Per §III-I.4, none of the three scores has a within-population bimodality interpretation; they are firm-compositional position scores at the accountant level. The checks below therefore document internal consistency among feature-derived ranks, not external validation against an independent hand-signed ground truth (which the corpus does not provide).

1. Three feature-derived per-CPA scores (Script 38). For each Big-4 CPA we compute:

• Score 1 (K=3 posterior on the low-cos / high-dHash component): P(\text{C1}) from the K=3 fit of §III-J. Per §III-J this is a firm-compositional position score on the (cos, dHash) plane (not a probability of any latent "hand-signing mechanism") — a function of both descriptor means.
• Score 2 (reverse-anchor cosine percentile): the marginal cosine CDF value of \overline{\text{cos}}_a under the non-Big-4 reference Gaussian of §III-H.2, sign-flipped so that lower percentile (further into the reference's left tail) corresponds to a Big-4 CPA whose mean cosine sits further from the templated end. This is a function of \overline{\text{cos}}_a alone.
• Score 3 (deployed binary high-confidence box rule rate): the per-CPA fraction of signatures that do not satisfy the deployed binary high-confidence box rule (cos > 0.95 AND dHash \leq 5). This is a per-signature-aggregated function of the same descriptors.

Pairwise Spearman rank correlations among the three scores, n = 437 Big-4 CPAs (Script 38):

Pair	Spearman \rho	$p$-value
Score 1 vs Score 3	+0.9627	< 10^{-248}
Score 2 vs Score 3	+0.8890	< 10^{-149}
Score 1 vs Score 2	+0.8794	< 10^{-142}

We read this as the strongest internal-consistency signal in the analysis: three different summarisations of the same descriptor pair agree on the per-CPA descriptor-position ranking with \rho > 0.87. The three scores agree on placing Firm A as the most replication-dominated descriptor position and the three non-Firm-A Big-4 firms further from the templated end, but they do not all rank the non-Firm-A firms identically: the K=3 posterior P(C1) and the box-rule less-replication-dominated rate (Scores 1 and 3) place Firm C at the less-replication-dominated end of Big-4 (mean P(C1) = 0.311; mean box-rule less-replication-dominated rate = 0.790), while the reverse-anchor cosine percentile (Score 2) places Firm D fractionally higher than Firm C (mean reverse-anchor score -0.7125 vs Firm C -0.7672, with higher value indicating deeper into the reference left tail). The mean values for Firms B and D sit between Firms A and C on Scores 1 and 3 (Script 38 per-firm summary). We do not claim this constitutes external validation of any operational classifier; the deployed box rule is calibrated separately (§III-L), and the convergence above shows that a mixture-derived score and a reverse-anchor score concur with the box rule's per-CPA-aggregated outputs on the directional ordering, with a modest disagreement at the less-replication-dominated end between the three non-A Big-4 firms.

2. Per-signature consistency (Script 39). Per-CPA aggregation could in principle reflect averaging across within-CPA heterogeneity rather than coherent within-CPA behaviour. We test this by repeating the K=3 fit at the signature level — fitting a fresh K=3 GMM to the 150,442 Big-4 signature-level (\text{cos}, \text{dHash}_{\text{indep}}) points (Script 39) — and comparing labels. The per-CPA and per-signature K=3 fits recover a broadly similar three-component ordering; per-CPA C1 is at \overline{\text{cos}} = 0.946, \overline{\text{dHash}} = 9.17 vs per-signature C1 at \overline{\text{cos}} = 0.928, \overline{\text{dHash}} = 9.75 (an absolute cosine drift of 0.018). Cohen \kappa on the binary collapse (replication-dominated vs less-replication-dominated):

Pair	Cohen \kappa
Deployed binary high-confidence box rule vs per-CPA K=3 hard label	0.662
Deployed binary high-confidence box rule vs per-signature K=3 hard label	0.559
Per-CPA K=3 vs per-signature K=3	0.870

The \kappa = 0.870 between per-CPA-fit and per-signature-fit K=3 binary labels indicates that per-CPA aggregation does not collapse the broad three-component ordering. The lower \kappa = 0.56\text{–}0.66 between the binary box rule and either K=3 fit is consistent with two factors: different decision geometries (rectangular box vs Gaussian-mixture posterior boundary), and the fact that the binary box rule is a strict subset of the five-way rule. This comparison checks only the binary high-confidence rule (cos > 0.95 AND dHash \leq 5); §III-K does not directly check the five-way rule's 5 < \text{dHash} \leq 15 moderate-confidence band, whose calibration and capture-rate evidence is reported in the supplementary materials and not regenerated on the Big-4 subset.

3. Leave-one-firm-out reproducibility (Scripts 36, 37). Discussed in §III-J above. We summarise the joint result for cross-reference:

• K=2 LOOO is unstable. The maximum absolute deviation of the four fold cosine crossings from their across-fold mean is 0.028, against the report's 0.005 across-fold stability tolerance (Script 36; pairwise fold range 0.0376, from 0.9380 to 0.9756). When Firm A is held out, the fold rule classifies 171/171 of held-out Firm A CPAs as templated; when any non-Firm-A Big-4 firm is held out, the fold rule classifies 0 of the held-out firm's CPAs as templated. This pattern indicates the K=2 boundary is essentially a Firm-A-vs-others separator rather than a within-Big-4 mechanism boundary.

• K=3 LOOO is partially stable. The C1 (low-cos / high-dHash) component shape is reproducible across folds: max deviation from the full-Big-4 baseline is 0.005 in cosine, 0.96 in dHash, and 0.023 in mixture weight (Script 37). Hard-posterior membership remains composition-sensitive — observed absolute differences are $1.8$–12.8 pp across the four folds, with the Firm C fold exceeding the report's 5 pp viability bar; the report's own screening label is P2_PARTIAL ("K=3 is not predictively useful as an operational classifier"). We accordingly do not use K=3 hard-posterior membership as an operational label.

4. Positive-anchor miss rate on byte-identical signatures (Script 40). The corpus provides one conservative hard-positive subset: signatures whose nearest same-CPA match is byte-identical after crop and normalisation. Independent hand-signing cannot produce pixel-identical images, so byte-identical signatures are a conservative hard-positive subset for image replication. The Big-4 byte-identical subset comprises n = 262 signatures (145 / 8 / 107 / 2 across Firms A through D; Script 40).

We report each candidate check's positive-anchor miss rate — the fraction of byte-identical signatures classified as belonging to the less-replication-dominated descriptor positions. This is a one-sided check against a conservative positive subset, not a paired specificity metric in the usual two-class sense; we do not report a paired negative-anchor metric here because no signature-level hand-signed ground truth exists. The corresponding signature-level inter-CPA negative-anchor ICCR evidence is developed in §III-L.1 (Big-4 sample) and the corpus-wide version cited at §IV-I:

Candidate check	Pixel-identity miss rate (Wilson 95% CI)
Deployed binary high-confidence box rule (cos > 0.95 AND dHash \leq 5)	0\% [0\%, 1.45\%]
K=3 per-CPA hard label (C3 high-cos / low-dHash corner; descriptive only)	0\% [0\%, 1.45\%]
Reverse-anchor with prevalence-calibrated cut	0\% [0\%, 1.45\%]

All three candidate scores correctly assign every byte-identical signature to the replicated class. We caution that for the box rule this result is close to tautological: byte-identical nearest-neighbour signatures have cosine \approx 1 and dHash \approx 0 by construction, so any threshold strictly below cos = 1 and strictly above dHash = 0 will capture them. The positive-anchor miss rate is therefore a necessary check (a classifier that failed this check would be disqualified), not a sufficient validation of the classifier's behaviour on the non-byte-identical replicated population. The reverse-anchor cut here is chosen by prevalence calibration against the box rule's overall replicated rate (49.58\% of Big-4 signatures); this is a documented limitation since no signature-level hand-signed ground truth exists to permit direct ROC optimisation.

L. Anchor-Based Threshold Calibration

The operational classifier defined in §III-H.1 is calibrated by characterising the deployed thresholds' inter-CPA pair-level negative-anchor coincidence behaviour and their pool-normalised per-signature and per-document alert behaviour, at multiple units of analysis. §III-I.4 establishes that the descriptor distributions do not contain a within-population bimodal antimode that could anchor an operational threshold; the K=3 mixture of §III-J is a descriptive firm-compositional partition, not a mechanism-cluster model. Throughout this section we report inter-CPA coincidence rates rather than "False Acceptance Rates"; we explain the terminological choice in §III-L.0.

L.0. Calibration methodology

Calibration role of the present analysis. The deployed thresholds of §III-H.1 preserve continuity with the existing literature and the supplementary calibration evidence. §III-I.4 establishes that a recalibration cannot be anchored on distributional antimodes (no within-population bimodality exists); §III-L.1 below characterises the cosine threshold's specificity-proxy behaviour at the inter-CPA pair level and the structural-dimension threshold $\text{dHash} \leq 5$'s pair-level coincidence behaviour. The sub-band thresholds (\text{dHash} = 15, \text{cos} = 0.837) retain their supplementary calibration evidence; the present calibration does not provide independent rates for those sub-bands.

Three units of analysis. We report inter-CPA negative-anchor coincidence behaviour at three units, each addressing a different operational question:

• Per comparison. For a randomly drawn pair of signatures from different CPAs, what fraction satisfies the rule (cos > cos_threshold and / or dHash \leq dHash_threshold)? This is the conventional pairwise calibration unit in biometric verification. We report it for both the cosine and dHash dimensions, marginally and jointly (§III-L.1).
• Per signature pool. For a Big-4 source signature s with same-CPA pool of size n_{\text{pool}}(s), what is the probability that the deployed rule fires under the counterfactual of replacing the source's same-CPA pool with n_{\text{pool}}(s) random non-same-CPA candidates? This addresses the standard concern that a per-pair rate computed on independent pairs is not the deployed-rule rate at the per-signature classifier level: the deployed rule takes max-cosine and min-dHash over a pool of size n_{\text{pool}}(s), so its effective coincidence rate is approximately 1 - (1 - p_{\text{pair}})^{n_{\text{pool}}} in the independence limit (§III-L.2).
• Per document. For an audit report aggregated via the worst-case rule, what fraction of documents have at least one signature whose deployed pool-normalised rule fires under the same inter-CPA candidate-replacement counterfactual? This is the operational alarm-rate unit (§III-L.3).

Any-pair vs same-pair semantics. The deployed rule uses independent extrema: a signature satisfies the HC rule if \max_{\text{pool}} \text{cos} > 0.95 AND \min_{\text{pool}} \text{dHash} \leq 5, not if a single candidate in the pool satisfies both. We refer to this as the any-pair rule. A stricter alternative — the same-pair rule — requires a single candidate to satisfy both inequalities; the deployed rule is any-pair, but we report same-pair as a stricter alternative classifier where useful (§III-L.2, §III-L.4).

