Paper A v13 rev9.2: slim abstract (rev9.1 review R7)

Trim the single-paragraph abstract 436 -> 358 words (~18%) and roughly halve the
closing caveat, per the reviewer's optional R7. All headline numbers preserved
(86,071 reports, 150,442 signatures, unimodality p = 0.35, 1.2%/17.5% chance rates,
82% vs 24-35% benchmark, 262 byte-identical); the honest-scope sentence is kept but
condensed to "between-accountant specificity proxy — not a within-accountant error
rate, nor a bound on one; no pipeline separation; no single-signature labels."

Rebuild docx + PDF.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01K35dXhb9XEM1mnYz6SSHpU

.gitignore

@@ -61,3 +61,5 @@ node_modules/
 
 # Sensitive/large data
 *.xlsx
+
+.serena/

.planning/STATE.md

@@ -1,49 +1,80 @@
 # STATE — Current snapshot
 
-**Date**: 2026-05-12
+**Date**: 2026-05-14
 **Active milestone**: Paper A v4.0 — Big-4 reframe
-**Active branch**: `paper-a-v4-big4` (12 commits ahead of `yolo-signature-pipeline`)
-**Active phase**: Phase 2 — Methodology rewrite, draft delivered, **awaiting user review of 5 open questions in `paper/v4/paper_a_methodology_v4_section_iii.md`** before Phase 3 begins
+**Active branch**: `paper-a-v4-big4` (41 commits ahead of `master`; fully pushed to `origin/paper-a-v4-big4` at `128a914`)
+**Active phase**: **Phase 5 — AI peer review COMPLETE; Phase 6 ready to begin**
 
-## Recently completed
+## Phase 5 closure summary (2026-05-14)
 
-**Phase 1 (Foundation, 7 spike + foundation scripts)**:
-- Script 32 (`e1d81e3`): non-Firm-A calibration verdict C
-- Script 33 (`8ac0988`): reverse-anchor PAPER_C_STRONG (directional ρ=+0.744)
-- Script 34 (`55f9f94`): Big-4 K=2 dip-test multimodal p<0.0001, bootstrap CI [0.974, 0.977] / [3.48, 3.97]
-- Script 35 (`55f9f94`): firm × cluster — Firm A 0% C1 / 82.5% C3, PwC 23.5% C1
-- Script 36 (`ccd9f23`): K=2 LOOO **UNSTABLE** (firm-mass conflation; max Δcos=0.028)
-- Script 37 (`92f1db8`): K=3 LOOO **PARTIAL** (component shape stable, membership ±5-13pp)
-- Script 38 (`bc36dcc`): convergence **STRONG** — 3 lenses pairwise ρ ≥ 0.879
-- Script 39 (`39575ce`): per-signature convergence **MODERATE** — κ=0.87 between per-CPA and per-sig K=3 fits
-- Script 40 (`338737d`): pixel-identity FAR = **0%** on n=262 ground-truth replicated
+**Convergence achieved**: 3/3 reviewers in Accept/Minor band across the round-2 cross-check.
 
-**Phase 2 (Methodology rewrite)**: §III-G..L draft delivered at `paper/v4/paper_a_methodology_v4_section_iii.md` (commit on the same branch). Single coherent rewrite covering 6 sub-sections (G/H/I/J/K/L); cross-references to all 9 spike scripts; 5 open questions flagged at end of draft for user decision.
+| Reviewer | Final round | Verdict |
+|---|---|---|
+| Gemini 3.1 Pro | round 2 | **Accept** (Phase 5 splice-ready as-is) |
+| Opus 4.7 | round 2 | Minor Revision (4 substantive findings → closed in round-4) |
+| codex GPT-5.5 | round 9 | Minor Revision (2 provenance findings → closed in round-5) |
 
-## Pending — Phase 2 user review (BEFORE Phase 3)
+**Original Phase 5 gate met**: Accept/Minor consensus from ≥2 of 3 reviewers. No empirical reruns required.
 
-5 decisions needed from user before Phase 3 (Results regeneration) starts:
+**Phase 5 fix rounds applied** (commits on this branch):
+1. `9604b27` — codex round-7 closeout copy-edit (candidate classifiers → candidate checks; refs [42]-[44] added; §II placeholder caveat removed; STATE.md refresh)
+2. `b884d39` — round-2 fixes (Opus M1: §IV K=3 mechanism-label reversion; M2: Table XV-B → XIX + cascade XIX → XX … XXV → XXVI; M3: "98-100%" within-firm semantic conflation; M4: duplicate §V-G heading; Gemini Table XV sample-size footnote)
+3. `4a6f9c5` — round-3 fixes (codex round-8 splice blockers: abstract trim 261 → 247 words; §IV-J Table XV footnote §IV-M.5 reclassification; §IV-I "§IV-M Table XVI" → "§IV-M Tables XXI-XXVI"; binary-collapse terminology cleanup)
+4. `d3ddf74` — round-4 fixes (Opus round-2 N1: Firm C 19,501 vs 19,122 denominator footnote; N2: composition-decomposition added as Table XXVII row 1; N3: Table XXVII numbered; N4: cross-firm hit matrix assumption disclosure)
+5. `128a914` — round-5 provenance patches (codex round-9 factual corrections: N1 "majority firm" → "1:1 tie-break to first-sorted firm" via Script 45 `np.argmax`; N2 row narrowed to Big-4-only evidence; non-Big-4 jittered-dHash range $[0.71, 1.00]$ → codex-verified $[0.38, 1.00]$ with read-only-spike provenance)
 
-1. §III-G scope justification — three-point argument enough, or add a fourth?
-2. §III-H Firm A phrasing — "case study of templated end" vs an alternative framing?
-3. §III-J K=3 vs K=2 selection — lean on LOOO (current draft) or strengthen BIC argument?
-4. §III-L hybrid classifier — keep inherited 5-way box rule, or commit to K=3 hard label as primary?
-5. Section IV table numbering scheme — confirm before Phase 3 builds tables.
+**Reviewer artifacts archived** (paper/):
+- `codex_review_gpt55_v4_round{7,8,9}.md`
+- `gemini_review_v4_round{1,2}.md`
+- `opus_review_v4_round{1,2}.md`
 
