pdf_signature_extraction/paper/paper_a_appendix_v3.md

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

The main text (Section III-I, Section IV-D.2) treats the Burgstahler-Dichev / McCrary discontinuity procedure [38], [39] as a density-smoothness diagnostic rather than as a threshold estimator. This appendix 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.

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 non-hand-signed mode (cosine \geq 0.975, dHash \leq 10) rather than between modes, which is the location pattern we would expect of a clean two-mechanism boundary.

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 per-signature similarity does not form a clean two-mechanism mixture.

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

Appendix B. Table-to-Script Provenance

For reproducibility, the following table maps each numerical table in Section IV to the analysis script that produces its underlying values and to the report file emitted by that script. Scripts are under signature_analysis/. Report artifact paths below are listed relative to the project's analysis report root, which is /Volumes/NV2/PDF-Processing/signature-analysis/ in our local deployment; replicators should rebase the paths to whatever report root they configure when invoking the scripts.

The table-to-script mapping above is intended as a navigation aid for replicators. All scripts run deterministically under the fixed random seeds documented in the supplementary materials; the artifact paths above were verified against the local deployment at the time of submission, and any reviewer reproduction step should re-emit the artifacts from the listed scripts rather than depend on the absolute path layout.