# Section IV. Results — v4.0 Draft v3.3 (post codex rounds 21–34)

> **Draft note (2026-05-12, v3.2; internal — remove before submission).** This file replaces the §IV-A through §IV-H block of `paper/paper_a_results_v3.md` (v3.20.0) with the Big-4 reframed structure. Section IV expands from 8 sub-sections in v3.20.0 to 12 sub-sections in v4.0 (A through L) to mirror the §III-G..L lineage. **Table-numbering scheme**: the v4 manuscript uses Tables V through XVIII (plus Table XV-B for document-level worst-case counts) for the new v4 Big-4 results; inherited v3.x tables are cited only as "v3.20.0 Table N" with their original v3 number and are *not* renumbered into the v4 sequence. No v4 Table IV is printed; the inherited v3.20.0 Table IV (per-firm detection counts) remains a v3.x reference rather than a v4 table. **Anonymisation**: the Big-4 firms are pseudonymously labelled Firm A through Firm D throughout the manuscript body; real names are not printed in v4 tables or prose. The v3 → v3.1 → v3.2 revision history is: v3 (post round 23) made the table-numbering scheme and anonymisation policy decisions and applied 14 presentation fixes; v3.1 (post round 24) tightened the close-out checklist; v3.2 (post round 25) finalises this draft note. Empirical anchors trace to Scripts 32–42 on branch `paper-a-v4-big4`; the §III provenance table covers the methodology-side citations and §IV adds new tables for the v4.0-specific results.

## A. Experimental Setup

The signature-detection and feature-extraction pipeline (§III-A through §III-F) was executed on the full TWSE MOPS audit-report corpus (90,282 PDFs spanning 2013–2023; §III-B). Detection and embedding ran on RTX 4090 (CUDA, deterministic forward inference, fixed seed); the v4.0 statistical analyses ran on Apple Silicon (MPS / CPU). Random seeds are fixed (`SEED = 42`) across the v4.0 spike scripts 32–42 for reproducibility. The signature_analysis SQLite snapshot at `/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db` is treated as frozen; no v4.0 result re-ingests source PDFs.

The v4.0 primary analyses (§IV-D through §IV-J) are scoped to the Big-4 sub-corpus (Firms A–D, $n = 437$ CPAs with $n_{\text{sig}} \geq 10$, totalling 150,442 signatures with both descriptors available) per the methodology choice articulated in §III-G. The §IV-K Full-Dataset Robustness section reports the full-dataset (686 CPAs) variant of the K=3 mixture + Paper A box-rule Spearman analysis as a cross-scope robustness check.

## B. Signature Detection Performance

The detection metrics are inherited unchanged from v3.20.0 §IV-B. v3.20.0 reports: VLM screening identified 86,072 documents with signature pages; 12 corrupted PDFs were excluded; YOLOv11n batch inference processed the remaining 86,071 documents; 85,042 of these yielded at least one signature detection; the total extracted-signature count is 182,328 (v3.20.0 Table III). Per-firm counts of detected signatures are reported in v3.20.0 Table IV. v4.0 does not renumber the v3.x detection tables into the v4 sequence; v3.20.0 Tables III and IV are cited by their original numbers.

The Big-4 subset of the detection output yields 150,442 signatures with both descriptors (cosine and independent dHash) successfully computed; this is the per-signature population used in all §IV v4 primary analyses (§IV-D through §IV-J).

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

The all-pairs intra-vs-inter class distribution analysis (KDE crossover at $\overline{\text{cos}} = 0.837$; v3.20.0 §IV-C, v3.20.0 Table V) is inherited unchanged. This analysis was computed on the full corpus (not Big-4-restricted) and remains the source of the Uncertain / Likely-hand-signed boundary used by the §III-L five-way per-signature classifier (cosine $\leq 0.837 \Rightarrow$ Likely-hand-signed, matching Script 42's `cos <= 0.837` rule definition). v4.0 makes no scope-specific re-derivation of this boundary; the all-pairs cross-class crossover is a corpus-wide reference and is not restated as a v4.0 finding. v3.20.0 Table V is cited by its original number and is not renumbered into the v4 sequence.