-Plus: any prose-level edits the user wants on the §III draft.
+## Phase 5 substantive findings catalogue
+
+**v4 methodological pivot** (unchanged through all reviewer rounds):
+- Distributional path to thresholds (K=3 / dip / antimode) abandoned; anchor-based ICCR calibration at 3 units adopted
+- "FAR" → "ICCR" throughout; inter-CPA-as-negative assumption disclosed as partially violated by within-firm template sharing
+- K=3 demoted to descriptive firm-compositional partition (§III-J line 90 retires "hand-leaning / mixed / replicated" mechanism labels)
+- Positioning: anchor-calibrated specificity-only screening framework with human-in-the-loop review; NOT a validated forensic detector
+
+**Empirical anchors** (all provenance-verified across reviewer panel):
+- Three feature-derived scores converge Spearman $\rho \geq 0.879$ (internal consistency; not external validation)
+- Anchor-based ICCRs: per-comparison $0.0006/0.0013/0.00014$; per-signature $0.11$; per-document $0.34$
+- Firm heterogeneity decisive: Firm A per-doc HC+MC alarm $0.62$ vs Firms B/C/D $0.09$–$0.16$; logistic OR $0.05/0.01/0.03$ relative to Firm A reference
+- Within-firm collision concentration under deployed any-pair rule: Firm A $98.8\%$ vs Firms B/C/D $76.7$–$83.7\%$; same-pair joint event saturates at $97.0$–$99.96\%$ within-firm at all four firms
+
+## Phase 6 — Partner Jimmy v4.0 review (READY TO BEGIN)
+
+**Pre-Phase-6 partner alignment** (2026-05-13 still open): partner asked whether firm heterogeneity could be framed as "statistically insignificant." **Decision: no** — heterogeneity is highly significant (40–62σ in logistic regression; all three AI reviewers independently confirmed the decisive framing). Confirm framing with partner before exporting DOCX.
+
+**Phase 6 tasks**:
+1. Confirm "statistically insignificant" framing rejection with partner
+2. Manuscript-splice assembly:
+   - Splice §III-G–§III-M (paper_a_methodology_v4_section_iii.md) onto v3.20.0 §III-A–§III-F into master `paper/paper_a_methodology_v3.md` body
+   - Splice §IV v3.3 (paper_a_results_v4_section_iv.md) into master `paper/paper_a_results_v3.md`
+   - Splice Phase 4 prose (Abstract / §I / §II / §V / §VI) into the master manuscript file
+   - **Strip internal-only blocks** during splice: Phase 4 line 3 draft note + lines 153-162 close-out checklist; §III line 3 + lines 434-447 cross-reference checklist + open-questions block; §IV line 3 + close-out checklist at line 365+
+   - Re-verify table numbering after splice (Table XXVII currently lives in §III between §IV-M.6's Table XXVI; confirm order in final master file)
+3. Export v4.0 DOCX via `paper/export_v3.py` (with author block fill)
+4. Ship to ~/Downloads
+5. Iterate on Jimmy's review comments
+6. Capture review artifact in `paper/partner_jimmy_v4_review.md`
+
+## Phase 7 — IEEE Access submission (pending Phase 6)
+
+1. iThenticate similarity check (target < 20%)
+2. IEEE eCF form
+3. Upload manuscript + cover letter via IEEE Access submission portal
+4. Capture confirmation number
 
 ## Blockers
 
-None.
-
-## Open questions deferred from spike
-
-- Bootstrap stability of cosine and dHash crossings *jointly* (not just marginally) — addressed in Phase 1 if time permits
-- K=2 vs K=3 final choice for §III-J — both reported, but operational classifier needs to commit to one (recommend K=2 for interpretability; K=3 in supplementary)
+None. Phase 5 closed; Phase 6 ready to begin pending partner-framing confirmation.
 
 ## Things to remember (per memory)
 
-- Provenance-verify all empirical claims against fresh sqlite/grep ([[feedback-provenance-fabrication]])
-- Don't mock the DB or use placeholders — every number must trace to a script + query
-- Partner Jimmy already proposed Big-4 direction (this is execution, not pitching a new direction)
-- Paper C standalone is shelved — folded into v4.0 §IV-K
+- Inter-CPA "FAR" is NOT true FAR; it's a coincidence rate (ICCR) under an assumption violated by within-firm template sharing — never write "FAR" or "specificity" without the disclaimer ([[feedback-inter-cpa-negative-anchor-assumption]])
+- Dip test on Big-4 dh is composition + integer artefact, not mechanism — §III-I.1 "dip justifies finite mixture" framing must NOT be used; K=3 is descriptive of firm composition ([[feedback-dip-test-composition-artifact]])
+- Provenance-verify all empirical claims against fresh sqlite/grep ([[feedback-provenance-fabrication]]) — codex round-9's DB-verification caught a "majority firm" inference in round-4 that turned out to be 1:1 ties resolved by `np.argmax` tie-break; round-5 corrected it
+- AI peer reviewers have accepted fabricated claims in the past; verify numbers against scripts, not against reviewer agreement ([[feedback-ai-review-provenance]]) — codex round-9's read-only rerun of the non-Big-4 jittered procedure exposed an unreproducible $[0.71, 1.00]$ range that round-5 corrected to $[0.38, 1.00]$
+- Paper C standalone is shelved — folded into v4.0 §IV-K (Light full-dataset robustness)