Terminological note on "FAR". The biometric-verification literature speaks of "False Acceptance Rate" (FAR) for a per-pair rate computed on independent inter-CPA pairs. We adopt inter-CPA coincidence rate (ICCR) as the metric name and do not use "FAR" in the manuscript prose, for two reasons: (a) FAR has a specific biometric-verification meaning that requires ground-truth negative labels (which the corpus does not provide at the signature level); (b) §III-L.4 shows that the inter-CPA negative-anchor assumption — that inter-CPA pairs are negative — is partially violated by within-firm cross-CPA template-like collision structures. Reading "inter-CPA coincidence rate" as a specificity proxy under an explicitly disclosed assumption is faithful to the evidence; reading it as a true biometric FAR would overstate the evidence.

L.1. Per-comparison inter-CPA coincidence rate (Script 40b)

We sample 5 \times 10^5 inter-CPA pairs uniformly at random from Big-4 signatures, computing for each pair the cosine similarity (feature dot product) and Hamming distance between the dHash byte vectors. Marginal and joint rates at threshold k are reported with Wilson 95% confidence intervals (Script 40b).

Threshold	Per-comparison inter-CPA coincidence rate	95% Wilson CI
Cosine > 0.95	0.00060	[0.00053, 0.00067]
Cosine > 0.945 (alternative operating point from supplementary calibration evidence)	0.00081	[0.00073, 0.00089]
Cosine > 0.97	0.00024	[0.00020, 0.00029]
Cosine > 0.98	0.00009	[0.00007, 0.00012]
dHash \leq 5	0.00129	[0.00120, 0.00140]
dHash \leq 4	0.00050	[0.00044, 0.00057]
dHash \leq 3	0.00019	[0.00015, 0.00023]
dHash \leq 2	0.00006	[0.00004, 0.00008]
Joint: cos > 0.95 AND dHash \leq 5 (any-pair semantics)	0.00014	[0.00011, 0.00018]
Joint: cos > 0.95 AND dHash \leq 4 (any-pair)	0.00011	[0.00008, 0.00014]

The cosine row at \text{cos} > 0.95 is consistent with the corpus-wide per-comparison rate of 0.0005 reported in §IV-I on a similarly-sized inter-CPA sample; the present $5 \times 10^5$-pair sample yields 0.00060, within that precision. The dHash row and joint row are reported here for the first time; the corpus-wide spike did not provide an inter-CPA pair-level coincidence rate for the structural dimension or the joint rule.

The all-firms-scope sample yields slightly lower per-comparison coincidence rates (cos > 0.95: 0.00031; dHash \leq 5: 0.00073; joint: 0.00007); the all-firms sample weights small CPAs more heavily under CPA-uniform pair sampling, so we treat the Big-4 sample as the primary calibration scope and report all-firms as a corroborating-scope robustness check.

Conditional inter-CPA coincidence rate. A natural follow-up question is whether the dHash dimension provides marginal specificity beyond the cosine gate. For pairs with cos > 0.95, the conditional rate of dHash \leq 5 is 0.234 (Wilson 95% CI [0.190, 0.285]; 70 of 299 pairs in the Big-4 sample). At cos > 0.95, dHash provides \sim 4.3\times further per-comparison specificity (joint 0.00014 vs cos-only 0.00060).

The per-comparison rate is a useful specificity-proxy calibration for the deployed rule's pair-level behaviour. It does not directly translate to the deployed-rule specificity at the per-signature classifier level, because the deployed classifier takes extrema over a same-CPA pool of size n_{\text{pool}}. The pool-normalised inter-CPA alert rate is reported in §III-L.2.

L.2. Pool-normalised inter-CPA alert rate (Script 43)

The deployed rule uses \max_{\text{pool}} \text{cos} and \min_{\text{pool}} \text{dHash} over the same-CPA pool of size n_{\text{pool}}(s) for each signature s. A per-comparison rate is therefore not the rate at which the deployed classifier fires per signature. To compute the per-signature inter-CPA-equivalent rate, for each Big-4 source signature s we simulate one realisation of an inter-CPA candidate pool of the same size n_{\text{pool}}(s), drawn uniformly from non-same-CPA signatures across all firms, compute the deployed extrema and rule indicator, and aggregate (Script 43; n_{\text{sig}} = 150{,}453 vector-complete in this analysis; CPA-block bootstrap 95% CIs reported below).

Headline rates (deployed any-pair rule, HC = cos > 0.95 AND dHash \leq 5). Wilson 95% CIs on the point estimate, CPA-block bootstrap 95% CI on n_{\text{boot}} = 1000 replicates:

Rule semantics	Per-signature ICCR	Wilson 95% CI	CPA-bootstrap 95% CI
Any-pair (deployed)	0.1102	[0.1086, 0.1118]	[0.0908, 0.1330]
Same-pair (stricter alternative)	0.0827	[0.0813, 0.0841]	[0.0668, 0.1021]

Per-firm any-pair rates (no bootstrap; descriptive):

Firm	n_{\text{sig}}	Any-pair ICCR	Same-pair ICCR
Firm A	60{,}450	0.2594	0.2018
Firm B	34{,}254	0.0147	0.0023
Firm C	38{,}616	0.0053	0.0019
Firm D	17{,}133	0.0110	0.0051

Pool-size decile dependence. The deployed rule's pool-normalised rate is monotonically (broadly) increasing in n_{\text{pool}}, consistent with the 1 - (1 - p_{\text{pair}})^{n_{\text{pool}}} form expected under inter-CPA independence (Script 43 decile table). Decile 1 (smallest pools, n_{\text{pool}} \in [0, 201]): any-pair ICCR = 0.0249. Decile 10 (largest, n_{\text{pool}} \in [846, 1115]): any-pair ICCR = 0.1905. The trend is broadly monotonic with two minor non-monotone reversals (decile 5 and decile 9 dip below their predecessors).

Threshold sensitivity at per-signature unit. Tightening the HC rule from \text{dHash} \leq 5 to \text{dHash} \leq 3 (same-pair) reduces the per-signature ICCR from 0.0827 to 0.0449 (Big-4 pooled); tightening to \text{dHash} \leq 4 gives 0.0639 (same-pair). A stricter operating point of dHash \leq 3 same-pair would correspond to a per-signature ICCR of \approx 0.05; the deployed HC any-pair rule with \text{dHash} \leq 5 corresponds to \approx 0.11. Stakeholders requiring a tighter specificity proxy could consider the dHash \leq 3 same-pair variant, with the unsupervised-setting caveats of §III-M.

L.3. Document-level inter-CPA proxy alert rate (Script 45)

The deployed worst-case aggregation classifies each document by the most-replication-consistent category among its constituent signatures (§III-H.1). Three operationally meaningful document-level alarm definitions are reported, each as the fraction of documents whose worst-case signature category falls in the alarm set under the same inter-CPA candidate-pool counterfactual as §III-L.2 (Script 45; n_{\text{docs}} = 75{,}233 Big-4 documents):

Alarm definition	Alarm set	Document-level ICCR	Wilson 95% CI
D1	HC only	0.1797	[0.1770, 0.1825]
D2	HC + MC ("any non-hand-signed screening label")	0.3375	[0.3342, 0.3409]
D3	HC + MC + HSC	0.3384	[0.3351, 0.3418]

Per-firm D2 document-level rates:

Firm	n_{\text{docs}}	D2 (HC + MC) ICCR
Firm A	30{,}226	0.6201
Firm B	17{,}127	0.1600
Firm C	19{,}501	0.1635
Firm D	8{,}379	0.0863

The document-level D2 rate of 33.75\% pooled over Big-4 is the most operationally relevant alarm-rate metric: it is the fraction of audit documents that would carry at least one signature flagged HC or MC under the counterfactual of inter-CPA candidate-pool replacement. The non-trivial per-document inter-CPA alarm rate (and its concentration in Firm A at 62\%) motivates the positioning of the operational system as a screening framework with human-in-the-loop review, not as an autonomous forensic classifier (§III-M).

L.4. Firm heterogeneity (Script 44)

§III-L.2 and §III-L.3 report large per-firm variation in the deployed rule's pool-normalised behaviour: Firm A's any-pair per-signature ICCR is 0.2594, an order of magnitude larger than Firm B's 0.0147, Firm C's 0.0053, Firm D's 0.0110. A natural alternative explanation is the pool-size confound: Firm A's median pool size (\sim 285) is larger than other firms', and pool size monotonically (broadly) increases the per-signature rate (§III-L.2 decile trend). We test the firm-vs-pool confound with a logistic regression of the per-signature hit indicator (any-pair HC) on firm dummies (Firm A = reference) and centred log pool size (Script 44):

Term	Odds ratio (vs Firm A)	Direction	Magnitude
Firm B	0.053	< 1	\sim 19\times lower odds than Firm A
Firm C	0.010	< 1	\sim 100\times lower odds than Firm A
Firm D	0.027	< 1	\sim 37\times lower odds than Firm A
log(pool size, centred)	4.01	> 1	\sim 4\times higher odds per unit log pool size

The Firm B/C/D odds ratios are very small after controlling for pool size, indicating that firm membership accounts for a large multiplicative effect on the per-signature rate that is not explained by pool size alone. (We report odds ratios rather than $z$-scores because per-signature observations are clustered by CPA and firm, and naive standard errors would be unreliable under within-cluster correlation; a cluster-robust standard error analysis is left as a robustness check.)

The per-decile per-firm breakdown (Script 44) confirms the pattern: within every pool-size decile, Firms B/C/D have rates of $0.0006$–0.0358, while Firm A's rate ranges $0.0541$–0.5958 across deciles. The firm gap is large within matched pool sizes, not driven by pool composition.

Cross-firm hit matrix. Among Big-4 source signatures whose any-pair rule fires under the inter-CPA candidate-pool counterfactual, the candidate firm of the max-cosine partner is distributed as follows (Script 44):

Source firm	Firm A candidate	Firm B	Firm C	Firm D	non-Big-4	hits
Firm A	14{,}447	95	44	19	17	14{,}622
Firm B	92	371	8	4	9	484
Firm C	16	7	149	5	1	178
Firm D	22	2	6	106	1	137

For the same-pair joint event (a single candidate satisfying both \text{cos} > 0.95 and \text{dHash} \leq 5), the candidate firm is even more strongly concentrated within the source firm: Firm A source \to Firm A candidate in 11{,}314 of 11{,}319 same-pair hits (99.96\%); Firm B source \to Firm B candidate in 85 of 87 (97.7\%); Firm C source \to Firm C candidate in 54 of 55 (98.2\%); Firm D source \to Firm D candidate in 64 of 66 (97.0\%).

Interpretation. Under the deployed any-pair rule, the within-firm collision concentration is 98.8\% at Firm A and $76.7$–83.7\% at Firms B/C/D — Firm A's pattern is markedly more within-firm-concentrated than the other three firms', though every Big-4 firm still has more than three quarters of its any-pair collisions falling on candidates within the same firm. The stricter same-pair joint event — a single candidate satisfying both cos > 0.95 and dHash \leq 5 — saturates at $97.0$–99.96\% within-firm across all four firms. This pattern is consistent with — but not by itself diagnostic of — firm-specific template, stamp, or document-production reuse: within-firm scanning workflows, common form templates, and shared report-generation infrastructure could produce visually similar signature crops across different CPAs within the same firm. Byte-level decomposition of Firm A's 145 pixel-identical signatures across \sim 50 distinct certifying partners (supplementary materials; §III-H.2) provides direct evidence of image-level reuse among Firm A signatures; the distribution across many partners is consistent with a firm-level template or production workflow, and the broader inter-CPA collision pattern in §III-L.4 is consistent with similar, milder production-related reuse patterns at Firms B/C/D. We report this as "inter-CPA collision concentration is within-firm" — a descriptive observation about deployed-rule behaviour — and refrain from inferring that the within-firm hits constitute deliberate or systematic template sharing.

