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Paper A v3.7: demote BD/McCrary to density-smoothness diagnostic; add Appendix A

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Implements codex gpt-5.4 recommendation (paper/codex_bd_mccrary_opinion.md,
"option (c) hybrid"): demote BD/McCrary in the main text from a co-equal
threshold estimator to a density-smoothness diagnostic, and add a
bin-width sensitivity appendix as an audit trail.

Why: the bin-width sweep (Script 25) confirms that at the signature
level the BD transition drifts monotonically with bin width (Firm A
cosine: 0.987 -> 0.985 -> 0.980 -> 0.975 as bin width widens 0.003 ->
0.015; full-sample dHash transitions drift from 2 to 10 to 9 across
bin widths 1 / 2 / 3) and Z statistics inflate superlinearly with bin
width, both characteristic of a histogram-resolution artifact. At the
accountant level the BD null is robust across the sweep. The paper's
earlier "three methodologically distinct estimators" framing therefore
could not be defended to an IEEE Access reviewer once the sweep was
run.

Added
- signature_analysis/25_bd_mccrary_sensitivity.py: bin-width sweep
  across 6 variants (Firm A / full-sample / accountant-level, each
  cosine + dHash_indep) and 3-4 bin widths per variant. Reports
  Z_below, Z_above, p-values, and number of significant transitions
  per cell. Writes reports/bd_sensitivity/bd_sensitivity.{json,md}.
- paper/paper_a_appendix_v3.md: new "Appendix A. BD/McCrary Bin-Width
  Sensitivity" with Table A.I (all 20 sensitivity cells) and
  interpretation linking the empirical pattern to the main-text
  framing decision.
- export_v3.py: appendix inserted into SECTIONS between conclusion
  and references.
- paper/codex_bd_mccrary_opinion.md: codex gpt-5.4 recommendation
  captured verbatim for audit trail.

Main-text reframing
- Abstract: "three methodologically distinct estimators" ->
  "two estimators plus a Burgstahler-Dichev/McCrary density-
  smoothness diagnostic". Trimmed to 243 words.
- Introduction: related-work summary, pipeline step 5, accountant-
  level convergence sentence, contribution 4, and section-outline
  line all updated. Contribution 4 renamed to "Convergent threshold
  framework with a smoothness diagnostic".
- Methodology III-I: section renamed to "Convergent Threshold
  Determination with a Density-Smoothness Diagnostic". "Method 2:
  BD/McCrary Discontinuity" converted to "Density-Smoothness
  Diagnostic" in a new subsection; Method 3 (Beta mixture) renumbered
  to Method 2. Subsections 4 and 5 updated to refer to "two threshold
  estimators" with BD as diagnostic.
- Methodology III-A pipeline overview: "three methodologically
  distinct statistical methods" -> "two methodologically distinct
  threshold estimators complemented by a density-smoothness
  diagnostic".
- Methodology III-L: "three-method analysis" -> "accountant-level
  threshold analysis (KDE antimode, Beta-2 crossing, logit-Gaussian
  robustness crossing)".
- Results IV-D.1 heading: "BD/McCrary Discontinuity" ->
  "BD/McCrary Density-Smoothness Diagnostic". Prose now notes the
  Appendix-A bin-width instability explicitly.
- Results IV-E: Table VIII restructured to label BD rows
  "(diagnostic only; bin-unstable)" and "(diagnostic; null across
  Appendix A)". Summary sentence rewritten to frame BD null as
  evidence for clustered-but-smoothly-mixed rather than as a
  convergence failure. Table cosine P5 row corrected from 0.941 to
  0.9407 to match III-K.
- Results IV-G.3 and IV-I.2: "three-method convergence/thresholds"
  -> "accountant-level convergent thresholds" (clarifies the 3
  converging estimates are KDE antimode, Beta-2, logit-Gaussian,
  not KDE/BD/Beta).
- Discussion V-B: "three-method framework" -> "convergent threshold
  framework".
- Conclusion: "three methodologically distinct methods" -> "two
  threshold estimators and a density-smoothness diagnostic";
  contribution 3 restated; future-work sentence updated.
- Impact Statement (archived): "three methodologically distinct
  threshold-selection methods" -> "two methodologically distinct
  threshold estimators plus a density-smoothness diagnostic" so the
  archived text is internally consistent if reused.

Discussion V-B / V-G already framed BD as a diagnostic in v3.5
(unchanged in this commit). The reframing therefore brings Abstract /
Introduction / Methodology / Results / Conclusion into alignment with
the Discussion framing that codex had already endorsed.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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# Appendix A. BD/McCrary Bin-Width Sensitivity
The main text (Sections III-I and IV-E) treats the Burgstahler-Dichev / McCrary discontinuity procedure [38], [39] as a *density-smoothness diagnostic* rather than as one of the threshold estimators whose convergence anchors the accountant-level threshold band.
This appendix documents the empirical basis for that framing by sweeping the bin width across six (variant, bin-width) panels: Firm A / full-sample / accountant-level, 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 |
| Accountant-level cosine_mean | 686 | 0.002 | no transition | — | — |
| Accountant-level cosine_mean | 686 | 0.005 | 0.9800 | -3.23 | +5.18 |
| Accountant-level cosine_mean | 686 | 0.010 | no transition | — | — |
| Accountant-level dHash_indep_mean| 686 | 0.2 | no transition | — | — |
| Accountant-level dHash_indep_mean| 686 | 0.5 | no transition | — | — |
| Accountant-level dHash_indep_mean| 686 | 1.0 | 3.0 | -2.00 | +3.24 |
-->
Two patterns are visible in Table A.I.
First, at the signature level 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, at the accountant level---the unit we rely on for primary threshold inference (Sections III-H, III-J, IV-E)---the procedure produces no significant transition at two of three cosine bin widths and two of three dHash bin widths, and the one marginal transition it does produce ($Z_\text{below} = -2.00$ in the dHash sweep at bin width $1.0$) sits exactly at the critical value for $\alpha = 0.05$.
This pattern is itself informative: it is consistent with *clustered-but-smoothly-mixed* accountant-level aggregates, in which the between-cluster boundary is gradual enough that a discontinuity-based test cannot reject the smoothness null at conventional significance.
Taken together, Table A.I shows (i) 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---and (ii) that the accountant-level BD/McCrary null is a robust finding that survives the bin-width sweep.
Both observations support the main-text decision to use BD/McCrary as a density-smoothness diagnostic rather than as a threshold estimator.
The accountant-level threshold band reported in Table VIII ($\text{cosine} \approx 0.975$ from the convergence of the KDE antimode, the Beta-2 crossing, and the logit-GMM-2 crossing) is therefore not adjusted to include any BD/McCrary location, and the absence of a BD transition at the accountant level is reported as itself evidence for the clustered-but-smooth interpretation in Section V-B.
Raw per-bin $Z$ sequences and $p$-values for every (variant, bin-width) panel are available in the supplementary materials (`reports/bd_sensitivity/bd_sensitivity.json`) produced by `signature_analysis/25_bd_mccrary_sensitivity.py`.