# Canonical number set — Paper A v13 rev9.1 (verified 2026-06-30)

Source of truth: `signature_analysis.db` (`/Volumes/NV2/PDF-Processing/signature-analysis/`).
Verified against scripts in `paper/v13_build/scripts/`. All firm-level signature counts and
within-accountant HC rates below are DB-reproduced to 2 decimals.

## ROOT CAUSE of the "Firm C/D two-snapshot" inconsistency

It is **NOT** a stale snapshot. It is **two live firm-assignment keys** used inconsistently:

| Key | Definition | C | D | total |
|---|---|---|---|---|
| **assigned** (canonical) | `assigned_accountant → accountants.firm` (registry firm) | 38,613 | 17,133 | **150,442** |
| excel | `excel_firm` column (source Excel metadata) | 38,993 | 16,752 | 150,441 |

Difference = **379 signatures with excel_firm=資誠(C) but registry firm=安永(D)** (one/few
cross-labeled accountants). A and B are identical under both keys. Because the headline
contrast pools B/C/D, the swap is invisible to it (FE/LOYO/bootstrap reproduce exactly under
either key); only **per-firm-separated tables** (IV, II-b, II-c, VI) are affected.

DECISION: standardize on the **assigned** key — it is what headline Table IV/II-b already use.

## "Analyzable signatures" definition (reproduces 150,442 exactly)

Analyzable = signature whose CPA has **≥2 signatures** (singleton CPAs have no same-accountant
partner). Per-firm singletons (A:2, B:6, C:3, D:0) exactly equal raw−Set A.

## Canonical signature counts (assigned key)

| Firm | Full 2013–23 | 2013–19 | 2020–23 |
|---|---|---|---|
| A | 60,448 | 36,550 | 23,898 |
| B | 34,248 | 19,677 | 14,571 |
| C | 38,613 | 22,449 | 16,164 |
| D | 17,133 | 9,945 | 7,188 |
| Big-4 | **150,442** | 88,621 | 61,821 |

Every row sums exactly. (excel-key alternatives, do NOT use for per-firm tables:
C 2020–23 = 16,485, D 2020–23 = 6,866 — these are what the stale Table II-c rows show.)

## Five-way breakdown (HC | MC | HSC | UN | LH | n), low cut = 0.8547

FULL 2013–2023 (Table IV):
- A: 81.70 | 10.76 | 0.05 | 7.35 | 0.14 | 60,448
- B: 34.56 | 35.88 | 0.29 | 28.95 | 0.32 | 34,248
- C: 23.75 | 41.44 | 0.38 | 33.97 | 0.47 | 38,613
- D: 24.51 | 29.33 | 0.22 | 45.28 | 0.66 | 17,133
- Overall: 49.58 | 26.47 | 0.21 | 23.42 | 0.32 | 150,442

2013–2019 (Table II-b):
- B: 29.04 | 39.31 | 0.39 | 30.91 | 0.35 | 19,677
- C: 21.59 | 42.09 | 0.37 | 35.53 | 0.43 | 22,449
- D: 22.01 | 29.67 | 0.20 | 47.35 | 0.76 | 9,945

2020–2023 (Table II-c) — **C/D are the fix**:
- A: 83.84 | 9.13 | 0.04 | 6.88 | 0.11 | 23,898
- B: 42.01 | 31.24 | 0.16 | 26.31 | 0.28 | 14,571
- C: **26.74 | 40.55 | 0.40 | 31.80 | 0.51 | 16,164**  (manuscript stale: 26.53/.../16,485)
- D: **27.98 | 28.85 | 0.24 | 42.42 | 0.51 | 7,188**   (manuscript stale: 28.53/.../6,866)

Per-firm period HC rates (§IV-C text / Fig 5 — already CORRECT in manuscript):
A 80.3→83.8, B 29.0→42.0, C 21.6→26.7, D 22.0→28.0.

## Table VI (any-pair vs same-pair HC) under assigned key

| firm | n | any-pair% | same-pair% |
|---|---|---|---|
| A | 60,448 | 81.7 | 57.3 |
| B | 34,248 | 34.6 | 9.0 |
| C | 38,613 | 23.7 | 5.3 |
| D | 17,133 | 24.5 | 7.7 |
| all | 150,442 | 49.6 | 27.3 |

(Manuscript currently shows excel-key: C 38,993, D 16,752, all 150,441, D any-pair 24.7.)

## Accountant counts (R2) — reviewer's "179 > 171 impossible" is a FALSE POSITIVE

Different keys, not a contradiction:
- Bootstrap/FE (`accountant_id`, is_valid): **A=179** (reproduces exactly), BCD=281 (manuscript 280).
- Table III / accountant-level partition (171/112/102/52 = 437): RESOLVED 2026-06-30 — this is the
  Big-4 accountants with **≥10 signatures** (registry key), reproduces EXACTLY on current DB.
  GMM recipe: accountant point = (mean max_similarity_to_same_accountant, mean min_dhash_independent);
  sklearn GaussianMixture(n_components=3, covariance_type='full', random_state=42, n_init=10);
  templated cluster = highest-cosine component (cos 0.983 / dHash 2.4). Shares reproduce to the digit:
  A=82.5% (141/171), B=0.0% (0/112), C=1.0% (1/102), D=1.9% (1/52). Script: scratchpad/gmm.py.
- Analyzable accountant count (registry name key, ≥2 sig) = 457 (A=178, B=119, C=107, D=53) — owns
  the 150,442 signatures; now stated in Table I and the §III-B prose.

THREE DISTINCT, ALL-REPRODUCIBLE accountant universes (the reviewer collapsed them → false "179>171"):
  457 (≥2 sig, owns 150,442) | 437 (≥10 sig, Table III GMM) | 459 = 179/280 (accountant_id bootstrap).
Manuscript edits applied 2026-06-30: Table I + §III-B prose → 457; §III-B prose + Table III caption
note the 437/≥10-sig subset. Bootstrap 179/280 stays (correct, key-invariant).

## F5 robustness (re-validated EXACTLY on current DB, key-invariant)
Firm+Year FE ORs: B=0.116, C=0.061, D=0.070. LOYO gap range [53.1, 54.9]pp, full-sample 53.7pp.