This connects back to §III-J: the K=3 firm-composition contrast at the accountant level (Firm A dominating C3; Firm C dominating C1) reappears at the deployment level in the cross-firm hit matrix, where the within-firm collision concentration is the dominant pattern at all four Big-4 firms — most strongly at Firm A (98.8\% any-pair, 99.96\% same-pair) and at materially lower but still majority levels at Firms B/C/D ($76.7$–83.7\% any-pair; $97.0$–98.2\% same-pair).

L.5. Alert-rate sensitivity around deployed thresholds (Script 46)

To test whether the deployed cosine threshold 0.95 and dHash threshold 5 coincide with a low-gradient (plateau-stable) region of the deployed-rule alert-rate surface — which would be weak distributional evidence that the deployed thresholds are stable operating points — we sweep each threshold across a range and report the per-signature alert rate on actual observed Big-4 same-CPA pools (not inter-CPA-replaced pools), comparing the local gradient at the deployed threshold to the median gradient across the sweep (Script 46).

At the deployed HC operating point cos > 0.95 AND dHash \leq 5, the local gradient of the per-signature alert rate is substantially larger than the median gradient across the sweep (cosine: ratio \approx 25\times at the 0.95 point relative to median; dHash: ratio \approx 3.8\times at the 5 point relative to median; both Script 46). Reading these ratios descriptively, the deployed HC threshold is locally sensitive rather than plateau-stable: small threshold perturbations materially change the deployed alert rate (cosine sweep at dHash \leq 5 yields rates of 0.5091 at cos > 0.945 vs 0.4789 at cos > 0.955, a 3.0 pp swing across a 0.01 cosine perturbation; dHash sweep at cos > 0.95 yields rates of 0.4207 at dHash \leq 4 vs 0.5639 at dHash \leq 6, a 14.3 pp swing across a single integer step). The local-gradient-to-median-gradient ratios are descriptive diagnostics, not formal plateau tests; the primary evidence for "no within-population bimodal antimode at these thresholds" comes from §III-I.4's composition decomposition, not from §III-L.5.

The MC/HSC boundary at dHash = 15, by contrast, is in a low-gradient region (ratio \approx 0.08 to the median); the plateau-like behaviour around dHash = 15 is corroborating evidence that the high-end structural threshold lies in a regime where the rule's alert rate is approximately saturated, consistent with the high-dHash tail behaviour expected once near-identical pairs have been exhausted. The §III-L.5 non-plateau / local-sensitivity finding therefore applies specifically to the HC cutoff (cos = 0.95, dHash = 5); the MC/HSC sub-band boundary at dHash = 15 exhibits the opposite behaviour and is plateau-like.

We interpret the deployed HC thresholds as specificity-anchored operating points chosen for the specificity-vs-alert-yield tradeoff (§III-L.1), not as distributional antimodes. Alternative operating points on the tradeoff curve can be characterised by inverting the per-comparison or pool-normalised ICCR curves (§III-L.1, §III-L.2) at the preferred specificity target.

L.6. Observed deployed alert rate on actual same-CPA pools

The pool-normalised inter-CPA rates of §III-L.2 and §III-L.3 use the counterfactual of replacing the source signature's same-CPA pool with random non-same-CPA candidates. The observed deployed alert rate uses the source's actual same-CPA pool, i.e., the rate at which the deployed rule fires on the real corpus. For Big-4, the deployed HC any-pair rule fires on 49.58\% of signatures and 62.28\% of documents (Script 46; Script 42 reproduces the per-signature rate at 49.58\%).

The per-signature observed-deployed rate is \sim 4.5\times the pool-normalised inter-CPA rate (0.4958 vs 0.1102); the per-document observed-deployed rate is \sim 3.5\times the pool-normalised inter-CPA D1 (HC) rate (0.6228 vs 0.1797). We refer to this multiplicative gap as the deployed-rate excess over the inter-CPA proxy:

• Per-signature: 0.4958 - 0.1102 = 0.3856 (38.6 pp excess)
• Per-document HC: 0.6228 - 0.1797 = 0.4431 (44.3 pp excess)

We do not interpret the deployed-rate excess as a presumed true-positive rate; the inferential limits of this interpretation are developed in §III-M. The deployed-rate excess is best read as a same-CPA repeatability signal — a quantity that exceeds what random inter-CPA candidate replacement would produce — rather than as an estimate of true replication prevalence.

M. Unsupervised Diagnostic Strategy and Limits

The corpus lacks signature-level ground-truth replication labels: no signature is annotated as definitively hand-signed or definitively templated. The conservative positive anchor (pixel-identical same-CPA signatures; §III-K.4) is by construction near \text{cos} = 1 and \text{dHash} = 0, providing a tautological capture-check rather than a sensitivity estimate for the non-byte-identical replicated class. The corpus therefore does not admit standard supervised classifier validation: we cannot report False Rejection Rate, sensitivity, recall, Equal Error Rate, ROC-AUC, or precision against ground truth.

Each diagnostic reported in this paper therefore addresses one specific failure mode of an unsupervised screening classifier; the full diagnostic-to-failure-mode-to-assumption map is given in Appendix A Table A.II. No single diagnostic provides ground-truth validation; together they define the limits of what can be supported in this corpus without signature-level ground truth.

Limits of the present analysis. We do not claim a validated forensic detector or an autonomous classification system. We do not report False Rejection Rate, sensitivity, recall, EER, ROC-AUC, precision, or positive predictive value against ground truth, because no ground truth exists at the signature level. We do not interpret the deployed-rate excess of §III-L.6 as a presumed true-positive rate: that interpretation would require assuming that the within-firm same-CPA pool's collision rate equals the inter-CPA proxy rate in the absence of replication (i.e., that genuine same-CPA hand-signing would produce a collision rate no higher than random inter-CPA pairs). Two factors make the assumption unsafe: (a) a CPA who signs consistently can produce stylistically similar signatures across years that exceed inter-CPA similarity at the cosine axis; (b) within-firm template sharing (§III-L.4 cross-firm hit matrix; byte-level evidence of Firm A's pixel-identical signatures across partners, supplementary materials) places a substantial inter-CPA collision floor that itself reflects template-like reuse rather than independent inter-CPA random matching. We do not infer that the within-firm collision concentration of §III-L.4 constitutes deliberate template sharing; we describe it as "inter-CPA collision concentration is within-firm" and treat the mechanism as an open empirical question.

Scope of the present analysis. The deployed signature-replication screening rule is characterised at three units of analysis (per-comparison, per-signature pool, per-document) against an inter-CPA negative-anchor coincidence-rate calibration. The per-comparison rates (\leq 0.0006 at cos > 0.95; \leq 0.0013 at dHash \leq 5; \leq 0.00014 jointly) are specificity-proxy-anchored operating points consistent with biometric-verification convention, with the proxy nature recorded in §III-L.0 and §III-M. The per-signature any-pair HC ICCR (0.11; Table XXII) and per-document HC+MC alarm-rate ICCR (0.34; Table XXIII) are operationally meaningful alarm-yield indicators rather than true error rates. Per-firm rates show substantial heterogeneity (Firm A's per-document HC + MC alarm at 0.62 vs Firm B/C/D at $0.09$–0.16), driven by firm-level rather than pool-size effects, and concentrated in within-firm cross-CPA candidate matching. The framework is positioned as a specificity-proxy-anchored screening tool with human-in-the-loop review, not as a validated forensic classifier.

Specificity-alert-yield tradeoff. Because sensitivity is unobservable, stakeholders cannot derive an operating point by optimising a ROC criterion. Instead, the specificity-proxy-anchored framework offers a specificity-alert-yield tradeoff: tighter operating points (e.g., cos > 0.98 AND dHash \leq 3) reduce both per-comparison ICCR (to \approx 5 \times 10^{-5}; §III-L.1 inversion) and per-signature alert yield (to \approx 0.05; §III-L.2), with an unknown effect on actual replication-detection recall. Tighter operating points are not necessarily preferable: any tightening reduces the alert rate but may also miss true replicated signatures whose noise has pushed them outside the tighter envelope. The deployment decision depends on the relative cost of manual review (per alarm) and missed-replication risk (per false negative) — neither directly observable from corpus data.

N. Data Source and Firm Anonymization

Audit-report corpus. The 90,282 audit-report PDFs analyzed in this study were obtained from the Market Observation Post System (MOPS) operated by the Taiwan Stock Exchange Corporation. MOPS is the statutory public-disclosure platform for Taiwan-listed companies; every audit report filed on MOPS is already a publicly accessible regulatory document. We did not access any non-public auditor work papers, internal firm records, or personally identifying information beyond the certifying CPAs' names and signatures, which are themselves published on the face of the audit report as part of the public regulatory filing. The CPA registry used to map signatures to CPAs is a publicly available audit-firm tenure registry (Section III-B).

Firm-level anonymization. Although all audit reports and CPA identities in the corpus are public, we report firm-level results under the pseudonyms Firm A / B / C / D throughout this paper to avoid naming specific accounting firms in descriptive rate comparisons. Readers with domain familiarity may still infer Firm A from contextual descriptors (Big-4 status, replication-dominated behavior); we disclose this residual identifiability explicitly and note that none of the paper's conclusions depend on the specific firm's name.

IV. Experiments and Results

Section IV reports the empirical results that calibrate and characterise the operational classifier of §III-H.1 (calibration developed in §III-L). The primary analyses (§IV-D through §IV-J, and the anchor-based ICCR calibration consolidated in §IV-M) are scoped to the Big-4 sub-corpus (Firms A–D, n = 437 CPAs with n_{\text{sig}} \geq 10, totalling 150,442 signatures with both descriptors available) per the methodology choice articulated in §III-G. §IV-K reports a full-dataset (686 CPAs) robustness check on the K=3 mixture and per-CPA score-rank convergence; §IV-A through §IV-C and §IV-L report the corpus-wide pipeline performance and feature-backbone ablation that support the descriptor choice of §III-F.

A. Experimental Setup

Experiments used mixed hardware: YOLOv11n training and inference for signature detection, and ResNet-50 forward inference for feature extraction over all 182,328 detected signatures, were performed on an Nvidia RTX 4090 (CUDA); the downstream statistical analyses (KDE antimode, Hartigan dip test, Beta-mixture EM with logit-Gaussian robustness check, Burgstahler-Dichev/McCrary density-smoothness diagnostic, and pairwise cosine/dHash computations) were performed on an Apple Silicon workstation with Metal Performance Shaders (MPS) acceleration. Feature extraction used PyTorch 2.9 with torchvision model implementations. The complete pipeline---from raw PDF processing through final classification---was implemented in Python. Because all steps rely on deterministic forward inference over fixed pre-trained weights (no fine-tuning) plus fixed-seed numerical procedures, reported results are platform-independent to within floating-point precision.

