# Section III. Methodology — v4.0 Draft v3 (post codex rounds 21 + 22)

> **Draft note (2026-05-12, v3).** This file replaces the §III-G through §III-L block of `paper/paper_a_methodology_v3.md` (v3.20.0). Sub-sections III-A through III-F (Pipeline / Data Collection / Page Identification / Detection / Feature Extraction / Dual-Method Descriptors) are unchanged from v3.20.0 and not reproduced here.
>
> **v2** incorporated codex gpt-5.5 round-21 review (`paper/codex_review_gpt55_v4_round1.md`, Major Revision); key revisions were: (i) the inherited five-way per-signature box rule restored as the **primary operational classifier** (§III-L), (ii) the K=3 Gaussian mixture positioned as **accountant-level descriptive characterisation** (§III-J), (iii) "convergent validation" softened to "convergent internal-consistency checks" since the three scores share underlying features (§III-K), (iv) the pixel-identity metric renamed from FAR to positive-anchor miss rate (§III-K), (v) five empirical/wording slips corrected.
>
> **v3** incorporates codex gpt-5.5 round-22 review (`paper/codex_review_gpt55_v4_round2.md`, Minor Revision); five narrow fixes applied: (1) §III-K per-firm ranking corrected to acknowledge that Score 2 (reverse-anchor) ranks Firm D fractionally above Firm C while Scores 1 and 3 rank Firm C highest; (2) "smallest scope" / "any single firm" language narrowed to "comparison scopes tested in Script 32"; (3) §III-L corrected so the §III-K Spearman correlations are explicitly tied to the K=3 *posterior* P(C1) rather than the K=3 hard label; (4) provenance table source for $n = 150{,}442$ updated to cite Script 39 directly; (5) "max fold-to-fold deviation" wording made precise — the $0.028$ value is the maximum absolute deviation from the across-fold mean, not the pairwise across-fold range ($0.0376$).
>
> Empirical anchors throughout reference Scripts 32–40 on branch `paper-a-v4-big4`; a provenance table appears at the end of this section listing every numerical claim with its script and report path.

## G. Unit of Analysis and Scope

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

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

We adopt one stipulation about same-CPA pair detectability:

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

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

**Scope: the Big-4 sub-corpus.** v4.0's primary analyses (§III-I, §III-J, §III-K, all §IV results except §IV-K) are restricted to the four largest accounting firms in Taiwan, pseudonymously labelled Firm A through Firm D throughout the manuscript. The Big-4 sub-corpus comprises 437 CPAs (171 / 112 / 102 / 52 across Firms A through D) with $n_{\text{sig}} \geq 10$ (Scripts 36, 38), totalling 150,442 Big-4 signatures with both descriptors available (Script 39 reports the explicit per-signature $n$ used in the signature-level K=3 fit). Restricting the analyses to Big-4 is a methodological choice driven by four considerations:

1. **Within-pool homogeneity for mixture characterisation.** Pooling Big-4 with mid- and small-firm CPAs introduces a heterogeneous tail of $\sim$249 CPAs distributed across multiple firms with idiosyncratic signing practices and small per-firm samples. The full-sample and Big-4-only calibrations *differ* in their fitted marginal crossings (full-sample published $\overline{\text{cos}}^* = 0.945$, $\overline{\text{dHash}}^* = 8.10$ from v3.x; Big-4-only $\overline{\text{cos}}^* = 0.975$, $\overline{\text{dHash}}^* = 3.76$ from Script 34; bootstrap 95% CIs $[0.974, 0.977]$ / $[3.48, 3.97]$, $n_{\text{boot}} = 500$); the offset is large compared to the Big-4 bootstrap CI half-width of $0.0015$. We report this as a *scope-dependent shift* rather than asserting a causal "mid/small-firm tail distorts" claim.