NEW_SESSION_HANDOFF.md

@@ -1,432 +0,0 @@
-# 新对话交接文档 - PP-OCRv5研究
-
-**日期**: 2025-10-29
-**前序对话**: PaddleOCR-Cover分支开发
-**当前分支**: `paddleocr-improvements` (稳定)
-**新分支**: `pp-ocrv5-research` (待创建)
-
----
-
-## 🎯 任务目标
-
-研究和实现 **PP-OCRv5** 的手写签名检测功能
-
----
-
-## 📋 背景信息
-
-### 当前状况
-
-✅ **已有稳定方案** (`paddleocr-improvements` 分支):
-- PaddleOCR 2.7.3 + OpenCV Method 3
-- 86.5%手写保留率
-- 区域合并算法工作良好
-- 测试: 1个PDF成功检测2个签名
-
-⚠️ **PP-OCRv5升级遇到问题**:
-- PaddleOCR 3.3.0 API完全改变
-- 旧服务器代码不兼容
-- 需要深入研究新API
-
-### 为什么要研究PP-OCRv5？
-
-**文档显示**: https://www.paddleocr.ai/main/en/version3.x/algorithm/PP-OCRv5/PP-OCRv5.html
-
-PP-OCRv5性能提升:
-- 手写中文检测: **0.706 → 0.803** (+13.7%)
-- 手写英文检测: **0.249 → 0.841** (+237%)
-- 可能支持直接输出手写区域坐标
-
-**潜在优势**:
-1. 更好的手写识别能力
-2. 可能内置手写/印刷分类
-3. 更准确的坐标输出
-4. 减少复杂的后处理
-
----
-
-## 🔧 技术栈
-
-### 服务器环境
-
-```
-Host: 192.168.30.36 (Linux GPU服务器)
-SSH: ssh gblinux
-目录: ~/Project/paddleocr-server/
-```
-
-**当前稳定版本**:
-- PaddleOCR: 2.7.3
-- numpy: 1.26.4
-- opencv-contrib-python: 4.6.0.66
-- 服务器文件: `paddleocr_server.py`
-
-**已安装但未使用**:
-- PaddleOCR 3.3.0 (PP-OCRv5)
-- 临时服务器: `paddleocr_server_v5.py` (未完成)
-
-### 本地环境
-
-```
-macOS
-Python: 3.14
-虚拟环境: venv/
-客户端: paddleocr_client.py
-```
-
----
-
-## 📝 核心问题
-
-### 1. API变更
-
-**旧API (2.7.3)**:
-```python
-from paddleocr import PaddleOCR
-ocr = PaddleOCR(lang='ch')
-result = ocr.ocr(image_np, cls=False)
-
-# 返回格式:
-# [[[box], (text, confidence)], ...]
-```
-
-**新API (3.3.0)** - ⚠️ 未完全理解:
-```python
-# 方式1: 传统方式 (Deprecated)
-result = ocr.ocr(image_np)  # 警告: Please use predict instead
-
-# 方式2: 新方式
-from paddlex import create_model
-model = create_model("???")  # 模型名未知
-result = model.predict(image_np)
-
-# 返回格式: ???
-```
-
-### 2. 遇到的错误
-
-**错误1**: `cls` 参数不再支持
-```python
-# 错误: PaddleOCR.predict() got an unexpected keyword argument 'cls'
-result = ocr.ocr(image_np, cls=False)  # ❌
-```
-
-**错误2**: 返回格式改变
-```python
-# 旧代码解析失败:
-text = item[1][0]       # ❌ IndexError
-confidence = item[1][1]  # ❌ IndexError
-```
-
-**错误3**: 模型名称错误
-```python
-model = create_model("PP-OCRv5_server")  # ❌ Model not supported
-```
-
----
-
-## 🎯 研究任务清单
-
-### Phase 1: API研究 (优先级高)
-
-- [ ] **阅读官方文档**
-  - PP-OCRv5完整文档
-  - PaddleX API文档
-  - 迁移指南 (如果有)
-
-- [ ] **理解新API**
-  ```python
-  # 需要搞清楚:
-  1. 正确的导入方式
-  2. 模型初始化方法
-  3. predict()参数和返回格式
-  4. 如何区分手写/印刷
-  5. 是否有手写检测专用功能
-  ```
-
-- [ ] **编写测试脚本**
-  - `test_pp_ocrv5_api.py` - 测试基础API调用
-  - 打印完整的result数据结构
-  - 对比v4和v5的返回差异
-
-### Phase 2: 服务器适配
-
-- [ ] **重写服务器代码**
-  - 适配新API
-  - 正确解析返回数据
-  - 保持REST接口兼容
-
-- [ ] **测试稳定性**
-  - 测试10个PDF样本
-  - 检查GPU利用率
-  - 对比v4性能
-
-### Phase 3: 手写检测功能
-
-- [ ] **查找手写检测能力**
-  ```python
-  # 可能的方式:
-  1. result中是否有 text_type 字段?
-  2. 是否有专门的 handwriting_detection 模型?
-  3. 是否有置信度差异可以利用?
-  4. PP-Structure 的 layout 分析?
-  ```
-
-- [ ] **对比测试**
-  - v4 (当前方案) vs v5
-  - 准确率、召回率、速度
-  - 手写检测能力
-
-### Phase 4: 集成决策
-
-- [ ] **性能评估**
-  - 如果v5更好 → 升级
-  - 如果改进不明显 → 保持v4
-
-- [ ] **文档更新**
-  - 记录v5使用方法
-  - 更新PADDLEOCR_STATUS.md
-
----
-
-## 🔍 调试技巧
-
-### 1. 查看完整返回数据
-
-```python
-import pprint
-result = model.predict(image)
-pprint.pprint(result)  # 完整输出所有字段
-
-# 或者
-import json
-print(json.dumps(result, indent=2, ensure_ascii=False))
-```
-
-### 2. 查找官方示例
-
-```bash
-# 在服务器上查找PaddleOCR安装示例
-find ~/Project/paddleocr-server/venv/lib/python3.12/site-packages/paddleocr -name "*.py" | grep example
-
-# 查看源码
-less ~/Project/paddleocr-server/venv/lib/python3.12/site-packages/paddleocr/paddleocr.py
-```
-
-### 3. 查看可用模型
-
-```python
-from paddlex.inference.models import OFFICIAL_MODELS
-print(OFFICIAL_MODELS)  # 列出所有支持的模型名
-```
-
-### 4. Web文档搜索
-