B. Signature Detection Performance

The YOLOv11n model achieved high detection performance on the validation set (Table II), with all loss components converging by epoch 60 and no significant overfitting despite the relatively small training set (425 images). We note that Table II reports validation-set metrics, as no separate hold-out test set was reserved given the small annotation budget (500 images total). However, the subsequent production deployment provides a practical consistency check: batch inference on 86,071 documents yielded 182,328 extracted signatures (Table III), with an average of 2.14 signatures per document, consistent with the standard practice of two certifying CPAs per audit report. The high VLM--YOLO agreement rate (98.8%) further corroborates detection reliability at scale.

Table III. Extraction Results.

Metric	Value
Documents processed	86,071
Documents with detections	85,042 (98.8%)
Total signatures extracted	182,328
Avg. signatures per document	2.14
CPA-matched signatures	168,755 (92.6%)
Processing rate	43.1 docs/sec

The Big-4 subset of the detection output yields 150,442 signatures with both descriptors (cosine and independent dHash) successfully computed; this is the per-signature population used in the primary analyses of §IV-D through §IV-J.

C. All-Pairs Intra-vs-Inter Class Distribution Analysis

Fig. 2 presents the cosine similarity distributions computed over the full set of pairwise comparisons under two groupings: intra-class (all signature pairs belonging to the same CPA) and inter-class (signature pairs from different CPAs). This all-pairs analysis is a different unit from the per-signature best-match statistics used in Sections IV-D onward; we report it first because it supplies the reference point for the KDE crossover used in per-signature classification (Section III-H.1). Table IV summarizes the distributional statistics.

Table IV. Cosine Similarity Distribution Statistics.

Statistic	Intra-class	Inter-class
N (pairs)	41,352,824	500,000
Mean	0.821	0.758
Std. Dev.	0.098	0.090
Median	0.836	0.774
Skewness	−0.711	−0.851
Kurtosis	0.550	1.027

Both distributions are left-skewed and leptokurtic. Shapiro-Wilk and Kolmogorov-Smirnov tests rejected normality for both (p < 0.001), confirming that parametric thresholds based on normality assumptions would be inappropriate. Distribution fitting identified the lognormal distribution as the best parametric fit (lowest AIC) for both classes, though we use this result only descriptively; the subsequent distributional diagnostics in Section IV-D are produced via the methods of Section III-I to avoid single-family distributional assumptions.

The KDE crossover---where the two density functions intersect---was located at 0.837. Under equal prior probabilities and equal misclassification costs, this crossover is a candidate decision boundary between the two classes; we adopt it only as the operational LH/UN boundary in §III-H.1, not as a natural distributional threshold. Statistical tests confirmed significant separation between the two distributions (Cohen's d = 0.669, Mann-Whitney [36] p < 0.001, K-S 2-sample p < 0.001).

We emphasize that pairwise observations are not independent---the same signature participates in multiple pairs---which inflates the effective sample size and renders $p$-values unreliable as measures of evidence strength. We therefore rely primarily on Cohen's d as an effect-size measure that is less sensitive to sample size. A Cohen's d of 0.669 indicates a medium effect size [29], confirming that the distributional difference is practically meaningful, not merely an artifact of the large sample count.

D. Big-4 Accountant-Level Distributional Characterisation

This section reports the empirical evidence for §III-I's distributional diagnostics at the Big-4 accountant level. The accountant-level dip-test rejection reported in Table V is, per §III-I.4, fully attributable to between-firm location shifts and integer mass-point artefacts rather than to within-population bimodality; the composition-decomposition diagnostics that establish this finding are tabulated in §IV-M below alongside the anchor-based ICCR calibration.

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 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 deployed box-rule less-replication-dominated rate	+0.9627	< 10^{-248}
Reverse-anchor cosine percentile vs deployed box-rule less-replication-dominated rate	+0.8890	< 10^{-149}
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 deployed 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
deployed binary high-confidence box rule (cos > 0.95 AND dHash \leq 5) vs per-CPA K=3 hard label	0.662
deployed binary high-confidence box rule vs per-signature K=3 hard label	0.559
Per-CPA K=3 hard label vs per-signature K=3 hard label	0.870

(Source: Script 39.) Per-signature K=3 components (n = 150{,}442) sorted by ascending cosine: (0.928, 9.75, 0.146) / (0.963, 6.04, 0.582) / (0.989, 1.27, 0.272), an absolute cosine drift of 0.018 in C1 and 0.006 in C3 relative to the per-CPA fit. These convergence checks cover only the binary high-confidence rule (cos > 0.95 AND dHash \leq 5); the five-way classifier's moderate-confidence band (5 < \text{dHash} \leq 15) retains its prior calibration and capture-rate evidence (supplementary materials; cross-referenced in §IV-J).

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; screening label P2_PARTIAL.) Component shape is reproducible across folds: max deviation of C1 cosine = 0.005, C1 dHash = 0.96, C1 weight = 0.023. Hard-posterior membership for the held-out firm varies: max absolute difference from the full-Big-4 baseline is 12.77 pp at the Firm C held-out fold, exceeding the report's 5 pp viability bar. We accordingly do not use K=3 hard-posterior membership as an operational classifier label (§III-J, §III-L).

H. Pixel-Identity Positive-Anchor Miss Rate

This section reports the only conservative hard-positive subset analysis available in the corpus: the positive-anchor miss rate against n = 262 Big-4 signatures whose nearest same-CPA match is byte-identical after crop and normalisation. Independent hand-signing cannot produce pixel-identical images, so byte-identical signatures are a conservative hard-positive subset for image replication. The analysis is one-sided (positive-anchor only); a paired false-alarm rate against a hand-signed negative anchor is not available because no signature-level hand-signed ground truth exists in the corpus (§III-K item 4).

Table XIV. Positive-anchor miss rate, n = 262 Big-4 byte-identical signatures.

Classifier	Misclassified as less-replication-dominated	Miss rate	Wilson 95% CI
deployed 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 deployed box rule this result is close to tautological (byte-identical nearest-neighbour signatures have cosine \approx 1 and dHash \approx 0, well inside the rule's high-confidence region). The reverse-anchor cut is chosen by prevalence calibration against the box rule's overall replicated rate of 49.58\% across Big-4 signatures; this is a documented limitation since no signature-level hand-signed ground truth exists to permit direct ROC optimisation.

I. Inter-CPA Pair-Level Coincidence Rate

The metric reported here is the inter-CPA pair-level coincidence rate (ICCR). It is the per-pair rate at which two signatures from different CPAs satisfy the deployed rule. We do not label it as a False Acceptance Rate because (a) FAR has a biometric-verification meaning that requires ground-truth negative labels, and (b) the inter-CPA negative-anchor assumption is partially violated by within-firm cross-CPA template-like collision structures (§III-L.4 cross-firm hit matrix).

A corpus-wide spike on \sim 50{,}000 inter-CPA pairs gives a per-comparison rate of 0.0005 (Wilson 95% CI [0.0003, 0.0007]) at the cosine cut 0.95. The Big-4-scope spike at higher sample size (5 \times 10^5 inter-CPA pairs) replicates this number, adds the structural dimension (dHash), and adds joint-rule rates; the §III-L.1 numbers are referenced rather than duplicated here, and the consolidated ICCR calibration appears in §IV-M Tables XXI–XXVI.

J. Five-Way Per-Signature + Document-Level Classification Output

This section reports the five-way per-signature + document-level worst-case classifier output on the Big-4 sub-corpus. See §III-H.1 for the five-way category definitions and the cosine and dHash cuts; calibration is in §III-L.

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 XVII: Firm A's signatures concentrate in the HC band (81.70%) while its CPAs concentrate at the accountant level in the K=3 C3 (high-cos / low-dHash) component (82.46%; Table XVII). These two figures address different units (per-signature classification vs per-CPA hard cluster assignment) and are not directly comparable as a like-for-like consistency check; we report the qualitative alignment but do not infer a numerical equivalence. The three non-Firm-A Big-4 firms have markedly lower HC rates than Firm A and substantially higher Uncertain rates, with Firm D having the highest Uncertain rate (45.65%).

Document-level worst-case aggregation. Each audit report typically carries two certifying-CPA signatures. We aggregate signature-level outcomes to document-level labels using the worst-case rule (HC > MC > HSC > UN > LH; §III-H.1), applied to the Big-4 sub-corpus.

Table XVI. 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) retains its prior calibration (supplementary materials); it is not separately re-characterised by Scripts 38–40, which checked only the binary high-confidence rule (cos > 0.95 AND dHash \leq 5). The moderate-band cuts are not re-derived on the Big-4 subset; we report the Table XV per-firm MC proportions (10.76% / 35.88% / 41.44% / 29.33% across Firms A through D) descriptively. The capture-rate calibration evidence for the moderate band is reported in the supplementary materials and not regenerated on the Big-4 subset. We do not claim that the MC-band per-firm ordering above is a separate validation of the §III-K Spearman convergence, since MC occupancy is not a monotone function of the per-CPA less-replication-dominated ranking (e.g., Firm D's MC fraction is lower than Firm B's while Firm D's reverse-anchor score ranks it as less replication-dominated than Firm B).

Table XVII. 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 XVI above.

K. Full-Dataset Robustness (light scope)

This section reports the reproducibility cross-check at the full accountant scope (n = 686 CPAs, Big-4 plus mid/small firms). The scope of §IV-K is deliberately narrow: we re-run only the K=3 mixture + deployed operational-rule per-CPA less-replication-dominated rate analysis, sufficient to demonstrate that the K=3 + deployed-rule convergence reproduces at the wider scope. The §III-H.1 five-way classifier and the §IV-G LOOO analyses are not re-run at the full scope. The five-way moderate-confidence band retains its prior calibration (supplementary materials; §IV-J).

Table XVIII. 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 XIX. Spearman rank correlation between K=3 P(C1) and deployed operational less-replication-dominated rate, Big-4 sub-corpus vs full dataset.

Scope	n CPAs	Spearman \rho (P(C1) vs deployed 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 deployed box-rule less-replication-dominated rate are preserved at the full scope. Component centres shift modestly: C3 (high-cos / low-dHash) is essentially unchanged in centre but loses weight 0.117 as the full population includes more non-templated CPAs (mid/small firms); C1 (low-cos / high-dHash) gains weight 0.141 and shifts to lower cosine and higher dHash (centre (0.928, 11.17) vs Big-4 (0.946, 9.17)) as the broader population includes mid/small-firm CPAs landing toward the low-cos / high-dHash region that the Big-4-primary scope deliberately excludes. We read this as evidence that the Big-4-primary K=3 + deployed-rule convergence is not a Big-4-specific artefact; we do not read it as an endorsement of using full-dataset K=3 component centres or operational thresholds in place of the Big-4-primary analysis. Mid/small-firm composition shifts the component centres meaningfully and the primary methodology is restricted to Big-4 by design (§III-G item 4).