2. **Statistical multimodality at the accountant level.** Within the Big-4 sub-corpus, the Hartigan dip test rejects unimodality on both axes (cosine $p = 0.0000$, dHash $p = 0.0000$ in the bootstrap-2000 implementation, i.e., no bootstrap replicate exceeded the observed statistic; reported here as $p < 5 \times 10^{-4}$; Script 34). No such rejection holds in the comparison scopes tested by Script 32: Firm A pooled alone ($p_{\text{cos}} = 0.992$, $p_{\text{dHash}} = 0.924$, $n = 171$); Firms B + C + D pooled (Script 32, `big4_non_A` subset: $p_{\text{cos}} = 0.998$, $p_{\text{dHash}} = 0.906$); all non-Firm-A pooled (Script 32, `all_non_A` subset: other Big-4 plus mid/small firms; $p_{\text{cos}} = 0.998$, $p_{\text{dHash}} = 0.907$). Among the comparison scopes we evaluated, Big-4 is the smallest scope at which the dip test supports applying a finite-mixture model to the per-CPA distribution; we did not separately test single-firm dip statistics for Firms B, C, or D.

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

4. **Restricted generalizability claim.** v4.0's primary claims are scoped to the Big-4 audit-report context; we do not assert that the same mixture structure or operational thresholds extend to mid/small firms. The 249 non-Big-4 CPAs enter only (a) as an external reference population in §III-H's reverse-anchor internal-consistency check and (b) as a robustness comparison in §IV-K. Generalisation beyond Big-4 is left as future work.

## H. Reference Populations

v4.0 distinguishes two reference populations in its calibration, replacing v3.x's single-anchor framing.

**Internal reference: Firm A as the templated-end case study.** Firm A is empirically the most digitally-replicated of the Big-4. In the Big-4 K=3 mixture (§III-J; Scripts 35, 38), Firm A accounts for 0% of the C1 hand-leaning component (cos $\approx 0.946$, dHash $\approx 9.17$, weight $\approx 0.143$), 17.5% of the C2 mixed component, and 82.5% of the C3 replicated component; the opposite pattern holds at Firm C (Script 35: 23.5% C1, 75.5% C2, 1.0% C3, hereafter referred to as "the Firm whose CPAs are most concentrated in C1"). The byte-level pair analysis reported in v3.x §IV-F.1 identifies 145 Firm A pixel-identical signatures at the signature level (Script 40 verifies the 145/262 split among Big-4 pixel-identical signatures); the additional details that v3.x attributes to this analysis (50 distinct Firm A partners of 180 registered; 35 byte-identical matches spanning different fiscal years) are inherited from the Script 28 / Appendix B byte-decomposition output and were not regenerated in the v4.0 spike scripts. We retain those v3.x details by reference and mark them in the provenance table as "inherited from v3 §IV-F.1 / Script 28."

In v4.0, Firm A is *not* the calibration anchor for the operational threshold. Firm A enters the Big-4 mixture on equal footing with Firms B through D; the K=3 components are derived from the joint Big-4 distribution (§III-J), not from Firm A alone. Firm A's role in the methodology is descriptive: it is the Big-4 firm whose CPAs are most concentrated in the high-cosine, low-dHash corner of the descriptor plane, and the byte-level pair evidence above provides the firm-level signature-reuse evidence that anchors §III-K's pixel-identity positive-anchor miss rate.

**External reference: non-Big-4 as the reverse-anchor reference for internal-consistency checking.** The 249 non-Big-4 CPAs ($n_{\text{sig}} \geq 10$, drawn from $\sim$30 mid- and small-firms) constitute a population strictly outside the Big-4 target. Their per-CPA $(\overline{\text{cos}}_a, \overline{\text{dHash}}_a)$ distribution defines a 2D Gaussian reference (fit by Minimum Covariance Determinant with support fraction 0.85 for robustness; Script 38). This reference is used in §III-K's reverse-anchor internal-consistency check: each Big-4 CPA's location relative to the reference centre, measured as the marginal cosine cumulative-distribution-function value under the reference, is one of three feature-derived scores that v4.0 uses as a cross-check on the inherited per-signature classifier. The reverse-anchor reference is *not* a positive or negative anchor for threshold derivation — its role is to provide a strictly out-of-target benchmark against which the within-Big-4 mixture-derived ranking can be internally cross-checked.