-重点查看:
-- https://github.com/PaddlePaddle/PaddleOCR
-- https://www.paddleocr.ai
-- https://github.com/PaddlePaddle/PaddleX
-
----
-
-## 📂 文件结构
-
-```
-/Volumes/NV2/pdf_recognize/
-├── CURRENT_STATUS.md          # 当前状态文档 ✅
-├── NEW_SESSION_HANDOFF.md     # 本文件 ✅
-├── PADDLEOCR_STATUS.md        # 详细技术文档 ✅
-├── SESSION_INIT.md            # 初始会话信息
-│
-├── paddleocr_client.py        # 稳定客户端 (v2.7.3) ✅
-├── paddleocr_server_v5.py     # v5服务器 (未完成) ⚠️
-│
-├── test_paddleocr_client.py           # 基础测试
-├── test_mask_and_detect.py            # 遮罩+检测
-├── test_opencv_separation.py          # Method 1+2
-├── test_opencv_advanced.py            # Method 3 (最佳) ✅
-├── extract_signatures_paddleocr_improved.py  # 完整Pipeline
-│
-└── check_rejected_for_missing.py      # 诊断脚本
-```
-
-**服务器端** (`ssh gblinux`):
-```
-~/Project/paddleocr-server/
-├── paddleocr_server.py        # v2.7.3稳定版 ✅
-├── paddleocr_server_v5.py     # v5版本 (待完成) ⚠️
-├── paddleocr_server_backup.py # 备份
-├── server_stable.log          # 当前运行日志
-└── venv/                      # 虚拟环境
-```
-
----
-
-## ⚡ 快速启动
-
-### 启动稳定服务器 (v2.7.3)
-
-```bash
-ssh gblinux
-cd ~/Project/paddleocr-server
-source venv/bin/activate
-python paddleocr_server.py
-```
-
-### 测试连接
-
-```bash
-# 本地Mac
-cd /Volumes/NV2/pdf_recognize
-source venv/bin/activate
-python test_paddleocr_client.py
-```
-
-### 创建新研究分支
-
-```bash
-cd /Volumes/NV2/pdf_recognize
-git checkout -b pp-ocrv5-research
-```
-
----
-
-## 🚨 注意事项
-
-### 1. 不要破坏稳定版本
-
-- `paddleocr-improvements` 分支保持稳定
-- 所有v5实验在新分支 `pp-ocrv5-research`
-- 服务器保留 `paddleocr_server.py` (v2.7.3)
-- 新代码命名: `paddleocr_server_v5.py`
-
-### 2. 环境隔离
-
-- 服务器虚拟环境可能需要重建
-- 或者用Docker隔离v4和v5
-- 避免版本冲突
-
-### 3. 性能测试
-
-- 记录v4和v5的具体指标
-- 至少测试10个样本
-- 包括速度、准确率、召回率
-
-### 4. 文档驱动
-
-- 每个发现记录到文档
-- API用法写清楚
-- 便于未来维护
-
----
-
-## 📊 成功标准
-
-### 最低目标
-
-- [ ] 成功运行PP-OCRv5基础OCR
-- [ ] 理解新API调用方式
-- [ ] 服务器稳定运行
-- [ ] 记录完整文档
-
-### 理想目标
-
-- [ ] 发现手写检测功能
-- [ ] 性能超过v4方案
-- [ ] 简化Pipeline复杂度
-- [ ] 提升准确率 > 90%
-
-### 决策点
-
-**如果v5明显更好** → 升级到v5，废弃v4
-**如果v5改进不明显** → 保持v4，v5仅作研究记录
-**如果v5有bug** → 等待官方修复，暂用v4
-
----
-
-## 📞 问题排查
-
-### 遇到问题时
-
-1. **先查日志**: `tail -f ~/Project/paddleocr-server/server_stable.log`
-2. **查看源码**: 在venv里找PaddleOCR代码
-3. **搜索Issues**: https://github.com/PaddlePaddle/PaddleOCR/issues
-4. **降级测试**: 确认v2.7.3是否还能用
-
-### 常见问题
-
-**Q: 服务器启动失败?**
-A: 检查numpy版本 (需要 < 2.0)
-
-**Q: 找不到模型?**
-A: 模型名可能变化，查看OFFICIAL_MODELS
-
-**Q: API调用失败?**
-A: 对比官方文档，可能参数格式变化
-
----
-
-## 🎓 学习资源
-
-### 官方文档
-
-1. **PP-OCRv5**: https://www.paddleocr.ai/main/en/version3.x/algorithm/PP-OCRv5/PP-OCRv5.html
-2. **PaddleOCR GitHub**: https://github.com/PaddlePaddle/PaddleOCR
-3. **PaddleX**: https://github.com/PaddlePaddle/PaddleX
-
-### 相关技术
-
-- PaddlePaddle深度学习框架
-- PP-Structure文档结构分析
-- 手写识别 (Handwriting Recognition)
-- 版面分析 (Layout Analysis)
-
----
-
-## 💡 提示
-
-### 如果发现内置手写检测
-
-可能的用法:
-```python
-# 猜测1: 返回结果包含类型
-for item in result:
-    text_type = item.get('type')  # 'printed' or 'handwritten'?
-
-# 猜测2: 专门的layout模型
-from paddlex import create_model
-layout_model = create_model("PP-Structure")
-layout_result = layout_model.predict(image)
-# 可能返回: text, handwriting, figure, table...
-
-# 猜测3: 置信度差异
-# 手写文字置信度可能更低
-```
-
-### 如果没有内置手写检测
-
-那么当前OpenCV Method 3仍然是最佳方案，v5仅提供更好的OCR准确度。
-
----
-
-## ✅ 完成检查清单
-
-研究完成后，确保:
-
-- [ ] 新API用法完全理解并文档化
-- [ ] 服务器代码重写并测试通过
-- [ ] 性能对比数据记录
-- [ ] 决策文档 (升级 vs 保持v4)
-- [ ] 代码提交到 `pp-ocrv5-research` 分支
-- [ ] 更新 `CURRENT_STATUS.md`
-- [ ] 如果升级: 合并到main分支
-
----
-
-**祝研究顺利！** 🚀
-
-有问题随时查阅:
-- `CURRENT_STATUS.md` - 当前方案详情
-- `PADDLEOCR_STATUS.md` - 技术细节和问题分析
-
-**最重要**: 记录所有发现，无论成功或失败，都是宝贵经验！