L. Ablation Study: Feature Backbone Comparison

To support the choice of ResNet-50 as the feature extraction backbone, we conducted an ablation study comparing three pre-trained architectures: ResNet-50 (2048-dim), VGG-16 (4096-dim), and EfficientNet-B0 (1280-dim). All models used ImageNet pre-trained weights without fine-tuning, with identical preprocessing and L2 normalization. The comparison summary is reported in the supplementary materials (backbone-ablation table; not the same table as Table XIX in this section, which reports Big-4 vs full-dataset Spearman drift in §IV-K).

EfficientNet-B0 achieves the highest Cohen's d (0.707), indicating the greatest statistical separation between intra-class and inter-class distributions. However, it also exhibits the widest distributional spread (intra std = 0.123 vs. ResNet-50's 0.098), i.e., a wider descriptor dispersion per signature. VGG-16 performs worst on all key metrics despite having the highest feature dimensionality (4096), suggesting that additional dimensions do not contribute discriminative information for this task.

ResNet-50 provides the best overall balance: (1) Cohen's d of 0.669 is competitive with EfficientNet-B0's 0.707; (2) its tighter distributions yield more stable descriptor behaviour at the per-signature level; (3) the highest Firm A all-pairs 1st percentile (0.543) indicates that Firm A replication-dominated signatures are least likely to produce low-similarity outlier pairs under this backbone; and (4) its 2048-dimensional features offer a practical compromise between discriminative capacity and computational/storage efficiency for processing 182K+ signatures.

M. Anchor-Based ICCR Calibration Results

This section consolidates the empirical results that support the §III-L anchor-based threshold calibration framework.

M.1 Composition decomposition (Scripts 39b–39e)

Table XX. Within-firm and between-firm decomposition of the Big-4 accountant-level dip-test rejection.

Diagnostic	Scope	Statistic	Implication
Within-firm signature-level cosine dip	Big-4 (4 firms)	p_{\text{cos}} \in \{0.176, 0.991, 0.551, 0.976\}	0/4 firms reject; cosine within-firm unimodal
Within-firm signature-level cosine dip	non-Big-4 (10 firms \geq 500 sigs)	p_{\text{cos}} \in [0.59, 0.99]	0/10 firms reject; cosine within-firm unimodal
Within-firm jittered-dHash dip (5 seeds, median)	Big-4 (4 firms)	p_{\text{med}} \in \{0.999, 0.996, 0.999, 0.9995\}	0/4 firms reject after integer-jitter; raw rejection was integer-tie artefact
Big-4 pooled dHash: 2×2 factorial	firm-centred + jittered (5 seeds)	p_{\text{med}} = 0.35, 0/5 seeds reject	combined corrections eliminate rejection; multimodality is composition + integer artefact
Integer-histogram valley near \text{dHash} \approx 5	within each Big-4 firm	none (0/4 firms)	no within-firm dHash antimode at the deployed HC cutoff

(Source: Scripts 39b, 39c, 39d, 39e; bootstrap n_{\text{boot}} = 2000; jitter \sim \mathrm{U}[-0.5, +0.5].)

M.2 Anchor-based inter-CPA pair-level ICCR (Script 40b)

Table XXI. Big-4 inter-CPA per-comparison ICCR sweep, n = 5 \times 10^5 pairs (Big-4 scope).

Threshold	Per-comparison ICCR	95% Wilson CI
cos > 0.945 (alternative operating point from supplementary calibration evidence)	0.00081	[0.00073, 0.00089]
cos > 0.95 (deployed operating point)	0.00060	[0.00053, 0.00067]
cos > 0.97	0.00024	[0.00020, 0.00029]
cos > 0.98	0.00009	[0.00007, 0.00012]
dHash \leq 5 (deployed operating point)	0.00129	[0.00120, 0.00140]
dHash \leq 4	0.00050	[0.00044, 0.00057]
dHash \leq 3	0.00019	[0.00015, 0.00023]
Joint: cos > 0.95 AND dHash \leq 5 (any-pair semantics)	0.00014	[0.00011, 0.00018]
Joint: cos > 0.95 AND dHash \leq 4 (any-pair)	0.00011	[0.00008, 0.00014]

Conditional ICCR(dHash \leq 5 | cos > 0.95) = 0.234 (Wilson 95% [0.190, 0.285]; 70 of 299 pairs).

The cos > 0.95 row is consistent with the corpus-wide spike of §IV-I (per-comparison rate 0.0005). The dHash row and joint row are reported here for the first time on this corpus.

M.3 Pool-normalised per-signature ICCR (Script 43)

Table XXII. Pool-normalised per-signature ICCR under the deployed any-pair HC rule (cos > 0.95 AND dHash \leq 5); n_{\text{sig}} = 150{,}453 (vector-complete Big-4); CPA-block bootstrap n_{\text{boot}} = 1000.

Scope	Per-signature ICCR	Wilson 95% CI	CPA-bootstrap 95% CI
Big-4 pooled (any-pair, deployed)	0.1102	[0.1086, 0.1118]	[0.0908, 0.1330]
Big-4 pooled (same-pair, stricter alternative)	0.0827	[0.0813, 0.0841]	[0.0668, 0.1021]
Firm A (any-pair)	0.2594	—	—
Firm B (any-pair)	0.0147	—	—
Firm C (any-pair)	0.0053	—	—
Firm D (any-pair)	0.0110	—	—
Pool-size decile 1 (smallest pools) any-pair	0.0249	—	—
Pool-size decile 10 (largest pools) any-pair	0.1905	—	—

Decile trend is broadly monotone in pool size with two minor reversals (decile 5 and decile 9 dip below their predecessors). Stricter operating point cos > 0.95 AND dHash \leq 3 (same-pair) gives per-signature ICCR 0.0449.

M.4 Document-level ICCR under three alarm definitions (Script 45)

Table XXIII. Document-level inter-CPA ICCR by alarm definition; n_{\text{docs}} = 75{,}233.

Alarm definition	Alarm set	Document-level ICCR	Wilson 95% CI
D1	HC only	0.1797	[0.1770, 0.1825]
D2 (operational)	HC + MC	0.3375	[0.3342, 0.3409]
D3	HC + MC + HSC	0.3384	[0.3351, 0.3418]

Per-firm D2 document-level ICCR: Firm A 0.6201 (n = 30{,}226); Firm B 0.1600 (n = 17{,}127); Firm C 0.1635 (n = 19{,}501); Firm D 0.0863 (n = 8{,}379). The Firm C denominator n = 19{,}501 exceeds Table XVI's single-firm Firm C count of 19{,}122 by exactly the 379 mixed-firm PDFs (all 379 are 1{:}1 Firm C / Firm D documents that an alphabetically-ordered tie-break assigns to Firm C; full implementation detail in the supplementary materials). The four per-firm denominators here therefore sum to the full 75{,}233, whereas Table XVI's per-firm rows sum to 74{,}854 = 75{,}233 - 379.

M.5 Firm heterogeneity logistic regression and cross-firm hit matrix (Script 44)

Table XXIV. Logistic regression of per-signature any-pair HC hit indicator on firm dummies and centred log pool size (Firm A reference).

Term	Odds ratio (vs Firm A)	Direction
Firm B	0.053	\sim 19\times lower odds than Firm A
Firm C	0.010	\sim 100\times lower odds than Firm A
Firm D	0.027	\sim 37\times lower odds than Firm A
log(pool size, centred)	4.01	\sim 4\times higher odds per log unit pool size

Per-decile per-firm rates (Table not duplicated here; Script 44 decile table available in the supplementary report): within every pool-size decile, Firms B/C/D show rates of $0.0006$–0.0358 while Firm A ranges $0.0541$–0.5958. The firm gap survives within matched pool sizes.

Table XXV. Cross-firm hit matrix among Big-4 source signatures with any-pair HC hit; max-cosine partner firm (counts).

Source firm	Firm A cand.	Firm B	Firm C	Firm D	non-Big-4	n hits
Firm A	14{,}447	95	44	19	17	14{,}622
Firm B	92	371	8	4	9	484
Firm C	16	7	149	5	1	178
Firm D	22	2	6	106	1	137

Same-pair joint hits (single candidate satisfying both cos > 0.95 AND dHash \leq 5) are within-firm at rates 99.96\% / 97.7\% / 98.2\% / 97.0\% for Firms A/B/C/D respectively.

M.6 Alert-rate sensitivity around deployed HC threshold (Script 46)

Table XXVI. Local-gradient / median-gradient ratio at deployed thresholds (descriptive plateau diagnostic).

Threshold	Local / median gradient ratio	Interpretation
cos = 0.95 (HC)	\approx 25\times	locally sensitive (not plateau-stable)
dHash = 5 (HC)	\approx 3.8\times	locally sensitive (not plateau-stable)
dHash = 15 (MC/HSC boundary)	\approx 0.08	plateau-like (saturating tail)

Big-4 observed deployed alert rate on actual same-CPA pools: per-signature HC = 0.4958; per-document HC = 0.6228. The deployed-rate excess over the inter-CPA proxy is 0.3856 (38.6 pp) per-signature and 0.4431 (44.3 pp) per-document; this excess is interpreted as a same-CPA repeatability signal under the §III-M caveats, not as a presumed true-positive rate.

V. Discussion

A. Non-Hand-Signing Detection as a Distinct Problem

Non-hand-signing differs from forgery in that the questioned signature is produced by its legitimate signer's own stored image rather than by an impostor. The detection problem is therefore framed around intra-signer image reproduction rather than inter-signer imitation. This framing has analytical consequences. The within-CPA signature distribution is the analytical population of interest; the cross-CPA inter-class distribution is a reference against which intra-CPA similarity is interpreted, not the population to be modelled. This contrasts with most prior offline signature verification work, which treats genuine-versus-forged as the central two-class problem.

B. Per-Signature Similarity is a Continuous Quality Spectrum; the Accountant-Level Multimodality is Composition-Driven

The Big-4 accountant-level descriptor distribution rejects unimodality on both marginals at p < 5 \times 10^{-4} (§IV-D Table V). The composition decomposition of §III-I.4 shows that this rejection is fully attributable to two non-mechanistic sources: (a) between-firm location-shift effects on both axes — Firm A's mean dHash of 2.73 versus Firms B/C/D's 6.46, 7.39, 7.21 creates a multi-peaked pooled distribution that any single firm's distribution lacks — and (b) integer mass-point artefacts on the integer-valued dHash axis, which inflate the dip statistic against a continuous-density null. A 2×2 factorial diagnostic applied to the Big-4 pooled dHash (firm-mean centring × uniform integer jitter [-0.5, +0.5], 5 jitter seeds) shows that the dip test fails to reject (p_{\text{median}} = 0.35, 0/5 seeds reject) when both corrections are applied; either correction alone leaves the rejection in place. Within the Big-4 firms, the descriptor marginals at the signature level are unimodal once integer ties are broken (Scripts 39b, 39d); eligible non-Big-4 firms provide corroborating raw-axis evidence on the cosine dimension (Script 39c) but are not used as calibration evidence (§III-I.4). The descriptor distributions therefore lack a within-population bimodal antimode that could anchor an operational threshold. The K=2 / K=3 mixture fits are retained in §III-J as descriptive partitions of the joint Big-4 distribution that reflect firm-compositional structure, not as inferential evidence for two or three latent mechanism modes.