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

## I. Distributional Characterisation at the Accountant Level

This section characterises the joint distribution of accountant-level descriptor means $(\overline{\text{cos}}_a, \overline{\text{dHash}}_a)$ across the 437 Big-4 CPAs of §III-G. Three diagnostic procedures are applied: a univariate unimodality test on each marginal axis, a 2D Gaussian mixture fit (developed in §III-J), and a density-smoothness diagnostic.

**1. Hartigan dip test on each marginal.** We apply the Hartigan & Hartigan dip test [37] to each of the two marginal distributions $\{\overline{\text{cos}}_a\}_{a=1}^{437}$ and $\{\overline{\text{dHash}}_a\}_{a=1}^{437}$, with bootstrap-based $p$-value estimation ($n_{\text{boot}} = 2000$). In both cases no bootstrap replicate exceeded the observed dip statistic, so the empirical $p$-value is bounded above by $5 \times 10^{-4}$; we report this in tables as $p < 5 \times 10^{-4}$ rather than $p = 0$ to reflect the bootstrap resolution (Script 34). For comparison, no rejection of unimodality holds in the comparison scopes tested in Script 32: Firm A pooled alone ($p_{\text{cos}} = 0.992$, $p_{\text{dHash}} = 0.924$, $n = 171$); Firms B + C + D pooled ($p_{\text{cos}} = 0.998$, $p_{\text{dHash}} = 0.906$, $n = 266$); all non-Firm-A CPAs pooled ($p_{\text{cos}} = 0.998$, $p_{\text{dHash}} = 0.907$, $n = 515$). Single-firm dip tests for Firms B, C, and D were not separately computed; the comparison scopes above sufficed to establish that no narrower-than-Big-4 *tested* scope rejected unimodality. The dip-test multimodality at the Big-4 level is the empirical justification for fitting a finite-mixture model in §III-J; without it, the mixture would be a forced fit on an essentially unimodal distribution.

**2. Mixture-model evidence.** A 2-component 2D Gaussian Mixture Model (full covariance, $n_{\text{init}} = 15$, fixed seed 42; Script 34) recovers components at $(\overline{\text{cos}}, \overline{\text{dHash}}) = (0.954, 7.14)$, weight $0.689$, and $(0.983, 2.41)$, weight $0.311$. The marginal crossings of the K=2 fit are $\overline{\text{cos}}^* = 0.9755$ and $\overline{\text{dHash}}^* = 3.755$, with bootstrap 95% confidence intervals $[0.9742, 0.9772]$ and $[3.48, 3.97]$ over $n_{\text{boot}} = 500$ resamples. The 3-component fit (§III-J) is BIC-preferred — using the convention that lower BIC is preferred, $\text{BIC}(K{=}3) - \text{BIC}(K{=}2) = -3.48$ (Script 36). The $\Delta$BIC magnitude is small in absolute terms; we do not treat $\Delta\text{BIC} = 3.5$ alone as decisive evidence for K=3, and the operational role of each fit is developed in §III-J and §III-K.

**3. Burgstahler-Dichev / McCrary density-smoothness diagnostic.** We apply the discontinuity test of [38, 39] as a *density-smoothness diagnostic* (rather than as a threshold estimator) on each marginal axis (cosine in bins of $0.002$, dHash in integer bins). At the Big-4 scope, the diagnostic identifies no significant transition on either marginal at $\alpha = 0.05$ (Script 34). Outside Big-4, the diagnostic does flag dHash transitions in some subsets (Script 32: `big4_non_A` dHash transition at $10.8$; `all_non_A` dHash transition at $6.6$; pre-2018 and post-2020 time-stratified variants also exhibit one or more dHash transitions), but no cosine transition is identified in any subset. The Big-4-scope null on both axes is consistent with the mixture-model evidence: the K=3 components overlap rather than separate sharply, so a local-discontinuity test does not flag a transition. We retain BD/McCrary in v4.0 as a non-parametric robustness diagnostic; the dHash transitions outside Big-4 are not used as operational thresholds because they are scope-dependent and lie within rather than between modes of the corresponding density.

## J. Mixture Model and Accountant-Level Characterisation

This section develops the K=2 and K=3 Gaussian mixture fits to the Big-4 accountant-level distribution and clarifies their role. **Both fits are descriptive characterisations of the joint Big-4 distribution; the operational per-signature classifier remains the inherited five-way box rule of §III-L.** Neither mixture is used to assign signature-level or document-level labels in the v4.0 primary analysis.