SESSION_CHECKLIST.md

@@ -1,195 +0,0 @@
-# Session Handoff Checklist ✓
-
-## Before You Exit This Session
-
-- [x] All documentation written
-- [x] Test results recorded (7/10 signatures, 70% recall)
-- [x] Session initialization files created
-- [x] .gitignore configured
-- [x] Commit guide prepared
-- [ ] **Git commit performed** (waiting for user approval)
-
-## Files Created for Next Session
-
-### Essential Files ⭐
-- [x] **SESSION_INIT.md** - Read this first in next session
-- [x] **NEW_SESSION_PROMPT.txt** - Copy-paste prompt template
-- [x] **PROJECT_DOCUMENTATION.md** - Complete 24KB history
-- [x] **HOW_TO_CONTINUE.txt** - Visual guide
-
-### Supporting Files
-- [x] README.md - Quick start guide
-- [x] COMMIT_SUMMARY.md - Git instructions
-- [x] README_page_extraction.md - Page extraction docs
-- [x] README_hybrid_extraction.md - Signature extraction docs
-- [x] .gitignore - Configured properly
-
-### Working Scripts
-- [x] extract_pages_from_csv.py - Tested (100 files)
-- [x] extract_signatures_hybrid.py - Tested (5 files, 70% recall)
-- [x] extract_handwriting.py - Component script
-
-## What's Working ✅
-
-| Component | Status | Details |
-|-----------|--------|---------|
-| Page extraction | ✅ Working | 100 files tested |
-| VLM name extraction | ✅ Working | 100% accurate on 5 files |
-| CV detection | ⚠️ Conservative | Finds 70% of signatures |
-| VLM verification | ✅ Working | 100% precision, no false positives |
-| Overall system | ✅ Working | 70% recall, 100% precision |
-
-## What's Not Working / Unknown ⚠️
-
-| Issue | Status | Next Steps |
-|-------|--------|------------|
-| Missing 30% signatures | Known | Tune CV parameters |
-| Text layer method | Untested | Need PDFs with text |
-| Large-scale performance | Unknown | Test with 100+ files |
-| Full dataset (86K) | Unknown | Estimate time & optimize |
-
-## Critical Context to Remember 🧠
-
-1. **VLM coordinates are unreliable** (32% offset on test file)
-   - Don't use VLM for location detection
-   - Use VLM for name extraction only
-
-2. **Name-based approach is the solution**
-   - VLM extracts names ✓
-   - CV finds locations ✓
-   - VLM verifies regions ✓
-
-3. **Test file with coordinate issue:**
-   - `201301_2458_AI1_page4.pdf`
-   - VLM found 2 names but coordinates pointed to blank areas
-   - Actual signatures at 26% (reported as 58% and 68%)
-
-## To Start Next Session
-
-### Simple Method (Recommended)
-```bash
-cat /Volumes/NV2/pdf_recognize/NEW_SESSION_PROMPT.txt
-# Copy output and paste to new Claude Code session
-```
-
-### Manual Method
-Tell Claude:
-> "I'm continuing the PDF signature extraction project at `/Volumes/NV2/pdf_recognize/`. Please read `SESSION_INIT.md` and `PROJECT_DOCUMENTATION.md` to understand the current state. I want to [choose option from SESSION_INIT.md]."
-
-## Quick Commands Reference
-
-### View Documentation
-```bash
-less /Volumes/NV2/pdf_recognize/SESSION_INIT.md
-less /Volumes/NV2/pdf_recognize/PROJECT_DOCUMENTATION.md
-```
-
-### Run Scripts
-```bash
-cd /Volumes/NV2/pdf_recognize
-source venv/bin/activate
-python extract_signatures_hybrid.py  # Main script
-```
-
-### Check Results
-```bash
-ls -lh /Volumes/NV2/PDF-Processing/signature-image-output/signatures/*.png
-```
-
-### View Session Handoff
-```bash
-cat /Volumes/NV2/pdf_recognize/HOW_TO_CONTINUE.txt
-```
-
-## What Can Be Improved (Future Work)
-
-### Priority 1: Increase Recall
-- Current: 70%
-- Target: 90%+
-- Method: Tune CV parameters in lines 178-214 of extract_signatures_hybrid.py
-
-### Priority 2: Scale Testing
-- Current: 5 files tested
-- Next: 100 files
-- Future: 86,073 files (full dataset)
-
-### Priority 3: Optimization
-- Current: ~24 seconds per PDF
-- Consider: Parallel processing, batch VLM calls
-
-### Priority 4: Text Layer Testing
-- Current: Untested (all PDFs are scanned)
-- Need: Find PDFs with searchable text layer
-
-## Verification Steps
-
-Before next session, verify files exist:
-```bash
-cd /Volumes/NV2/pdf_recognize
-
-# Check essential docs
-ls -lh SESSION_INIT.md PROJECT_DOCUMENTATION.md NEW_SESSION_PROMPT.txt
-
-# Check working scripts
-ls -lh extract_pages_from_csv.py extract_signatures_hybrid.py
-
-# Check test results
-ls /Volumes/NV2/PDF-Processing/signature-image-output/signatures/*.png | wc -l
-# Should show: 7 (the 7 verified signatures)
-```
-
-## Known Good State
-
-### Environment
-- Python: 3.9+ with venv
-- Ollama: http://192.168.30.36:11434
-- Model: qwen2.5vl:32b
-- Working directory: /Volumes/NV2/pdf_recognize/
-
-### Test Data
-- 5 PDFs processed
-- 7 signatures extracted
-- All verified (100% precision)
-- 3 signatures missed (70% recall)
-
-### Output Files
-```
-201301_1324_AI1_page3_signature_張志銘.png (33 KB)
-201301_1324_AI1_page3_signature_楊智惠.png (37 KB)
-201301_2061_AI1_page5_signature_廖阿甚.png (87 KB)
-201301_2458_AI1_page4_signature_周寶蓮.png (230 KB)
-201301_2923_AI1_page3_signature_黄瑞展.png (184 KB)
-201301_3189_AI1_page3_signature_黄益辉.png (24 KB)
-201301_3189_AI1_page3_signature_黄辉.png (84 KB)
-```
-
-## Git Status (Pre-Commit)
-
-Files staged for commit:
-- [ ] extract_pages_from_csv.py
-- [ ] extract_signatures_hybrid.py
-- [ ] extract_handwriting.py
-- [ ] README.md
-- [ ] PROJECT_DOCUMENTATION.md
-- [ ] README_page_extraction.md
-- [ ] README_hybrid_extraction.md
-- [ ] .gitignore
-
-**Waiting for:** User to review docs and approve commit
-
-## Session Health Check ✓
-
-- [x] All scripts working
-- [x] Test results documented
-- [x] Issues identified and recorded
-- [x] Next steps defined
-- [x] Session continuity files created
-- [x] Git commit prepared
-
-**Status:** ✅ Ready for handoff
-
----
-
-**Last Updated:** October 26, 2025
-**Session End:** Ready for next session
-**Next Action:** User reviews docs → Git commit → Continue work

paper/codex_review_gpt55_v4_round7.md

@@ -16,6 +16,8 @@ The remaining issues are packaging / copy-edit blockers, not empirical blockers:
 