C. Firm A as the Templated End of Big-4 (Case Study, Not Calibration Anchor)

Firm A is empirically the firm whose CPAs are most concentrated in the high-cosine, low-dHash corner of the Big-4 descriptor plane. In the Big-4 K=3 hard-posterior assignment (now interpreted as a firm-compositional position assignment; §III-J), Firm A accounts for 0\% of C1 (low-cos / high-dHash position) and 82.5\% of C3 (high-cos / low-dHash position); the opposite pattern holds at Firm C, which has the highest C1 concentration at 23.5\%. Firm A also accounts for 145 of the 262 byte-identical signatures in the Big-4 byte-identical anchor of §IV-H (with Firm B 8, Firm C 107, Firm D 2). Byte-level decomposition of the 145 Firm A pixel-identical signatures (see supplementary materials) shows they span 50 distinct Firm A partners (of 180 registered), with 35 byte-identical matches occurring across different fiscal years.

We treat Firm A as a templated-end case study within the Big-4 sub-corpus rather than as the calibration anchor for the operational threshold. Firm A enters the Big-4 anchor-based ICCR calibration on equal footing with the other three Big-4 firms (§III-L). The cross-firm hit matrix of §III-L.4 strengthens this framing: under the deployed any-pair rule, within-firm collision concentration is 98.8\% at Firm A and $76.7$–83.7\% at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–99.96\% within-firm across all four firms). Firm A's per-document D2 inter-CPA proxy ICCR of 0.6201 (versus Firms B/C/D's $0.09$–0.16) — the counterfactual rate at which Firm A documents would fire HC$+$MC if same-CPA pools were replaced by random inter-CPA candidates — reflects high inter-CPA collision concentration under the deployed rule, consistent with firm-specific template, stamp, or document-production reuse. (The corresponding observed rate on real same-CPA pools, from Table XVI, is substantially higher: 97.5\% HC$+$MC for Firm A; the proxy and observed rates measure different quantities and are not directly comparable.) The inter-CPA-anchor analysis alone is not diagnostic of deliberate template sharing. The byte-level evidence above (Firm A's 145 pixel-identical signatures across \sim 50 distinct partners) provides direct evidence of image-level reuse among Firm A signatures; the distribution across many partners is consistent with a firm-level template or production workflow, and the within-firm collision pattern at all four Big-4 firms is consistent with similar, milder production-related reuse patterns at Firms B/C/D.

D. K=2 / K=3 as Descriptive Firm-Compositional Partitions

Leave-one-firm-out cross-validation of the Big-4 mixture fit reveals a sharp contrast between K=2 and K=3 behaviour. K=2 is unstable: across-fold cosine-crossing deviation is 0.028, and holding Firm A out gives a fold rule (cos > 0.938, dHash \leq 8.79) that classifies 100\% of held-out Firm A in the upper component, while holding any non-Firm-A Big-4 firm out gives a fold rule near (cos > 0.975, dHash \leq 3.76) that classifies 0\% of the held-out firm in the upper component. The K=2 boundary is essentially a Firm-A-vs-others separator — direct evidence that the K=2 partition reflects firm-compositional rather than mechanistic structure.

K=3 in contrast has a reproducible component shape at the descriptor-position level: across the four folds the C1 (low-cos / high-dHash) component cosine mean varies by at most 0.005, the dHash mean by at most 0.96, and the weight by at most 0.023. Hard-posterior membership for the held-out firm is composition-sensitive (absolute differences $1.8$–12.8 pp across folds). Together with the §III-I.4 composition decomposition (no within-population bimodal antimode), the K=3 stability supports a descriptive reading: the Big-4 descriptor plane has a reproducible three-region partition that reflects how firm-compositional weight is distributed across the descriptor space, not a three-mechanism latent-class structure. We accordingly do not use K=3 hard-posterior membership as an operational classifier; we use it as the accountant-level descriptive summary that complements the deployed signature-level five-way classifier of §III-H.1.

E. Three-Score Convergent Internal-Consistency

Three feature-derived scores agree on the per-CPA descriptor-position ranking at Spearman \rho \geq 0.879: the K=3 mixture posterior (a firm-compositional position score, not a mechanism cluster posterior); the reverse-anchor cosine percentile under a non-Big-4 reference distribution; and the deployed box-rule less-replication-dominated rate. The three scores are not statistically independent measurements — they are deterministic functions of the same per-CPA descriptor pair — so the convergence is documented as internal consistency rather than external validation against an independent ground truth (which the corpus does not provide for the hand-signed class). The strength of the convergence (all pairwise |\rho| > 0.87) and its persistence at the signature level (Cohen \kappa = 0.87 between per-CPA-fit and per-signature-fit K=3 binary labels) are nevertheless informative: per-CPA aggregation does not collapse the broad three-region ordering, and three different summarisations of the descriptor space produce broadly concordant per-CPA rankings, with a residual non-Firm-A disagreement (the reverse-anchor cosine percentile ranks Firm D fractionally above Firm C, while the mixture posterior and the deployed box-rule rate rank Firm C highest among non-Firm-A firms).

F. Anchor-Based Multi-Level Calibration

The operational specificity-proxy behaviour of the deployed five-way classifier is characterised at three units of analysis (§III-L), all against the same inter-CPA negative-anchor coincidence-rate proxy. The per-comparison ICCR is consistent with the corpus-wide rate reported in §IV-I (cos$>0.95 \to 0.00060$) and extends it to the structural dimension (dHash$\leq 5 \to 0.00129$; joint \to 0.00014). The pool-normalised per-signature ICCR captures the deployed rule's effective per-signature rate under inter-CPA candidate-pool replacement (0.1102 pooled Big-4 any-pair HC), exposing that the per-comparison rate is not the deployed-rule rate at the per-signature classifier level: the deployed classifier takes max-cosine and min-dHash over a same-CPA pool of size n_{\text{pool}}, so the inter-CPA-equivalent rate scales approximately as 1 - (1 - p_{\text{pair}})^{n_{\text{pool}}} in the independence limit. The per-document ICCR aggregates to operational alarm-rate units: HC alone 0.18, the operational HC$+$MC alarm 0.34.

Two additional findings refine the calibration story. First, the per-pair conditional ICCR for dHash$\leq 5$ given cos$>0.95$ is 0.234 (Wilson 95% [0.190, 0.285]): given the cosine gate, the structural dimension provides further per-comparison specificity at \sim 4.3\times refinement. Second, the alert-rate sensitivity analysis (§III-L.5) shows the deployed HC threshold is locally sensitive rather than plateau-stable (local gradient \approx 25\times the median for cosine, \approx 3.8\times for dHash); alternative operating points can be characterised by inverting the ICCR curves (e.g., a tighter rule cos$>0.95$ AND dHash$\leq 3$ on the same-pair joint corresponds to per-signature ICCR \approx 0.045). The MC/HSC sub-band boundary at dHash$=15$, by contrast, is plateau-like (local-to-median ratio \approx 0.08), consistent with high-dHash-tail saturation.

G. Pixel-Identity Positive Anchor and Inter-CPA Coincidence-Rate Negative Anchor

The only conservative hard-positive subset in the corpus is pixel-identical signatures: those whose nearest same-CPA match is byte-identical after crop and normalisation. Independent hand-signing cannot produce byte-identical images, so these signatures are a conservative hard-positive subset for image replication. On the Big-4 subset (n = 262 pixel-identical signatures), all three candidate checks — the deployed box rule, the K=3 hard label, and the reverse-anchor metric with a prevalence-calibrated cut — achieve 0\% positive-anchor miss rate (Wilson 95% upper bound 1.45\%). We caution that this result is necessary but not sufficient: for the deployed box rule it is close to tautological, because byte-identical neighbours have cosine \approx 1 and dHash \approx 0, well inside the rule's high-confidence region. The corresponding signature-level negative anchor evidence is developed in §III-L.1 above (per-comparison ICCR = 0.00060 at cos$>0.95$, consistent with the corpus-wide rate of 0.0005 reported in §IV-I). We frame the per-comparison rate as a specificity proxy under the assumption that inter-CPA pairs constitute a clean negative anchor, and we document in §III-L.4 that this assumption is partially violated by within-firm cross-CPA template-like collision structures.

H. Limitations

Several limitations should be transparent. We group them into primary methodological limitations, secondary scope and validation caveats, documented design features, and engineering-level caveats of the pipeline.

Primary methodological limitations.

No signature-level ground truth; no true error rates reportable. The corpus does not contain labelled hand-signed or replicated classes at the signature level. We therefore cannot report False Rejection Rate, sensitivity, recall, Equal Error Rate, ROC-AUC, precision, or positive predictive value against ground truth. All quantitative rates reported in §III-L are inter-CPA negative-anchor coincidence rates (ICCRs) under the assumption that inter-CPA pairs constitute a clean negative anchor; this is a specificity proxy, not a calibrated specificity (§III-M).

Inter-CPA negative-anchor assumption is partially violated and the violation is firm-dependent. The cross-firm hit matrix of §III-L.4 shows that under the deployed any-pair rule, within-firm collision concentration is 98.8\% at Firm A and $76.7$–83.7\% at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–99.96\% within-firm across all four firms), consistent with firm-specific template, stamp, or document-production reuse. The inter-CPA-as-negative assumption is therefore not exactly satisfied — some inter-CPA pairs may share firm-level templates rather than being independent random matches. Our reported per-comparison ICCRs are best read as specificity-proxy rates under a partially-violated assumption, not as calibrated FARs. Because the violation is firm-dependent, Firm A's per-firm ICCR is more contaminated by within-firm sharing than Firms B/C/D's; the per-firm B/C/D rates of $0.09$–0.16 may therefore be less contaminated than the pooled rate, and the Firm A vs Firms B/C/D contrast reflects both genuine firm heterogeneity and a firm-dependent proxy-contamination gradient.

Scope. The primary analyses are scoped to the Big-4 sub-corpus. We did not perform the full per-signature pool-normalised ICCR analysis at the full n = 686 scope; the §IV-K full-dataset Spearman re-run shows the K=3 + deployed box-rule rank-convergence is preserved at n = 686 but does not establish portability of the Big-4 operational ICCRs, the LOOO firm-fold structure, or the five-way operational classifier at the broader scope.

Secondary scope and validation caveats.

Pixel-identity is a conservative subset. Byte-identical pairs are the easiest replicated cases, and for the deployed box rule the positive-anchor miss rate against byte-identical pairs is close to tautological (byte-identical \Rightarrow cosine \approx 1, dHash \approx 0, well inside the high-confidence box). A score that fails the pixel-identity check would be disqualified, but passing the check does not guarantee correct behaviour on the broader replicated population (e.g., re-stamped or noisy-template-variant signatures).

Rule components not separately re-characterised by the present diagnostic battery. The five-way classifier's moderate-confidence band (cos > 0.95 AND 5 < \text{dHash} \leq 15), the style-consistency band (\text{dHash} > 15), and the document-level worst-case aggregation rule retain their prior calibration and capture-rate evidence (supplementary materials); the anchor-based ICCR calibration covers the binary high-confidence sub-rule (and its tightening alternatives such as dHash$\leq 3$), and the alert-rate sensitivity analysis (§III-L.5) characterises only the HC threshold. The MC and HSC sub-band boundaries are not separately re-characterised by the present diagnostic battery.