**K=2 fit.** Two components at $(\overline{\text{cos}}, \overline{\text{dHash}}) = (0.954, 7.14)$ ("hand-leaning"), weight $0.689$, and $(0.983, 2.41)$ ("replicated"), weight $0.311$ (Script 34). $\text{BIC}(K{=}2) = -1108.45$. Marginal crossings: $\overline{\text{cos}}^* = 0.9755$, $\overline{\text{dHash}}^* = 3.755$.

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

| Component | $\overline{\text{cos}}$ | $\overline{\text{dHash}}$ | weight | descriptive label |
|---|---|---|---|---|
| C1 | 0.9457 | 9.17 | 0.143 | hand-leaning |
| C2 | 0.9558 | 6.66 | 0.536 | mixed |
| C3 | 0.9826 | 2.41 | 0.321 | replicated |

$\text{BIC}(K{=}3) = -1111.93$, lower than $K{=}2$ by $3.48$ (mild support for K=3 under standard BIC interpretation but not by itself decisive).

**Why we report both K=2 and K=3.** Leave-one-firm-out cross-validation (§III-K) shows that K=2 is unstable across folds: holding Firm A out gives a fold rule cos $> 0.938$ AND dHash $\leq 8.79$, while holding any single non-Firm-A Big-4 firm out gives a fold rule near cos $> 0.975$ AND dHash $\leq 3.76$ (Script 36). The maximum absolute deviation of the four fold cosine crossings from their across-fold mean is $0.028$ (the corresponding pairwise across-fold range is $0.0376$, from $0.9380$ for the held-out-Firm-A fold to $0.9756$ for the held-out-EY fold; Script 36 stability summary). The $0.028$ value is $5.6\times$ the report's $0.005$ across-fold *stability tolerance* — not the bootstrap CI; the full-Big-4 bootstrap cosine half-width is the much smaller $0.0015$. K=3 in contrast has a *reproducible component shape*: across the four folds the C1 cosine mean varies by at most $0.005$, the C1 dHash mean by at most $0.96$, and the C1 weight by at most $0.025$ (Script 37). K=3 hard-posterior membership for the held-out firm is more composition-sensitive — for Firm C the held-out C1 rate is $36.3\%$ vs the full-Big-4 baseline of $23.5\%$, an absolute difference of $12.8$ pp; for Firm A the held-out C1 rate is $4.7\%$ vs baseline $0.0\%$; the report's own legend classifies this pattern as `P2_PARTIAL`, with the explicit interpretation that "the C1 cluster exists but membership is not well-predicted by the held-out fit" and "K=3 is not predictively useful as an operational classifier" (Script 37 verdict legend).

We take the joint K=2/K=3 LOOO evidence as supporting the following descriptive claims, all of which are used in §III-K and §V but none of which underwrites a v4.0 operational classifier:

- The Big-4 K=2 marginal crossing $(0.975, 3.76)$ is essentially a firm-mass separator between Firm A and Firms B + C + D, not a within-Big-4 mechanism boundary.
- The Big-4 K=3 mixture exhibits a reproducible three-component shape across LOOO folds; a hand-leaning component (C1) exists at $\overline{\text{cos}} \approx 0.946$, $\overline{\text{dHash}} \approx 9.17$ with weight $\approx 0.14$.
- Hard-posterior membership in C1 is composition-sensitive (max absolute deviation $12.8$ pp across LOOO folds, exceeding the report's $5$ pp viability bar); K=3 is therefore not used to assign operational hand-leaning labels to CPAs in v4.0.

The operational signature-level classifier remains the inherited five-way box rule of §III-L, calibrated as in v3.x. Cross-checks between the inherited rule and the K=3 mixture appear in §III-K.

## K. Convergent Internal-Consistency Checks

The mixture characterisation of §III-J is supported by three feature-derived per-CPA scores and a hard-ground-truth subset analysis. We caution at the outset that the three scores are **not statistically independent measurements** — all three are deterministic functions of the same per-CPA descriptor means $(\overline{\text{cos}}_a, \overline{\text{dHash}}_a)$ — so their high pairwise rank correlations are partly a mechanical consequence of shared inputs. The checks below therefore document **internal consistency among feature-derived ranks**, not external validation against an independent hand-signed ground truth (which the corpus does not provide).