 ## Round-26 finding closure table
 
+Note: the round-26 file contains 11 labelled Major rows and 15 labelled Minor rows; this table covers every labelled row. The prompt's 9 Major / 12 Minor tally appears to merge duplicate themes.
+
 ### Major findings
 
 | # | Round-26 finding | v2 status | Round-27 note |

paper/codex_review_gpt55_v4_round8.md

@@ -0,0 +1,138 @@
+# Paper A Round 28 Review — codex GPT-5.5 v4 round 8
+
+Reviewer: gpt-5.5
+Date: 2026-05-14
+Target: paper/v4/paper_a_prose_v4_phase4.md + paper/v4/paper_a_methodology_v4_section_iii.md + paper/v4/paper_a_results_v4_section_iv.md (post round-2 fixes, commit b884d39)
+Prior reviewer artifacts: paper/codex_review_gpt55_v4_round7.md; paper/gemini_review_v4_round1.md; paper/opus_review_v4_round1.md
+Round-2 commit reviewed: b884d39
+
+## Verdict
+
+Minor Revision.
+
+Round-2 closes the empirical substance of Opus M2-M4 and the core of M1/M3: the deployed any-pair vs same-pair within-firm collision semantics are now separated in the body, the §IV K=3 mechanism-label regression is mostly repaired, §V headings now run A-H, and the Table XV-B cascade has been applied in the public §IV table sequence.
+
+However, the round-2 pass introduced or exposed several splice blockers: the abstract is now 261 words against the stated <=250 target; the new §IV-J Table XV sample-size footnote incorrectly says §IV-M.5 uses n=150,442 even though Script 44 / Tables XXIV-XXV use the 150,453 vector-complete substrate; §IV-I still points the ICCR calibration reader to "§IV-M Table XVI" although the relevant tables are now XXI-XXVI; and internal draft notes/checklists still contain stale Table XV-B / >=97% / hand-leaning language. No new statistical work is required, but the manuscript is not ready for splice until those are patched.
+
+## Round-1 panel finding closure table
+
+| Source | Finding | Current status | Evidence / note |
+|---|---|---|---|
+| codex r7 M1 | Abstract "Three independent feature-derived scores" | CLOSED | Current prose uses "Three feature-derived scores" and the shared-input caveat at paper/v4/paper_a_prose_v4_phase4.md:37 and :97; §III states the scores are not statistically independent at paper/v4/paper_a_methodology_v4_section_iii.md:113. |
+| codex r7 M2 | §I overclaimed Big-4 scope | CLOSED | Big-4 primary scope is explicit at §III-G, paper/v4/paper_a_methodology_v4_section_iii.md:19 and Phase 4 paper/v4/paper_a_prose_v4_phase4.md:39; corrected M3 language does not invalidate this closure. |
+| codex r7 M3 | Stale §III-D cross-reference | CLOSED | Pipeline / validation references now point to §III-L / §III-M / §III-I.4, e.g. paper/v4/paper_a_prose_v4_phase4.md:29. |
+| codex r7 M4 | §I repeated independent-score error | CLOSED | Phase 4 describes internal consistency, not independent validation, at paper/v4/paper_a_prose_v4_phase4.md:37 and :97. |
+| codex r7 M5 | §II not submission-ready as standalone | PARTIAL | §II still contains a review-pass note saying only the v4 paragraph is reproduced, paper/v4/paper_a_prose_v4_phase4.md:65. This is a splice-packaging issue. |
+| codex r7 M6 | Refs [42]-[44] absent / placeholders | SUPERSEDED | My round-7 claim was wrong against the current reference file: [42]-[44] are present at paper/paper_a_references_v3.md:87-91, and §II cites them at paper/v4/paper_a_prose_v4_phase4.md:67. |
+| codex r7 M7 | §V reified CPA mechanism labels | CLOSED | §V now uses descriptor-position language and explicitly rejects latent mechanism classes at paper/v4/paper_a_prose_v4_phase4.md:81 and :93. |
+| codex r7 M8 | §V speculative within-CPA unimodality explanation | CLOSED | §V-B restricts the result to composition + integer artefacts, paper/v4/paper_a_prose_v4_phase4.md:81. |
+| codex r7 M9 | §V limitations incomplete | CLOSED | §V-H lists nine v4 limitations plus five inherited v3.20.0 limitations at paper/v4/paper_a_prose_v4_phase4.md:111-139. |
+| codex r7 M10 | §V full-dataset scope overread | CLOSED | Scope limitation is explicit at paper/v4/paper_a_prose_v4_phase4.md:117 and §IV-K is narrow at paper/v4/paper_a_results_v4_section_iv.md:232-254. |
+| codex r7 M11 | §VI overclaimed cross-scope pipeline reproducibility | CLOSED | §VI now limits cross-scope support to K=3/Spearman robustness and leaves full ICCR generalisation to future work, paper/v4/paper_a_prose_v4_phase4.md:147-149. Opus M3 does not reopen this. |
+| codex r7 m1 | "candidate classifiers" wording | CLOSED | Current §V-G uses "candidate checks", paper/v4/paper_a_prose_v4_phase4.md:107. |
+| codex r7 m2 | Abstract word-count margin | OPEN | Round-2 M3 rewrite pushed the abstract to 261 words by `wc -w`, while the draft target at paper/v4/paper_a_prose_v4_phase4.md:9 is <=250. |
+| codex r7 m3 | Abstract omitted operational output | CLOSED | Abstract includes ICCR units and operational HC+MC per-document alarm, paper/v4/paper_a_prose_v4_phase4.md:11. |
+| codex r7 m4 | Contribution 4 overclaimed narrower scopes | CLOSED | Contribution 4 now says the threshold path is unsupported by composition decomposition, paper/v4/paper_a_prose_v4_phase4.md:49. |