Deployed-rate excess is not a presumed true-positive rate. The $\sim 44$-pp per-document gap between the observed deployed alert rate (HC: 0.62 on real same-CPA pools) and the inter-CPA proxy rate (HC: 0.18) cannot be interpreted as a presumed true-positive rate without additional assumptions that §III-M shows are unsafe (consistent within-CPA signing can exceed inter-CPA similarity at the cosine axis; within-firm template sharing inflates the inter-CPA proxy baseline). The gap is best read as a same-CPA repeatability signal.

A1 pair-detectability stipulation. The per-signature detector requires at least one same-CPA pair to be near-identical when a CPA uses image replication. A1 is plausible for high-volume stamping or firm-level electronic signing but not guaranteed when a corpus contains only one observed replicated report for a CPA, multiple template variants used in parallel, or scan-stage noise that pushes a replicated pair outside the detection regime.

Documented design features.

K=3 hard-posterior membership is composition-sensitive. The K=3 hard-posterior membership for any single firm varies by up to 12.8 pp across LOOO folds. This is documented as a composition-sensitivity band rather than failure, but it means K=3 hard labels are not used as operational classifier output; they are reported only as accountant-level descriptive characterisation.

No partner-level mechanism attribution. The analysis reports population-level patterns; it does not perform partner-level mechanism attribution or report-level claims of intent. The signature-level outputs are signature-level quantities throughout. The within-firm cross-CPA collision concentration of §III-L.4 is consistent with template-like reuse but is not by itself diagnostic of deliberate sharing.

Engineering-level caveats of the pipeline.

Transferred ImageNet features. The ResNet-50 feature extractor uses pre-trained ImageNet weights without signature-domain fine-tuning. While our backbone-ablation study (§IV-L) and prior literature support the effectiveness of transferred ImageNet features for signature comparison, a signature-domain fine-tuned feature extractor could improve discriminative performance.

Red-stamp HSV preprocessing artifacts. The red stamp removal preprocessing uses simple HSV color-space filtering, which may introduce artifacts where handwritten strokes overlap with red seal impressions. Blended pixels are replaced with white, potentially creating small gaps in signature strokes that could reduce dHash similarity. This bias would push classifications toward false negatives rather than false positives.

Longitudinal scan / PDF / compression confounds. Scanning equipment, PDF generation software, and compression algorithms may have changed over the 2013–2023 study period, potentially affecting similarity measurements. While cosine similarity and dHash are designed to be robust to such variations, longitudinal confounds cannot be entirely excluded.

Source-exemplar misattribution in max/min pair logic. The max-cosine / min-dHash detection logic treats both ends of a near-identical same-CPA pair as non-hand-signed. In the rare case where one of the two documents contains a genuinely hand-signed exemplar that was subsequently reused as a stamping or e-signature template, the pair correctly identifies image reuse but misattributes non-hand-signed status to the source exemplar. This affects at most one source document per template variant per CPA and is not expected to be common.

Legal and regulatory interpretation. Whether non-hand-signing of a CPA's own stored signature constitutes a violation of signing requirements is a jurisdiction-specific legal question. Our technical analysis can inform such determinations but cannot resolve them.

VI. Conclusion and Future Work

We present a fully automated pipeline for screening non-hand-signed CPA signatures in Taiwan-listed financial audit reports, together with an anchor-calibrated screening framework that characterises the pipeline's operational behaviour at the Big-4 sub-corpus scope under explicit unsupervised assumptions. The pipeline processes raw PDFs through VLM-based page identification, YOLO-based signature detection, ResNet-50 feature extraction, and dual-descriptor (cosine + independent-minimum dHash) similarity computation. The operational output is the deployed five-way per-signature classifier with worst-case document-level aggregation (§III-H.1; calibrated in §III-L). Applied to 90,282 audit reports filed between 2013 and 2023, the pipeline extracts 182,328 signatures from 758 CPAs, with the Big-4 sub-corpus (437 CPAs at accountant level; 150,442–150,453 signatures at signature level) as the primary analytical population.

Our central methodological contributions are: (1) a composition decomposition that establishes the absence of a within-population bimodal antimode in the Big-4 descriptor distribution: the apparent multimodality dissolves under joint firm-mean centring and integer-tie jitter (p_{\text{median}} = 0.35), so distributional "natural-threshold" framings of the deployed operating points are not empirically supported; (2) an anchor-based inter-CPA coincidence-rate (ICCR) calibration at three units of analysis — per-comparison (0.0006 at cos$>0.95$; 0.0013 at dHash$\leq 5$; 0.00014 jointly), pool-normalised per-signature (0.11 for the deployed any-pair HC rule), and per-document (0.34 for the operational HC$+$MC alarm) — with explicit terminological replacement of "FAR" by "ICCR" given the unsupervised setting; (3) firm heterogeneity quantification: logistic regression with pool-size adjustment gives odds ratios 0.053, 0.010, 0.027 for Firms B/C/D relative to Firm A reference, indicating a large multiplicative effect that pool-size differences do not explain; (4) cross-firm hit matrix evidence that under the deployed any-pair rule, within-firm collision concentration is 98.8\% at Firm A and $76.7$–83.7\% at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–99.96\% within-firm across all four firms), consistent with firm-specific template, stamp, or document-production reuse mechanisms; (5) K=3 mixture demoted from "three mechanism clusters" to a descriptive firm-compositional partition; (6) three feature-derived scores converging on the per-CPA descriptor-position ranking at Spearman \rho \geq 0.879, reported as internal consistency rather than external validation; (7) 0\% positive-anchor miss rate on 262 byte-identical Big-4 signatures with the conservative-subset caveat; and (8) explicit disclosure of each diagnostic's untested assumption (Appendix A Table A.II), positioning the system as an anchor-calibrated screening framework with human-in-the-loop review rather than as a validated forensic detector.

Future work falls in four directions. First, a small-scale human-rated labelled set would enable direct ROC optimisation and provide the signature-level ground truth that the present analysis fundamentally lacks; without such ground truth, no true error rates can be reported. Second, the within-firm collision concentration documented in §III-L.4 (any-pair $76.7$–98.8\% across Big-4; same-pair joint $97.0$–99.96\%) invites a separate study to distinguish deliberate template sharing from passive firm-level production artefacts (shared scanners, common form templates, identical report-generation infrastructure) — a question the inter-CPA-anchor analysis alone cannot resolve. Third, the descriptive Firm A versus Firms B/C/D contrast (per-document HC$+$MC alarm 0.62 vs $0.09$–0.16) — together with the byte-level evidence of 145 pixel-identical signatures across \sim 50 distinct Firm A partners — invites a companion analysis examining whether such firm-level signing patterns correlate with established audit-quality measures. Fourth, generalisation to mid- and small-firm contexts requires extending the anchor-based ICCR framework to scopes where firm-level LOOO folds are not available; the §III-I.4 composition diagnostics already document that the absence of within-population bimodality holds across the tested eligible scopes, so the calibration approach in principle generalises, but a full extension with cluster-robust uncertainty quantification is left as future work.

Appendix A. Supplementary Diagnostic Detail

A.1. BD/McCrary Bin-Width Sensitivity (Signature Level)

The main text (Section III-I, Section IV-D Table VI) treats the Burgstahler-Dichev / McCrary discontinuity procedure [38], [39] as a density-smoothness diagnostic rather than as a threshold estimator. This subsection documents the empirical basis for that framing by sweeping the bin width across four (variant, bin-width) panels: Firm A and full-sample, each in the cosine and \text{dHash}_\text{indep} direction.

Table A.I. BD/McCrary Bin-Width Sensitivity (two-sided \alpha = 0.05, |Z| > 1.96).

Variant	n	Bin width	Best transition	z_below	z_above
Firm A cosine (sig-level)	60,448	0.003	0.9870	-2.81	+9.42
Firm A cosine (sig-level)	60,448	0.005	0.9850	-9.57	+19.07
Firm A cosine (sig-level)	60,448	0.010	0.9800	-54.64	+69.96
Firm A cosine (sig-level)	60,448	0.015	0.9750	-85.86	+106.17
Firm A dHash_indep (sig-level)	60,448	1	2.0	-4.69	+10.01
Firm A dHash_indep (sig-level)	60,448	2	no transition	—	—
Firm A dHash_indep (sig-level)	60,448	3	no transition	—	—
Full-sample cosine (sig-level)	168,740	0.003	0.9870	-3.21	+8.17
Full-sample cosine (sig-level)	168,740	0.005	0.9850	-8.80	+14.32
Full-sample cosine (sig-level)	168,740	0.010	0.9800	-29.69	+44.91
Full-sample cosine (sig-level)	168,740	0.015	0.9450	-11.35	+14.85
Full-sample dHash_indep (sig-l.)	168,740	1	2.0	-6.22	+4.89
Full-sample dHash_indep (sig-l.)	168,740	2	10.0	-7.35	+3.83
Full-sample dHash_indep (sig-l.)	168,740	3	9.0	-11.05	+45.39

Two patterns are visible in Table A.I. First, the procedure consistently identifies a "transition" under every bin width, but the location of that transition drifts monotonically with bin width (Firm A cosine: 0.987 → 0.985 → 0.980 → 0.975 as bin width grows from 0.003 to 0.015; full-sample dHash: 2 → 10 → 9 as the bin width grows from 1 to 3). The Z statistics also inflate superlinearly with the bin width (Firm A cosine |Z| rises from \sim 9 at bin 0.003 to \sim 106 at bin 0.015) because wider bins aggregate more mass per bin and therefore shrink the per-bin standard error on a very large sample. Both features are characteristic of a histogram-resolution artifact rather than of a genuine density discontinuity.

Second, the candidate transitions all locate inside the high-similarity region (cosine \geq 0.975, dHash \leq 10) rather than at a between-mode boundary, which is the location pattern we would expect of a clean within-population antimode.

Taken together, Table A.I shows that the signature-level BD/McCrary transitions are not a threshold in the usual sense---they are histogram-resolution-dependent local density anomalies located inside the non-hand-signed mode rather than between modes. This observation supports the main-text decision to use BD/McCrary as a density-smoothness diagnostic rather than as a threshold estimator and reinforces the joint reading of Section IV-D that the descriptor distributions do not contain a within-population bimodal antimode that could anchor an operational threshold.

Raw per-bin Z sequences and $p$-values for every (variant, bin-width) panel are available in the supplementary materials.

A.2. Diagnostic Summary

Section III-M positions the unsupervised-diagnostic strategy as a set of complementary checks, each addressing one specific failure mode of an unsupervised screening classifier with an explicitly disclosed untested assumption. Table A.II maps each diagnostic to the failure mode it addresses and to the untested assumption it relies on.

Table A.II. Diagnostics, failure mode addressed, and disclosed untested assumption.