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

- **Score 1 (mixture posterior):** $P(\text{C1}_{\text{hand-leaning}})$ from the K=3 fit of §III-J — a function of both descriptor means.
- **Score 2 (reverse-anchor cosine percentile):** the marginal cosine CDF value of $\overline{\text{cos}}_a$ under the non-Big-4 reference Gaussian of §III-H, sign-flipped so that lower percentile (further into the reference's left tail) corresponds to a more hand-leaning Big-4 CPA. This is a function of $\overline{\text{cos}}_a$ alone.
- **Score 3 (Paper A box-rule hand-leaning rate):** the per-CPA fraction of signatures that do **not** satisfy the inherited binary high-confidence box rule (cos $> 0.95$ AND dHash $\leq 5$). This is a per-signature-aggregated function of the same descriptors.

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

| Pair | Spearman $\rho$ | $p$-value |
|---|---|---|
| Score 1 vs Score 3 | $+0.963$ | $< 10^{-248}$ |
| Score 2 vs Score 3 | $+0.889$ | $< 10^{-149}$ |
| Score 1 vs Score 2 | $+0.879$ | $< 10^{-142}$ |

We read this as the strongest internal-consistency signal in v4.0: three different summarisations of the same descriptor pair agree on the per-CPA hand-leaning ranking with $\rho > 0.87$. The three scores agree on placing Firm A as the most replication-dominated and the three non-Firm-A Big-4 firms as more hand-leaning, but they do not all rank the non-Firm-A firms identically: the K=3 posterior P(C1) and the box-rule hand-leaning rate (Scores 1 and 3) place Firm C at the most-hand-leaning end of Big-4 (mean P(C1) $= 0.311$; mean box-rule hand-leaning rate $= 0.790$), while the reverse-anchor cosine percentile (Score 2) places Firm D fractionally higher than Firm C (mean reverse-anchor score $-0.7125$ vs Firm C $-0.7672$, with higher value indicating deeper into the reference left tail). The mean values for Firms B (KPMG) and D (EY) sit between Firms A and C on Scores 1 and 3 (Script 38 per-firm summary). We do not claim this constitutes external validation of any operational classifier; the inherited box rule is calibrated separately (§III-L), and the convergence above shows that a mixture-derived score and a reverse-anchor score concur with the box rule's per-CPA-aggregated outputs on the directional ordering, with a modest disagreement at the most-hand-leaning end between the three non-A Big-4 firms.

**2. Per-signature consistency (Script 39).** Per-CPA aggregation could in principle reflect averaging across within-CPA heterogeneity rather than coherent within-CPA behaviour. We test this by repeating the K=3 fit at the signature level — fitting a fresh K=3 GMM to the 150,442 Big-4 signature-level $(\text{cos}, \text{dHash}_{\text{indep}})$ points (Script 39) — and comparing labels. The per-CPA and per-signature K=3 fits recover a broadly similar three-component ordering; per-CPA C1 is at $\overline{\text{cos}} = 0.946$, $\overline{\text{dHash}} = 9.17$ vs per-signature C1 at $\overline{\text{cos}} = 0.928$, $\overline{\text{dHash}} = 9.75$ (an absolute cosine drift of $0.018$). Cohen $\kappa$ on the binary collapse (replicated vs not-replicated):

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

The Script 39 report verdict is `SIG_CONVERGENCE_MODERATE`. The $\kappa = 0.870$ between per-CPA-fit and per-signature-fit K=3 binary labels indicates that per-CPA aggregation does not collapse the broad three-component ordering. The lower $\kappa = 0.56\text{–}0.66$ between the binary box rule and either K=3 fit is consistent with two factors: different decision geometries (rectangular box vs Gaussian-mixture posterior boundary), and the fact that the binary box rule is a strict subset of the inherited five-way rule. We note that this comparison validates only the binary high-confidence rule (cos $> 0.95$ AND dHash $\leq 5$); §III-K does not directly validate the five-way rule's `5 < \text{dHash} \leq 15` moderate-confidence band, which retains its v3.x interpretation and capture-rate evaluation (§IV-F).