## D. Big-4 Accountant-Level Distributional Characterisation

This section reports the empirical evidence for §III-I's distributional diagnostics at the Big-4 accountant level. All numbers below are direct re-statements from Scripts 32 / 34. The accountant-level dip-test rejection reported in Table V is, per §III-I.4 (Scripts 39b–39e), fully attributable to between-firm location shifts and integer mass-point artefacts rather than to within-population bimodality; the v4-new composition-decomposition diagnostics that establish this finding are tabulated in §IV-M below alongside the anchor-based ICCR calibration.

**Table V.** Hartigan dip-test results, accountant-level marginals (Big-4 primary; comparison scopes from Script 32).

| Population | $n$ CPAs | $p_{\text{cos}}$ | $p_{\text{dHash}}$ | Interpretation |
|---|---|---|---|---|
| **Big-4 pooled (primary)** | 437 | $< 5 \times 10^{-4}$ | $< 5 \times 10^{-4}$ | reject unimodality on both axes |
| Firm A pooled alone | 171 | 0.992 | 0.924 | unimodal |
| Firms B + C + D pooled | 266 | 0.998 | 0.906 | unimodal |
| All non-Firm-A pooled | 515 | 0.998 | 0.907 | unimodal |

Bootstrap implementation: $n_{\text{boot}} = 2000$; for the Big-4 cells, no bootstrap replicate exceeded the observed dip statistic, so the empirical $p$-value is bounded above by the bootstrap resolution $1 / 2000 = 5 \times 10^{-4}$ (Script 34 reports this as $p = 0.0000$; we report $p < 5 \times 10^{-4}$ to reflect the resolution). Single-firm dip statistics for Firms B, C, and D were not separately computed.

**Table VI.** Burgstahler-Dichev / McCrary density-smoothness diagnostic on accountant-level marginals (cosine in 0.002 bins; dHash in integer bins; $\alpha = 0.05$, two-sided).

| Population | Cosine: significant transition? | dHash: significant transition? |
|---|---|---|
| **Big-4 pooled (primary)** | none ($p > 0.05$) | none ($p > 0.05$) |
| Firm A pooled alone | none | none |
| Firms B + C + D pooled | none | one transition at $\overline{\text{dHash}} = 10.8$ |
| All non-Firm-A pooled | none | one transition at $\overline{\text{dHash}} = 6.6$ |

The Big-4-scope null on both axes is consistent with the §IV-E mixture evidence: the K=3 components overlap in their tails rather than separating sharply, so a local-discontinuity test does not flag a transition. Outside Big-4, dHash transitions appear in some subsets but no cosine transition is identified in any tested subset (Script 32 sweeps; pre-2018 and post-2020 stratified variants exhibit dHash transitions at varying locations). These off-Big-4 dHash transitions are scope-dependent and are not used as v4.0 operational thresholds; we do not claim a specific structural interpretation for them without an explicit bin-width sensitivity sweep at those scopes.

## E. Big-4 K=2 / K=3 Mixture Fits

This section reports the K=2 and K=3 2D Gaussian mixture fits to the Big-4 accountant-level distribution and the bootstrap stability of their marginal crossings.

**Table VII.** Big-4 K=2 mixture components (descriptive partition; not mechanism clusters per §III-J) and marginal-crossing bootstrap 95% confidence intervals.

| K=2 component | $\overline{\text{cos}}$ | $\overline{\text{dHash}}$ | weight |
|---|---|---|---|
| K=2-a (low-cos / high-dHash position) | 0.954 | 7.14 | 0.689 |
| K=2-b (high-cos / low-dHash position) | 0.983 | 2.41 | 0.311 |

Marginal crossings (point + bootstrap 95% CI, $n_{\text{boot}} = 500$):

| Axis | Point | Bootstrap median | 95% CI | CI half-width |
|---|---|---|---|---|
| cos | 0.9755 | 0.9754 | $[0.9742, 0.9772]$ | 0.0015 |
| dHash | 3.755 | 3.763 | $[3.476, 3.969]$ | 0.246 |

$\text{BIC}(K{=}2) = -1108.45$ (Script 34).

