# 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.