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

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

- *K=3 LOOO is partially stable.* The C1 hand-leaning component shape is reproducible across folds: max deviation from the full-Big-4 baseline is $0.005$ in cosine, $0.96$ in dHash, and $0.025$ in mixture weight (Script 37). Hard-posterior membership remains composition-sensitive — observed absolute differences are $1.8$–$12.8$ pp across the four folds, with the Firm C fold exceeding the report's $5$ pp viability bar; the report's own verdict is `P2_PARTIAL` ("K=3 is not predictively useful as an operational classifier"). We accordingly do not use K=3 hard-posterior membership as an operational label.

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

We report each candidate classifier's *positive-anchor miss rate* — the fraction of byte-identical signatures misclassified as hand-leaning. This is a one-sided check against a conservative positive subset, **not a false-alarm rate in the usual two-class sense**; we do not report a paired false-alarm rate because no signature-level hand-signed ground truth exists. The corresponding signature-level false-alarm-rate evidence comes from the v3.x inter-CPA negative anchor (§III-J inherited; Table X), which retains its v3.x interpretation:

| Candidate classifier | Pixel-identity miss rate (Wilson 95% CI) |
|---|---|
| Inherited Paper A binary high-confidence box rule (cos $> 0.95$ AND dHash $\leq 5$) | $0\%$ $[0\%, 1.45\%]$ |
| K=3 per-CPA hard label (C3 = replicated; descriptive only) | $0\%$ $[0\%, 1.45\%]$ |
| Reverse-anchor with prevalence-calibrated cut | $0\%$ $[0\%, 1.45\%]$ |

All three candidate scores correctly assign every byte-identical signature to the replicated class. We caution that for the inherited box rule this result is close to tautological: byte-identical nearest-neighbour signatures have cosine $\approx 1$ and dHash $\approx 0$ by construction, so any threshold strictly below cos $= 1$ and strictly above dHash $= 0$ will capture them; v3.x discussed this conservative-subset caveat at length (v3 §III-J item 1, V-F). The positive-anchor miss rate is therefore a necessary check (a classifier that *failed* this check would be disqualified), not a sufficient validation of the classifier's behaviour on the non-byte-identical replicated population. The reverse-anchor cut here is chosen by prevalence calibration against the inherited box rule's overall replicated rate ($49.58\%$ of Big-4 signatures; Script 40); this is a documented v4.0 limitation since no signature-level hand-signed ground truth exists to permit direct ROC optimisation.

## L. Signature- and Document-Level Classification

The v4.0 operational classifier is the inherited v3.x five-way per-signature box rule, retained unchanged for two reasons: (a) it preserves continuity with the v3.x literature and its established interpretation, and (b) the convergent internal-consistency checks of §III-K show that the box rule's per-CPA-aggregated outputs agree at $\rho \geq 0.96$ with a mixture-derived score and at $\rho \geq 0.89$ with a reverse-anchor score, supporting continued use without recalibration.

**Per-signature five-way classifier.** Operational thresholds are anchored on whole-sample Firm A percentile heuristics as in v3.x: cos $> 0.95$ (Firm A P7.5) for the cosine dimension and dHash$_{\text{indep}} \leq 5$ / $> 15$ for the structural dimension. All dHash references refer to the *independent-minimum* dHash defined in §III-G. We assign each signature to one of five signature-level categories using convergent evidence from both descriptors:

1. **High-confidence non-hand-signed:** Cosine $> 0.95$ AND $\text{dHash}_{\text{indep}} \leq 5$. Both descriptors converge on strong replication evidence.

2. **Moderate-confidence non-hand-signed:** Cosine $> 0.95$ AND $5 < \text{dHash}_{\text{indep}} \leq 15$. Feature-level evidence is strong; structural similarity is present but below the high-confidence cutoff.

3. **High style consistency:** Cosine $> 0.95$ AND $\text{dHash}_{\text{indep}} > 15$. High feature-level similarity without structural corroboration — consistent with a CPA who signs very consistently but not via image reproduction.