**Table VIII.** Big-4 K=3 mixture components (descriptive firm-compositional partition per §III-J; not mechanism clusters).

| K=3 component | $\overline{\text{cos}}$ | $\overline{\text{dHash}}$ | weight | descriptive position |
|---|---|---|---|---|
| C1 | 0.9457 | 9.17 | 0.143 | low-cos / high-dHash corner |
| C2 | 0.9558 | 6.66 | 0.536 | central region |
| C3 | 0.9826 | 2.41 | 0.321 | high-cos / low-dHash corner |

$\text{BIC}(K{=}3) = -1111.93$, lower than $K{=}2$ by $3.48$ (mild support; not by itself decisive). The full-fit K=3 baseline above is reproduced in Scripts 35, 37, and 38 with identical hyperparameters; Script 37 additionally fits K=3 on each leave-one-firm-out training set (those fold-specific components differ from the full-fit baseline by design and are reported separately in §IV-G Table XIII). Operational use of the K=2 / K=3 fits is governed by §III-J and §III-L; §IV-G reports the LOOO reproducibility evidence that motivates reporting both fits descriptively.

## F. Convergent Internal-Consistency Checks

This section reports the empirical evidence for §III-K's three-score internal-consistency analysis. We re-emphasise the §III-K caveat: the three scores are deterministic functions of the same per-CPA descriptor pair $(\overline{\text{cos}}_a, \overline{\text{dHash}}_a)$ and are *not statistically independent measurements*. The pairwise correlations document internal consistency among feature-derived ranks rather than external validation against an independent ground truth.

**Table IX.** Per-CPA Spearman rank correlations among three feature-derived scores, Big-4, $n = 437$.

| Score pair | Spearman $\rho$ | $p$-value |
|---|---|---|
| K=3 P(C1) vs Paper A box-rule less-replication-dominated rate | $+0.9627$ | $< 10^{-248}$ |
| Reverse-anchor cosine percentile vs Paper A box-rule less-replication-dominated rate | $+0.8890$ | $< 10^{-149}$ |
| K=3 P(C1) vs Reverse-anchor cosine percentile | $+0.8794$ | $< 10^{-142}$ |

(Source: Script 38.) Reverse-anchor reference: 2D Gaussian fit by MCD (support fraction 0.85) on $n = 249$ non-Big-4 CPAs; reference centre $\overline{\text{cos}} = 0.935$, $\overline{\text{dHash}} = 9.77$.

**Table X.** Per-firm summary across the three feature-derived scores, Big-4.

| Firm | $n$ CPAs | mean $P(\text{C1})$ | mean reverse-anchor score | mean Paper A less-replication-dominated rate |
|---|---|---|---|---|
| Firm A | 171 | 0.0072 | $-0.9726$ | 0.1935 |
| Firm B | 112 | 0.1410 | $-0.8201$ | 0.6962 |
| Firm C | 102 | 0.3110 | $-0.7672$ | 0.7896 |
| Firm D | 52 | 0.2406 | $-0.7125$ | 0.7608 |

(Source: Script 38 per-firm summary; reverse-anchor score is sign-flipped so that *higher* values indicate deeper into the reference left tail = less replication-dominated relative to the non-Big-4 reference.)

The three scores agree on placing Firm A as the most replication-dominated and the three non-Firm-A firms as less replication-dominated. The K=3 posterior P(C1) and the box-rule less-replication-dominated rate (Score 1 and Score 3) place Firm C at the least-replication-dominated end of Big-4; the reverse-anchor cosine percentile (Score 2) ranks Firm D fractionally above Firm C. This residual within-Big-4-non-A disagreement is a design feature of the reverse-anchor metric: Score 2 measures only the marginal cosine percentile under the non-Big-4 reference, so a firm with a slightly higher cosine but a markedly different dHash distribution (Firm D vs Firm C) can score higher on Score 2 while scoring lower on Scores 1 and 3, both of which use both descriptors.

**Table XI.** Per-signature Cohen $\kappa$ (binary collapse, replication-dominated vs less-replication-dominated), $n = 150{,}442$ Big-4 signatures.

| Pair | Cohen $\kappa$ |
|---|---|
| Paper A binary high-confidence box rule (cos $> 0.95$ AND dHash $\leq 5$) vs per-CPA K=3 hard label | 0.662 |
| Paper A binary high-confidence box rule vs per-signature K=3 hard label | 0.559 |
| Per-CPA K=3 hard label vs per-signature K=3 hard label | 0.870 |

(Source: Script 39; verdict label `SIG_CONVERGENCE_MODERATE`.) Per-signature K=3 components ($n = 150{,}442$) sorted by ascending cosine: $(0.928, 9.75, 0.146)$ / $(0.963, 6.04, 0.582)$ / $(0.989, 1.27, 0.272)$, an absolute cosine drift of $0.018$ in C1 and $0.006$ in C3 relative to the per-CPA fit. These convergence checks cover only the binary high-confidence rule (cos $> 0.95$ AND dHash $\leq 5$); the five-way classifier's moderate-confidence band ($5 < \text{dHash} \leq 15$) inherits its v3.x calibration and capture-rate evaluation (§IV-J).