+| codex r7 m5 | Contribution 8 overclaimed full-dataset check | CLOSED | Current contribution 8 is limited to annotation-free positive-anchor and unsupervised validation ceiling, paper/v4/paper_a_prose_v4_phase4.md:57. |
+| codex r7 m6 | "External validation" too broad | CLOSED | Current language is specificity-proxy / annotation-free / unsupervised-ceiling, e.g. paper/v4/paper_a_prose_v4_phase4.md:57 and :113. |
+| codex r7 m7 | §II LOOO sufficiency wording | CLOSED | §II frames LOOO as composition-sensitivity, not operational-classifier sufficiency, paper/v4/paper_a_prose_v4_phase4.md:67. |
+| codex r7 m8 | "Inherits and confirms" too strong | CLOSED | §V-B says v4 strengthens/extends by decomposition but does not overclaim direct validation, paper/v4/paper_a_prose_v4_phase4.md:79-81. |
+| codex r7 m9 | Firm A byte-level provenance | CLOSED | Inherited Script 28 provenance is stated at paper/v4/paper_a_prose_v4_phase4.md:85 and §III-H at paper/v4/paper_a_methodology_v4_section_iii.md:37. |
+| codex r7 m10 | Firm A alone did not anchor §IV-H | CLOSED | Pixel-identity anchor is Big-4 n=262 with all four firms listed, paper/v4/paper_a_results_v4_section_iv.md:143-153. |
+| codex r7 m11 | "Published box rule" not traceable | CLOSED | Current text uses inherited Paper A / v3.x box rule language, e.g. paper/v4/paper_a_results_v4_section_iv.md:215. |
+| codex r7 m12 | "Same per-CPA ranking" too strong | CLOSED | Residual Firm D/Firm C disagreement is disclosed at paper/v4/paper_a_prose_v4_phase4.md:97 and §IV-F at paper/v4/paper_a_results_v4_section_iv.md:102. |
+| codex r7 m13 | §V "candidate classifiers" residue | CLOSED | Replaced with "candidate checks" at paper/v4/paper_a_prose_v4_phase4.md:107. |
+| codex r7 m14 | Future-work audit-quality contrast needed caveat | CLOSED | Future work now keeps the Firm A vs B/C/D contrast descriptive, paper/v4/paper_a_prose_v4_phase4.md:149. |
+| codex r7 m15 | Conclusion underplayed operational output | CLOSED | §VI opens with five-way classifier and worst-case aggregation, paper/v4/paper_a_prose_v4_phase4.md:145. |
+| codex r7 new issue 1 | §II citation gap | SUPERSEDED | Refs [42]-[44] exist at paper/paper_a_references_v3.md:87-91. |
+| codex r7 new issue 2 | Stale close-out metadata | OPEN | Phase 4 close-out still says 243-244 words and "§V-G Limitations", paper/v4/paper_a_prose_v4_phase4.md:157-160; current abstract count is 261 and limitations are §V-H. |
+| Gemini M1 | Reject "statistically insignificant" firm heterogeneity framing | CLOSED | Current text says firm effects are large and not pool-size explained at paper/v4/paper_a_methodology_v4_section_iii.md:259-268 and paper/v4/paper_a_prose_v4_phase4.md:35. |
+| Gemini M2 | codex refs [42]-[44] error | CLOSED | References are present at paper/paper_a_references_v3.md:87-91. |
+| Gemini M3 | ICCR disclaimer adequacy | CLOSED | FAR is explicitly reframed as ICCR / specificity proxy at paper/v4/paper_a_methodology_v4_section_iii.md:185 and paper/v4/paper_a_results_v4_section_iv.md:159. |
+| Gemini M4 | K=3 demotion language | CLOSED for main body | §III-J and §V-D are correct at paper/v4/paper_a_methodology_v4_section_iii.md:76-90 and paper/v4/paper_a_prose_v4_phase4.md:89-93. Residual public wording "replicated vs not-replicated" in Table XI is flagged below as minor terminology residue. |
+| Gemini M5 | Feature-derived score caveat | CLOSED | Shared-input caveat appears in §III-K, §IV-F, and §V-E: paper/v4/paper_a_methodology_v4_section_iii.md:113; paper/v4/paper_a_results_v4_section_iv.md:79; paper/v4/paper_a_prose_v4_phase4.md:97. |
+| Gemini m1 | Internal draft notes/checklists | OPEN | Present in §III, §IV, and Phase 4: paper/v4/paper_a_methodology_v4_section_iii.md:3 and :431-447; paper/v4/paper_a_results_v4_section_iv.md:3 and :365-374; paper/v4/paper_a_prose_v4_phase4.md:3 and :153-162. |
+| Gemini m2 | Table XV-B vs XIX numbering | CLOSED in public body | Public document-level table is now Table XIX at paper/v4/paper_a_results_v4_section_iv.md:192 and §IV-M cascades through XX-XXVI at :266, :280, :300, :317, :329, :340, :353. Internal notes still stale. |
+| Gemini m3 | Word count note | OPEN | The note remains stale at paper/v4/paper_a_prose_v4_phase4.md:157 and the current abstract is over target. |
+| Gemini new issue | Table XV sample-size nuance | PARTIAL | A pointer to §III-G was added at paper/v4/paper_a_results_v4_section_iv.md:177, but the footnote misclassifies §IV-M.5 as n=150,442; Script 44 / Tables XXIV-XXV use the 150,453 vector-complete substrate per §III-G, paper/v4/paper_a_methodology_v4_section_iii.md:31. |
+| Opus M1 | §IV K=3 mechanism-label reversion | CLOSED for named tables/prose; MINOR RESIDUE | Tables IX/X/XIV/XVI/XVII now use descriptor-position or less-replication-dominated language at paper/v4/paper_a_results_v4_section_iv.md:81-100, :145-153, :217-226, :234-254. Public residue: Table XI still says "binary collapse, replicated vs not-replicated" at :104; internal §III open question still says hand-leaning at paper/v4/paper_a_methodology_v4_section_iii.md:445. |