4. **Uncertain:** Cosine between the all-pairs intra/inter KDE crossover (0.837) and 0.95 without sufficient convergent evidence in either direction.

5. **Likely hand-signed:** Cosine below the all-pairs KDE crossover threshold.

The conventions about these thresholds (cosine 0.95 as an operating point chosen for capture-vs-FAR tradeoff against the inter-CPA negative anchor, with Wilson 95% inter-CPA FAR of $0.0005$ at the operating point; cosine 0.837 as the all-pairs KDE crossover; dHash 5 and 15 as the upper tail of the high-similarity mode and the structural-similarity ceiling respectively) are inherited from v3.x §III-K and retain their v3.x calibration and capture-rate evidence (v3.x Tables IX, XI, XII, XII-B).

**Document-level aggregation.** Each audit report typically carries two certifying-CPA signatures. We aggregate signature-level outcomes to document-level labels using the v3.x worst-case rule: the document inherits the *most-replication-consistent* signature label among the two signatures (rank order: High-confidence $>$ Moderate-confidence $>$ Style-consistency $>$ Uncertain $>$ Likely-hand-signed). The aggregation rule reflects the detection goal of flagging any potentially non-hand-signed report.

**K=3 as accountant-level characterisation, not classifier.** The K=3 mixture of §III-J is reported in §IV as an accountant-level descriptive summary alongside the per-signature five-way classifier. We do not assign signature-level or document-level labels from the K=3 mixture in any v4.0 result table; the K=3 hard label is used for the accountant-level cluster cross-tabulation (Script 35), and the K=3 *posterior* P(C1) is used (as the continuous Score 1) in the internal-consistency Spearman correlations of §III-K.

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## Provenance table for numerical claims in §III-G through §III-L

| Claim | Value | Source | Notes |
|---|---|---|---|
| Big-4 CPA count, $n_{\text{sig}} \geq 10$ | $437$ ($171/112/102/52$) | Script 36 sample sizes; Script 38 per-firm summary | direct |
| Big-4 signature count | $150{,}442$ | Script 39 (per-signature K=3 fit explicitly cites this $n$) | direct |
| Non-Big-4 reference CPA count | $249$ | Script 38 reference population | direct |
| Big-4 K=2 marginal crossings $(0.9755, 3.755)$ | direct | Script 34; Script 36 §A | direct |
| Bootstrap 95% CI cosine $[0.9742, 0.9772]$ | direct | Script 34; Script 36 §A | $n_{\text{boot}} = 500$ |
| Bootstrap 95% CI dHash $[3.48, 3.97]$ | direct | Script 34; Script 36 §A | $n_{\text{boot}} = 500$ |
| Bootstrap CI half-width $0.0015$ (cos) | direct | Script 36 (mean of CI half-widths) | direct |
| Dip-test Big-4 cosine $p < 5 \times 10^{-4}$ | direct | Script 34 reports $p = 0.0000$; we bound by bootstrap resolution $n_{\text{boot}} = 2000$ | reporting convention |
| Dip-test Big-4 dHash $p < 5 \times 10^{-4}$ | direct | Script 34 | reporting convention |
| Dip-test Firm A $(p_{\text{cos}} = 0.992, p_{\text{dHash}} = 0.924)$ | direct | Script 32 §`firm_A` | direct |
| Dip-test `big4_non_A` $(0.998, 0.906)$ | direct | Script 32 §`big4_non_A` | direct |
| Dip-test `all_non_A` $(0.998, 0.907)$ | direct | Script 32 §`all_non_A` | direct |
| K=3 component centers / weights | $(0.9457, 9.17, 0.143)$ / $(0.9558, 6.66, 0.536)$ / $(0.9826, 2.41, 0.321)$ | Script 35 / Script 38 | direct |
| $\Delta\text{BIC}(K{=}3, K{=}2) = -3.48$ | direct | Script 34 (BIC K=2 = $-1108.45$; Script 36 reports BIC K=3 = $-1111.93$) | direct (arithmetic) |
| K=2 LOOO max cosine deviation $0.028$ | direct | Script 36 stability summary | direct |
| K=2 LOOO Firm A held-out $171/171$ replicated | direct | Script 36 fold table | direct |
| K=3 C1 component shape drift (cos $0.005$, dHash $0.96$, weight $0.025$) | direct | Script 37 stability summary | direct |
| K=3 LOOO held-out C1 absolute differences $1.8$–$12.8$ pp | direct | Script 37 held-out prediction check | direct |
| Three-score pairwise Spearman ($0.963$, $0.889$, $0.879$) | direct | Script 38 correlations | direct |
| Per-CPA / per-signature K=3 Cohen $\kappa$ ($0.662$, $0.559$, $0.870$) | direct | Script 39 kappa table | direct |
| Per-CPA / per-signature K=3 C1 center drift $0.018$ (cosine) | derived | $\lvert 0.9457 - 0.9280 \rvert$; Script 39 components | direct |
| Pixel-identity Big-4 subset $n = 262$ ($145/8/107/2$) | direct | Script 40 sample | direct |
| Positive-anchor miss rate $0\%$ on $n = 262$ (Wilson upper $1.45\%$) | direct | Script 40 results table | direct |
| Inter-CPA FAR $0.0005$ at cos $> 0.95$ (Wilson 95% $[0.0003, 0.0007]$) | direct | v3 §IV-F.1 / Table X (inherited) | inherited from v3 |
| Firm A byte-identical $145$ pixel-identical signatures in Big-4 subset | direct | Script 40 sample breakdown | direct |
| Firm A byte-identical "50 distinct partners of 180; 35 cross-year" | inherited | v3 §IV-F.1 / Script 28 / Appendix B byte-decomposition output | **inherited from v3; not regenerated in v4.0 spike scripts** |
| Big-4 K=3 per-firm C1 hard-assignment ($0\%$ / $8.9\%$ / $23.5\%$ / $11.5\%$) | direct | Script 35 firm × cluster cross-tab | direct |