## G. Leave-One-Firm-Out Reproducibility

This section reports the firm-level cross-validation evidence motivating §III-J's "K=3 descriptive, not operational" framing.

**Table XII.** K=2 leave-one-firm-out across the four Big-4 folds.

| Held-out firm | $n_{\text{train}}$ | $n_{\text{held}}$ | Fold rule (cos cut, dHash cut) | Held-out classified as templated by fold rule |
|---|---|---|---|---|
| Firm A | 266 | 171 | cos $> 0.9380$ AND dHash $\leq 8.79$ | $171 / 171 = 100.00\%$ ($95\%$ Wilson $[97.80\%, 100.00\%]$) |
| Firm B | 325 | 112 | cos $> 0.9744$ AND dHash $\leq 3.98$ | $0 / 112 = 0\%$ ($95\%$ Wilson $[0\%, 3.32\%]$) |
| Firm C | 335 | 102 | cos $> 0.9752$ AND dHash $\leq 3.75$ | $0 / 102 = 0\%$ ($95\%$ Wilson $[0\%, 3.63\%]$) |
| Firm D | 385 | 52 | cos $> 0.9756$ AND dHash $\leq 3.74$ | $0 / 52 = 0\%$ ($95\%$ Wilson $[0\%, 6.88\%]$) |

(Source: Script 36.) Across-fold cosine crossing: pairwise range $[0.9380, 0.9756]$, range = $0.0376$; max absolute deviation from the across-fold mean is $0.028$. This exceeds the report's $0.005$ across-fold stability tolerance by $5.6\times$ and is much larger than the full-Big-4 bootstrap CI half-width of $0.0015$. Together with the all-or-nothing held-out classification pattern (Firm A held out $\Rightarrow$ all held-out CPAs templated; any non-Firm-A firm held out $\Rightarrow$ none templated), this indicates the K=2 boundary is essentially a Firm-A-vs-others separator rather than a within-Big-4 mechanism boundary.

**Table XIII.** K=3 leave-one-firm-out: C1 component shape and held-out membership.

| Held-out firm | C1 cos (fit) | C1 dHash (fit) | C1 weight (fit) | Held-out C1 hard-label rate | Full-Big-4 baseline C1% | Absolute difference |
|---|---|---|---|---|---|---|
| Full-Big-4 baseline | 0.9457 | 9.17 | 0.143 | — | — | — |
| Firm A held out | 0.9425 | 10.13 | 0.145 | $4.68\%$ | $0.00\%$ | $4.68$ pp |
| Firm B held out | 0.9441 | 9.16 | 0.127 | $7.14\%$ | $8.93\%$ | $1.76$ pp |
| Firm C held out | 0.9504 | 8.41 | 0.126 | $36.27\%$ | $23.53\%$ | $12.77$ pp |
| Firm D held out | 0.9439 | 9.29 | 0.120 | $17.31\%$ | $11.54\%$ | $5.81$ pp |

(Source: Script 37; verdict label `P2_PARTIAL`.) Component shape is reproducible across folds: max deviation of C1 cosine = $0.005$, C1 dHash = $0.96$, C1 weight = $0.023$. Hard-posterior membership for the held-out firm varies: max absolute difference from the full-Big-4 baseline is $12.77$ pp at the Firm C held-out fold, exceeding the report's $5$ pp viability bar. We accordingly do not use K=3 hard-posterior membership as an operational classifier label (§III-J, §III-L).

## H. Pixel-Identity Positive-Anchor Miss Rate

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

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

| Classifier | Misclassified as less-replication-dominated | Miss rate | Wilson 95% CI |
|---|---|---|---|
| Paper A binary high-confidence box rule (cos $> 0.95$ AND dHash $\leq 5$) | $0 / 262$ | $0\%$ | $[0\%, 1.45\%]$ |
| K=3 per-CPA hard label (C3 = high-cos / low-dHash; descriptive) | $0 / 262$ | $0\%$ | $[0\%, 1.45\%]$ |
| Reverse-anchor (prevalence-calibrated cut) | $0 / 262$ | $0\%$ | $[0\%, 1.45\%]$ |

(Source: Script 40.) Per-firm breakdown of the byte-identical subset: Firm A 145; Firm B 8; Firm C 107; Firm D 2. All three candidate scores correctly assign every byte-identical signature to the replicated class.