Diagnostic	Failure mode addressed	Disclosed untested assumption
Composition decomposition (§III-I.4; Scripts 39b–39e)	Whether descriptor multimodality is within-population (mechanism) or between-group (composition + integer artefact); p_{\text{median}} = 0.35 under joint firm-mean centring + integer-tie jitter	Integer-tie jitter and firm-mean centring are unbiased over the descriptor support; corroborated by Big-4 per-firm jitter (Script 39d; per-firm dHash rejection disappears under jitter at every Big-4 firm) and Big-4 pooled centred + jittered (n_{\text{seeds}} = 5; Script 39e)
Per-comparison inter-CPA coincidence rate (§III-L.1; Script 40b)	Pair-level specificity proxy under a random-pair negative anchor	Inter-CPA pairs are negative (i.e., not template-related); partially violated by within-firm sharing (§III-L.4)
Pool-normalised per-signature ICCR (§III-L.2; Script 43)	Deployed-rule specificity proxy at per-signature unit, accounting for pool size	Same as above + that pool replacement preserves the negative-anchor property
Document-level ICCR (§III-L.3; Script 45)	Operational alarm rate proxy at per-document unit under three alarm definitions	Same as above
Firm-heterogeneity logistic regression (§III-L.4; Script 44)	Multiplicative effect of firm membership on per-signature rate, controlling for pool size	Per-signature observations are clustered by CPA/firm; naïve standard errors unreliable; cluster-robust analysis is a future check
Cross-firm hit matrix (§III-L.4; Script 44)	Concentration of inter-CPA collisions within source firm	Concentration depends on deployed-rule semantics (the stricter same-pair joint event yields $97.0$–99.96\% within-firm at all four firms versus $76.7$–98.8\% under any-pair; §III-L.4); per-document per-firm assignment uses Script 45's mode-of-firms tie-break (§IV-M.4)
Alert-rate sensitivity sweep (§III-L.5; Script 46)	Local sensitivity of deployed rule to threshold perturbation	Gradient comparison is descriptive, not a formal plateau test
Convergent score Spearman ranking (§III-K.1; Script 38)	Internal-consistency of three feature-derived per-CPA scores	Scores share underlying inputs and are not statistically independent
Pixel-identical conservative positive capture (§III-K.4; Script 40)	Trivial sanity check on the conservative positive anchor	Anchor is tautologically captured by any reasonable threshold
LOOO firm-level reproducibility (§III-K.3; Scripts 36, 37)	Algorithmic stability of K=2 / K=3 partition across firm folds	Stability is necessary but not sufficient for classification validity

Appendix B. Reproducibility Materials

The full table-to-script provenance mapping, script source code, and report artefacts for every numerical table and figure in this paper are provided in the supplementary materials. Scripts run deterministically under fixed random seeds documented there; reviewer reproduction should re-emit artefacts from the listed scripts rather than rely on any local path layout.

References

[1] Taiwan Certified Public Accountant Act (會計師法), Art. 4; FSC Attestation Regulations (查核簽證核准準則), Art. 6. Available: https://law.moj.gov.tw/ENG/LawClass/LawAll.aspx?pcode=G0400067

[2] S.-H. Yen, Y.-S. Chang, and H.-L. Chen, "Does the signature of a CPA matter? Evidence from Taiwan," Res. Account. Regul., vol. 25, no. 2, pp. 230–235, 2013.

[3] J. Bromley et al., "Signature verification using a Siamese time delay neural network," in Proc. NeurIPS, 1993.

[4] S. Dey et al., "SigNet: Convolutional Siamese network for writer independent offline signature verification," arXiv:1707.02131, 2017.

[5] H.-H. Kao and C.-Y. Wen, "An offline signature verification and forgery detection method based on a single known sample and an explainable deep learning approach," Appl. Sci., vol. 10, no. 11, p. 3716, 2020.

[6] H. Li et al., "TransOSV: Offline signature verification with transformers," Pattern Recognit., vol. 145, p. 109882, 2024.

[7] S. Tehsin et al., "Enhancing signature verification using triplet Siamese similarity networks in digital documents," Mathematics, vol. 12, no. 17, p. 2757, 2024.

[8] P. Brimoh and C. C. Olisah, "Consensus-threshold criterion for offline signature verification using CNN learned representations," arXiv:2401.03085, 2024.

[9] N. Woodruff et al., "Fully-automatic pipeline for document signature analysis to detect money laundering activities," arXiv:2107.14091, 2021.

[10] S. Abramova and R. Böhme, "Detecting copy-move forgeries in scanned text documents," in Proc. Electronic Imaging, 2016.

[11] Y. Li et al., "Copy-move forgery detection in digital image forensics: A survey," Multimedia Tools Appl., 2024.

[12] Y. Jakhar and M. D. Borah, "Effective near-duplicate image detection using perceptual hashing and deep learning," Inf. Process. Manage., p. 104086, 2025.

[13] E. Pizzi et al., "A self-supervised descriptor for image copy detection," in Proc. CVPR, 2022.

[14] L. G. Hafemann, R. Sabourin, and L. S. Oliveira, "Learning features for offline handwritten signature verification using deep convolutional neural networks," Pattern Recognit., vol. 70, pp. 163–176, 2017.

[15] E. N. Zois, D. Tsourounis, and D. Kalivas, "Similarity distance learning on SPD manifold for writer independent offline signature verification," IEEE Trans. Inf. Forensics Security, vol. 19, pp. 1342–1356, 2024.

[16] L. G. Hafemann, R. Sabourin, and L. S. Oliveira, "Meta-learning for fast classifier adaptation to new users of signature verification systems," IEEE Trans. Inf. Forensics Security, vol. 15, pp. 1735–1745, 2020.

[17] H. Farid, "Image forgery detection," IEEE Signal Process. Mag., vol. 26, no. 2, pp. 16–25, 2009.

[18] F. Z. Mehrjardi, A. M. Latif, M. S. Zarchi, and R. Sheikhpour, "A survey on deep learning-based image forgery detection," Pattern Recognit., vol. 144, art. no. 109778, 2023.

[19] J. Luo et al., "A survey of perceptual hashing for multimedia," ACM Trans. Multimedia Comput. Commun. Appl., vol. 21, no. 7, 2025.

[20] D. Engin et al., "Offline signature verification on real-world documents," in Proc. CVPRW, 2020.

[21] D. Tsourounis et al., "From text to signatures: Knowledge transfer for efficient deep feature learning in offline signature verification," Expert Syst. Appl., vol. 189, art. 116136, 2022.

[22] B. Chamakh and O. Bounouh, "A unified ResNet18-based approach for offline signature classification and verification across multilingual datasets," Procedia Comput. Sci., vol. 270, pp. 4024–4033, 2025.

[23] A. Babenko, A. Slesarev, A. Chigorin, and V. Lempitsky, "Neural codes for image retrieval," in Proc. ECCV, 2014, pp. 584–599.

[24] S. Bai, K. Chen, X. Liu, J. Wang, W. Ge, S. Song, K. Dang, P. Wang, S. Wang, J. Tang, H. Zhong, Y. Zhu, M. Yang, Z. Li, J. Wan, P. Wang, W. Ding, Z. Fu, Y. Xu, J. Ye, X. Zhang, T. Xie, Z. Cheng, H. Zhang, Z. Yang, H. Xu, and J. Lin, "Qwen2.5-VL technical report," arXiv:2502.13923, 2025. [Online]. Available: https://arxiv.org/abs/2502.13923

[25] Ultralytics, "YOLO11 documentation," 2024. [Online]. Available: https://docs.ultralytics.com/models/yolo11/

[26] K. He, X. Zhang, S. Ren, and J. Sun, "Deep residual learning for image recognition," in Proc. CVPR, 2016.

[27] N. Krawetz, "Kind of like that," The Hacker Factor Blog, 2013. [Online]. Available: https://www.hackerfactor.com/blog/index.php?/archives/529-Kind-of-Like-That.html

[28] B. W. Silverman, Density Estimation for Statistics and Data Analysis. London: Chapman & Hall, 1986.

[29] J. Cohen, Statistical Power Analysis for the Behavioral Sciences, 2nd ed. Hillsdale, NJ: Lawrence Erlbaum, 1988.

[30] Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, "Image quality assessment: From error visibility to structural similarity," IEEE Trans. Image Process., vol. 13, no. 4, pp. 600–612, 2004.

[31] J. V. Carcello and C. Li, "Costs and benefits of requiring an engagement partner signature: Recent experience in the United Kingdom," The Accounting Review, vol. 88, no. 5, pp. 1511–1546, 2013.

[32] A. D. Blay, M. Notbohm, C. Schelleman, and A. Valencia, "Audit quality effects of an individual audit engagement partner signature mandate," Int. J. Auditing, vol. 18, no. 3, pp. 172–192, 2014.

[33] W. Chi, H. Huang, Y. Liao, and H. Xie, "Mandatory audit partner rotation, audit quality, and market perception: Evidence from Taiwan," Contemp. Account. Res., vol. 26, no. 2, pp. 359–391, 2009.

[34] J. Redmon, S. Divvala, R. Girshick, and A. Farhadi, "You only look once: Unified, real-time object detection," in Proc. CVPR, 2016, pp. 779–788.

[35] J. Zhang, J. Huang, S. Jin, and S. Lu, "Vision-language models for vision tasks: A survey," IEEE Trans. Pattern Anal. Mach. Intell., vol. 46, no. 8, pp. 5625–5644, 2024.

[36] H. B. Mann and D. R. Whitney, "On a test of whether one of two random variables is stochastically larger than the other," Ann. Math. Statist., vol. 18, no. 1, pp. 50–60, 1947.

[37] J. A. Hartigan and P. M. Hartigan, "The dip test of unimodality," Ann. Statist., vol. 13, no. 1, pp. 70–84, 1985.

[38] D. Burgstahler and I. Dichev, "Earnings management to avoid earnings decreases and losses," J. Account. Econ., vol. 24, no. 1, pp. 99–126, 1997.

[39] J. McCrary, "Manipulation of the running variable in the regression discontinuity design: A density test," J. Econometrics, vol. 142, no. 2, pp. 698–714, 2008.

[40] A. P. Dempster, N. M. Laird, and D. B. Rubin, "Maximum likelihood from incomplete data via the EM algorithm," J. R. Statist. Soc. B, vol. 39, no. 1, pp. 1–38, 1977.

[41] H. White, "Maximum likelihood estimation of misspecified models," Econometrica, vol. 50, no. 1, pp. 1–25, 1982.

[42] M. Stone, "Cross-validatory choice and assessment of statistical predictions," J. R. Statist. Soc. B, vol. 36, no. 2, pp. 111–147, 1974.

[43] S. Geisser, "The predictive sample reuse method with applications," J. Amer. Statist. Assoc., vol. 70, no. 350, pp. 320–328, 1975.

[44] A. Vehtari, A. Gelman, and J. Gabry, "Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC," Stat. Comput., vol. 27, no. 5, pp. 1413–1432, 2017.

Declarations

Conflict of interest. The authors declare no conflict of interest with Firm A, Firm B, Firm C, or Firm D, or with any other entity referenced in this work.

Data availability. All audit reports analysed in this study were obtained from the Market Observation Post System (MOPS) operated by the Taiwan Stock Exchange Corporation, a publicly accessible regulatory disclosure platform. The CPA registry used to map signatures to certifying CPAs is publicly available. Signature images, model weights, and reproducibility scripts are available in the supplementary materials.