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## Cross-reference index (author working checklist; remove before submission)

- **Big-4 sub-corpus definition** (§III-G) — 437 CPAs, 150,442 signatures.
- **Reference populations** (§III-H) — Firm A as templated-end case study; non-Big-4 ($n = 249$) as reverse-anchor reference (less-replicated population).
- **Distributional characterisation** (§III-I) — Big-4 dip-test multimodality ($p < 5 \times 10^{-4}$); BD/McCrary null at Big-4 scope; mixture support.
- **K=3 components, descriptive only** (§III-J) — C1 hand-leaning, C2 mixed, C3 replicated; LOOO supports descriptive use, not operational classification.
- **Convergent internal-consistency** (§III-K) — three feature-derived scores ($\rho \geq 0.879$, not independent measurements); per-signature K=3 ($\kappa = 0.87$ vs per-CPA fit); K=2 LOOO unstable, K=3 LOOO partial; pixel-identity miss rate $0\%$ on $n = 262$.
- **Per-document classifier** (§III-L) — inherited five-way rule retained as primary; K=3 demoted to characterisation only.

## Open questions remaining for partner / reviewer

1. **Five-way rule validation against the moderate-confidence band.** §III-K's $\kappa$ evidence covers only the binary high-confidence rule (cos $> 0.95$ AND dHash $\leq 5$). The moderate-confidence band (cos $> 0.95$ AND $5 < \text{dHash} \leq 15$) inherits its v3.x calibration and capture-rate evidence (Tables IX, XI, XII). Is this inheritance sufficient, or should v4.0 add a Big-4-specific capture-rate analysis for the moderate band in §IV-F?

2. **Anonymisation of within-Big-4 firm contrasts.** §III-H states that Firm C is the firm most concentrated in C1 hand-leaning at $23.5\%$ (Script 35). The within-Big-4 ordering by hand-leaning concentration is informative for the §V discussion. v3.x reports under pseudonyms throughout. Confirm that we maintain pseudonyms consistently in §IV–V even when discussing the specific Firm C / Firm B / Firm D hand-leaning rates.

3. **Section IV table numbering.** Defer until §III final accepted by partner / reviewer; results numbering should mirror §III flow (sample/scope → mixture characterisation → convergent checks → LOOO → pixel-identity → signature/document classification → full-dataset robustness).