We caution that for the Paper A box rule this result is close to tautological (byte-identical nearest-neighbour signatures have cosine $\approx 1$ and dHash $\approx 0$, well inside the rule's high-confidence region); v3.20.0 §V-F discusses this conservative-subset caveat at length and we retain that discussion. The reverse-anchor cut is chosen by *prevalence calibration* against the inherited box rule's overall replicated rate of $49.58\%$ across Big-4 signatures; this is a documented v4.0 limitation since no signature-level hand-signed ground truth exists to permit direct ROC optimisation.

## I. Inter-CPA Pair-Level Coincidence Rate (Big-4 spike + inherited corpus-wide)

The signature-level inter-CPA pair-level coincidence-rate analysis (reported in v3.20.0 §IV-F.1, Table X as "FAR") is inherited and extended in v4.0. v4.0 retroactively reframes the metric as **inter-CPA pair-level coincidence rate (ICCR)** rather than "False Acceptance Rate" because the corpus does not provide signature-level ground-truth negative labels; the inter-CPA negative-anchor assumption underpinning the metric is itself partially violated by within-firm cross-CPA template-like collision structures (§III-L.4). The v3.20.0 corpus-wide spike on $\sim 50{,}000$ inter-CPA pairs reported a per-comparison rate of $0.0005$ (Wilson 95% CI $[0.0003, 0.0007]$) at the cosine cut $0.95$.

v4.0 additionally reports the §III-L.1 Big-4-scope spike at higher sample size ($5 \times 10^5$ inter-CPA pairs; Script 40b), which replicates and extends the v3 result and adds the structural dimension (dHash) and joint-rule rates. The §III-L.1 numbers are referenced rather than duplicated here; the consolidated v4-new ICCR calibration appears in §IV-M Tables XXI–XXVI.

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

This section reports the §III-L five-way per-signature + document-level worst-case classifier output on the Big-4 sub-corpus. The five-way category definitions are inherited unchanged from v3.20.0 §III-K (now §III-L); see §III-L for the cosine and dHash cuts.

**Table XV.** Five-way per-signature category counts, Big-4 sub-corpus, $n = 150{,}442$ classified.

| Category | Long name | $n$ signatures | % of classified |
|---|---|---|---|
| HC | High-confidence non-hand-signed | 74,593 | 49.58% |
| MC | Moderate-confidence non-hand-signed | 39,817 | 26.47% |
| HSC | High style consistency | 314 | 0.21% |
| UN | Uncertain | 35,480 | 23.58% |
| LH | Likely hand-signed | 238 | 0.16% |

(Source: Script 42; 11 of 150,453 loaded Big-4 signatures lacked one or both descriptors and were excluded. The $150{,}442$ vs $150{,}453$ distinction — descriptor-complete vs vector-complete — recurs across §IV: descriptor-complete analyses (§IV-D through §IV-J, all using accountant-level aggregates or per-signature category counts derived from the same 150,442-signature substrate) use $n = 150{,}442$; vector- or pair-recomputed analyses (§IV-M.2 Table XXI, §IV-M.3 Table XXII, §IV-M.5 Tables XXIV–XXV; Scripts 40b, 43, 44) use $n = 150{,}453$ because their pair- or pool-level computations load all vector-complete signatures including those failing the descriptor-complete filter. See §III-G for the sample-size reconciliation.)

**Per-firm five-way breakdown (% within firm).**

| Firm | HC | MC | HSC | UN | LH | total signatures |
|---|---|---|---|---|---|---|
| Firm A | 81.70% | 10.76% | 0.05% | 7.42% | 0.07% | 60,448 |
| Firm B | 34.56% | 35.88% | 0.29% | 29.09% | 0.18% | 34,248 |
| Firm C | 23.75% | 41.44% | 0.38% | 34.21% | 0.22% | 38,613 |
| Firm D | 24.51% | 29.33% | 0.22% | 45.65% | 0.29% | 17,133 |

(Source: Script 42 per-firm cross-tab.) The per-firm pattern qualitatively aligns with the K=3 cluster cross-tab of Table XVI: Firm A's signatures concentrate in the HC band (81.70%) while its CPAs concentrate at the accountant level in the K=3 C3 (high-cos / low-dHash) component (82.46%; Table XVI). These two figures address different units (per-signature classification vs per-CPA hard cluster assignment) and are not directly comparable as a like-for-like consistency check; we report the qualitative alignment but do not infer a numerical equivalence. The three non-Firm-A Big-4 firms have markedly lower HC rates than Firm A and substantially higher Uncertain rates, with Firm D having the highest Uncertain rate (45.65%).

**Document-level worst-case aggregation.** Each audit report typically carries two certifying-CPA signatures. We aggregate signature-level outcomes to document-level labels using the v3.20.0 worst-case rule (HC > MC > HSC > UN > LH; §III-L). v4.0 does not change this aggregation rule; only the population over which it is computed changes (Big-4 subset).

**Table XIX.** Document-level worst-case category counts, Big-4 sub-corpus, $n = 75{,}233$ unique PDFs.

| Category | Long name | $n$ documents | % |
|---|---|---|---|
| HC | High-confidence non-hand-signed | 46,857 | 62.28% |
| MC | Moderate-confidence non-hand-signed | 19,667 | 26.14% |
| HSC | High style consistency | 167 | 0.22% |
| UN | Uncertain | 8,524 | 11.33% |
| LH | Likely hand-signed | 18 | 0.02% |

(Source: Script 42 document-level table; 379 of 75,233 PDFs carried signatures from more than one Big-4 firm and are reported in the single-firm-PDF per-firm breakdown of the script CSV but pooled into the overall counts here.)

**Per-firm document-level breakdown (single-firm PDFs only).**

| Firm | HC | MC | HSC | UN | LH | total docs |
|---|---|---|---|---|---|---|
| Firm A | 27,600 | 1,857 | 7 | 758 | 4 | 30,226 |
| Firm B | 8,783 | 6,079 | 57 | 2,202 | 6 | 17,127 |
| Firm C | 7,281 | 8,660 | 77 | 3,099 | 5 | 19,122 |
| Firm D | 3,100 | 2,838 | 22 | 2,416 | 3 | 8,379 |

(Source: Script 42; mixed-firm PDFs $n = 379$ excluded from the per-firm rows but included in the overall counts above.)

The five-way **moderate-confidence non-hand-signed** band (cos $> 0.95$ AND $5 < \text{dHash} \leq 15$) inherits its v3.x calibration; it is **not separately validated by Scripts 38–40**, which evaluated only the binary high-confidence rule (cos $> 0.95$ AND dHash $\leq 5$). v4.0 does not re-derive the moderate-band cuts on the Big-4 subset; we report the Table XV per-firm MC proportions (10.76% / 35.88% / 41.44% / 29.33% across Firms A through D) descriptively. The v3.20.0 capture-rate calibration evidence for the moderate band (v3.20.0 Tables IX, XI, XII, XII-B) is carried into v4.0 by reference and not regenerated on the Big-4 subset. We do not claim that the MC-band per-firm ordering above is a separate validation of the §III-K Spearman convergence, since MC occupancy is not a monotone function of the per-CPA less-replication-dominated ranking (e.g., Firm D's MC fraction is lower than Firm B's while Firm D's reverse-anchor score ranks it as less replication-dominated than Firm B).

**Table XVI.** Firm × K=3 cluster cross-tabulation, Big-4 sub-corpus.

| Firm | $n$ | C1 (low-cos / high-dHash) | C2 (central) | C3 (high-cos / low-dHash) | C1 % | C3 % |
|---|---|---|---|---|---|---|
| Firm A | 171 | 0 | 30 | 141 | $0.00\%$ | $82.46\%$ |
| Firm B | 112 | 10 | 102 | 0 | $8.93\%$ | $0.00\%$ |
| Firm C | 102 | 24 | 77 | 1 | $23.53\%$ | $0.98\%$ |
| Firm D | 52 | 6 | 45 | 1 | $11.54\%$ | $1.92\%$ |

(Source: Script 35.) The cross-tab is the accountant-level descriptive output of the K=3 mixture (§III-J / §IV-E). It is reported here as a complement to the five-way per-signature classifier (Table XV), not as an operational classifier output. Reading: Firm A's CPAs are concentrated in the C3 (high-cos / low-dHash) component (no Firm A CPAs in C1); Firm C has the highest C1 (low-cos / high-dHash) concentration of the Big-4 (C1 fraction $23.5\%$); Firms B and D sit between A and C on the K=3 hard-label ordering, broadly consistent with the per-firm Spearman ordering of Table X (with the within-Big-4-non-A reverse-anchor disagreement noted there).

**Document-level worst-case aggregation outputs are reported in Table XIX above.**

## K. Full-Dataset Robustness (light scope)

This section reports the v4.0 reproducibility cross-check at the full accountant scope ($n = 686$ CPAs, Big-4 plus mid/small firms). The scope of §IV-K is deliberately narrow: we re-run only the K=3 mixture + Paper A operational-rule per-CPA less-replication-dominated rate analysis, sufficient to demonstrate that the v4.0 K=3 + Paper A convergence reproduces at the wider scope. The §III-L five-way classifier and the §IV-G LOOO analyses are not re-run at the full scope. The five-way moderate-confidence band is documented as inherited from v3.x calibration in §IV-J.

**Table XVII.** K=3 component comparison, Big-4 sub-corpus vs full dataset.

| K=3 component | Big-4 (n=437) cos / dHash / weight | Full (n=686) cos / dHash / weight | Drift Big-4 → Full |
|---|---|---|---|
| C1 (low-cos / high-dHash) | 0.9457 / 9.17 / 0.143 | 0.9278 / 11.17 / 0.284 | $\lvert\Delta\rvert$ cos 0.018, dHash 1.99, wt 0.141 |
| C2 (central) | 0.9558 / 6.66 / 0.536 | 0.9535 / 6.99 / 0.512 | $\lvert\Delta\rvert$ cos 0.002, dHash 0.33, wt 0.024 |
| C3 (high-cos / low-dHash) | 0.9826 / 2.41 / 0.321 | 0.9826 / 2.40 / 0.205 | $\lvert\Delta\rvert$ cos 0.000, dHash 0.01, wt 0.117 |

(Source: Script 41; full-dataset $\text{BIC}(K{=}3) = -792.31$ vs Big-4 $\text{BIC}(K{=}3) = -1111.93$; BIC values are not directly comparable across different $n$ and are reported only for completeness.)

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

| Scope | $n$ CPAs | Spearman $\rho$ (P(C1) vs Paper A less-replication-dominated rate) | $p$-value |
|---|---|---|---|
| Big-4 (primary) | 437 | $+0.9627$ | $< 10^{-248}$ |
| Full dataset | 686 | $+0.9558$ | $< 10^{-300}$ |
| $\lvert\rho_{\text{full}} - \rho_{\text{Big-4}}\rvert$ | — | $0.0069$ | — |

(Source: Script 41.)

**Reading.** The K=3 component ordering and the strong Spearman convergence between K=3 P(C1) and the Paper A box-rule less-replication-dominated rate are preserved at the full scope. Component centres shift modestly: C3 (high-cos / low-dHash) is essentially unchanged in centre but loses weight $0.117$ as the full population includes more non-templated CPAs (mid/small firms); C1 (low-cos / high-dHash) gains weight $0.141$ and shifts to lower cosine and higher dHash (centre $(0.928, 11.17)$ vs Big-4 $(0.946, 9.17)$) as the broader population includes mid/small-firm CPAs landing toward the low-cos / high-dHash region that the Big-4-primary scope deliberately excludes. We read this as evidence that the Big-4-primary K=3 + Paper A convergence is not a Big-4-specific artefact; we do **not** read it as an endorsement of using full-dataset K=3 component centres or operational thresholds in place of the Big-4-primary analysis. Mid/small-firm composition shifts the component centres meaningfully and the v4.0 primary methodology is restricted to Big-4 by design (§III-G item 4).

## L. Feature Backbone Ablation (inherited from v3.20.0 §IV-I)

The feature-backbone ablation (v3.20.0 Table XVIII; backbone replacement of ResNet-50 with alternative ImageNet-pretrained backbones to verify that the §III-E embedding choice is not load-bearing) is inherited unchanged. v3.20.0 Table XVIII is cited by its original v3 number and is **not** the same table as the v4 Table XVIII (which reports the Big-4 vs full-dataset Spearman drift in §IV-K). v4.0 makes no scope-specific re-derivation of the ablation; the analysis is a methodological-stability check on the embedding stage and is corpus-wide rather than Big-4-restricted.

## M. v4-New Anchor-Based ICCR Calibration Results

This section consolidates the v4-new empirical results that support the §III-L anchor-based threshold calibration framework. Numbers below are direct re-statements from the spike scripts cited per row; the corresponding §III provenance table entries appear in §III's provenance table.

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

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

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

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

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

**Table XXI.** Big-4 inter-CPA per-comparison ICCR sweep, $n = 5 \times 10^5$ pairs (Big-4 scope; v4 new).

| Threshold | Per-comparison ICCR | 95% Wilson CI |
|---|---|---|
| cos $> 0.945$ (v3.x published "natural threshold") | $0.00081$ | $[0.00073, 0.00089]$ |
| cos $> 0.95$ (inherited operating point) | $0.00060$ | $[0.00053, 0.00067]$ |
| cos $> 0.97$ | $0.00024$ | $[0.00020, 0.00029]$ |
| cos $> 0.98$ | $0.00009$ | $[0.00007, 0.00012]$ |
| dHash $\leq 5$ (inherited operating point) | $0.00129$ | $[0.00120, 0.00140]$ |
| dHash $\leq 4$ | $0.00050$ | $[0.00044, 0.00057]$ |
| dHash $\leq 3$ | $0.00019$ | $[0.00015, 0.00023]$ |
| Joint: cos $> 0.95$ AND dHash $\leq 5$ (any-pair semantics) | $0.00014$ | — |
| Joint: cos $> 0.95$ AND dHash $\leq 4$ (any-pair) | $0.00011$ | — |

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

The cos $> 0.95$ row replicates v3.20.0 §IV-F.1 Table X (v3 reported $0.0005$ under prior "FAR" terminology). The dHash row and joint row are v4 new.

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

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

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

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

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

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

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

Per-firm D2 document-level ICCR: Firm A $0.6201$ ($n = 30{,}226$); Firm B $0.1600$ ($n = 17{,}127$); Firm C $0.1635$ ($n = 19{,}501$); Firm D $0.0863$ ($n = 8{,}379$). The Firm C denominator $n = 19{,}501$ exceeds Table XIX's single-firm Firm C count of $19{,}122$ by exactly the $379$ mixed-firm PDFs: all $379$ are $1{:}1$ Firm C / Firm D mixed-firm documents, and Script 45's mode-of-firms implementation (`np.argmax` over `np.unique`'s alphabetically-sorted firm counts) returns the first-sorted firm on ties, which assigns these tied documents to Firm C rather than to Firm D. The four per-firm denominators here therefore sum to the full $75{,}233$, whereas Table XIX's per-firm rows sum to $74{,}854 = 75{,}233 - 379$.

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

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

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

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

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

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

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

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

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

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

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

---

## Phase 3 close-out checklist

The following items remain after codex rounds 21–24 and before §IV is sent to partner Jimmy for v4.0 review:

1. **Table XV per-signature category counts** — RESOLVED (v2 of §IV draft, Script 42 output). Per-signature, per-firm, document-level, and per-firm-document tables now populated.
2. **Table renumbering finalisation.** The v4 table sequence as of v3.2 is Tables V–XVIII plus Table XV-B (no v4 Table IV is printed); inherited v3.x tables such as capture-rate Tables IX, XI, XII and the backbone-ablation v3.20.0 Table XVIII are kept by reference and cited as "v3.20.0 Table N" rather than reproduced as v4-numbered tables. A final pass should confirm whether the target journal accepts the Table XV-B letter suffix; if not, XV-B can be renumbered to a sequential XIX with §IV-J text adjusted accordingly.
3. **§IV-A to §IV-C content audit.** Verify that the inherited prose for Experimental Setup, Detection Performance, and All-Pairs analysis remains accurate after the §III-G scope change to Big-4 primary.
4. **Open question carry-over from §III v3.** Codex round-22 open questions on five-way moderate-band validation, firm anonymisation policy, and §IV table numbering are addressed in this v3 of §IV: (a) five-way moderate band documented as inherited from v3.x in §IV-J with Big-4 per-firm proportions reported descriptively (Table XV); (b) firm anonymisation maintained throughout §IV (Firm A–D used consistently; real names removed in v3); (c) §IV table numbering set provisionally and to be finalised at Phase 3 close-out.

5. **Internal author notes (this checklist + §III's cross-reference index + both files' draft-note headers).** These are author working artefacts and should be moved to a separate notes file or stripped before partner / submission packaging.