+| Opus M2 | Table XV-B cascade | CLOSED in public body | Table XIX replaces XV-B at paper/v4/paper_a_results_v4_section_iv.md:192 and §IV-M is XX-XXVI at :266-353. No public Table XV-B reference remains; only internal notes at :3 and :370. |
+| Opus M3 | "98-100% within source firm" semantic conflation | CLOSED in body; ABSTRACT SHORT FORM | Body locations now separate deployed any-pair 98.8% / 76.7-83.7% from same-pair 97.0-99.96% at paper/v4/paper_a_prose_v4_phase4.md:35, :53, :87, :115, :147, :149 and §III at paper/v4/paper_a_methodology_v4_section_iii.md:99, :283, :285. The abstract uses a rounded any-pair-only 77-99% headline at paper/v4/paper_a_prose_v4_phase4.md:11, which is not misleading but omits the same-pair subrange. |
+| Opus M4 | Duplicate §V-G heading | CLOSED | §V headings now run A-H: paper/v4/paper_a_prose_v4_phase4.md:73, :77, :83, :89, :95, :99, :105, :109. |
+| Opus M5 | Stale "seven limitations" close-out note | OPEN internal | Phase 4 checklist still says "seven limitations" and "§V-G Limitations" at paper/v4/paper_a_prose_v4_phase4.md:160; the actual limitations heading is §V-H and has 14 items. |
+| Opus M6 | §IV-M composition table partial vs §III factorial | PARTIAL / LOW | §IV-M.1 remains a summary table at paper/v4/paper_a_results_v4_section_iv.md:266-276, while full factorial detail is in §III-I.4 at paper/v4/paper_a_methodology_v4_section_iii.md:61-68. This is acceptable if §IV-D points readers to a summary, but current §IV-D says diagnostics are "tabulated in §IV-M" at paper/v4/paper_a_results_v4_section_iv.md:23. |
+| Opus M7 | Mixed Spearman precision | OPEN / COPY-EDIT | §III reports 0.963/0.889/0.879 at paper/v4/paper_a_methodology_v4_section_iii.md:123-127, while §IV uses 0.9627/0.8890/0.8794 at paper/v4/paper_a_results_v4_section_iv.md:81-87. |
+| Opus minor 1 | Abstract word-count metadata | OPEN | Current abstract is 261 words; close-out note still says 243-244 at paper/v4/paper_a_prose_v4_phase4.md:157. |
+| Opus minor 2 | Internal draft notes | OPEN | Same as Gemini m1. |
+| Opus minor 3 | §IV-J Table XV-B pointer | CLOSED in body | Body now says Table XIX at paper/v4/paper_a_results_v4_section_iv.md:228. |
+| Opus minor 4 | Mixed decimal / percentage notation | OPEN / COPY-EDIT | Still mixed by design, e.g. 0.34 at paper/v4/paper_a_prose_v4_phase4.md:11 and 33.75% at :33. |
+| Opus minor 5 | v3.x §IV-F.1 cross-reference check | OPEN / SPLICE | v3.x §IV-F.1 references remain at paper/v4/paper_a_methodology_v4_section_iii.md:37 and paper/v4/paper_a_prose_v4_phase4.md:85. Verify during master splice. |
+| Opus minor 6 | Firm A 50/180 inherited provenance | CLOSED / DISCLOSED | Provenance is disclosed as inherited, not regenerated, at paper/v4/paper_a_methodology_v4_section_iii.md:37. |
+| Opus minor 7 | "FAR throughout" historical exception | PARTIAL | v4 framing disclaims FAR, but historical "FAR" appears in paper/v4/paper_a_results_v4_section_iv.md:159 and paper/v4/paper_a_methodology_v4_section_iii.md:185. It is correctly caveated, not an empirical issue. |
+| Opus minor 8 | MC band proportions | CLOSED | §IV-J proportions match Table XV rows at paper/v4/paper_a_results_v4_section_iv.md:181-186 and prose at :215. |
+| Opus minor 9 | §V-G item count | OPEN internal | Same as Opus M5. |
+| Opus minor 10 | LOOO range 1.8-12.8 pp | CLOSED | §IV Table XIII supports 1.76-12.77 pp, paper/v4/paper_a_results_v4_section_iv.md:131-139. |
+| Opus minor 11 | Abstract 98-100 public statement | SUPERSEDED | Replaced by rounded any-pair 77-99% at paper/v4/paper_a_prose_v4_phase4.md:11. |
+| Opus new issue 1 | Human-in-the-loop not operationalised | OPEN / COPY-EDIT | The positioning remains in Abstract and §III-M, paper/v4/paper_a_prose_v4_phase4.md:11 and paper/v4/paper_a_methodology_v4_section_iii.md:334, but no concrete review workflow is specified. |
+| Opus new issue 2 | Feature-derived caveat breaks down in §IV | CLOSED for main §IV | §IV tables and prose were repaired, except the Table XI binary-collapse label noted above. |
+| Opus new issue 3 | §III-M nine-tool table unnumbered | OPEN / COPY-EDIT | The validation table at paper/v4/paper_a_methodology_v4_section_iii.md:318-329 remains unnumbered. |
+| Opus new issue 4 | §I pipeline-step vs framework-element framing | OPEN / COPY-EDIT | The eight-item enumeration remains at paper/v4/paper_a_prose_v4_phase4.md:29. |
+
+## Round-2 induced issues
+
+1. **Abstract now exceeds the target word count.** `sed -n '11p' paper/v4/paper_a_prose_v4_phase4.md | wc -w` returns 261. The draft note at paper/v4/paper_a_prose_v4_phase4.md:9 sets an IEEE Access <=250 target, and the close-out note at :157 is stale.
+
+2. **The new Table XV sample-size footnote is partly wrong.** The pointer to §III-G is useful, but paper/v4/paper_a_results_v4_section_iv.md:177 says §IV-M.5 uses n=150,442. §III-G says Scripts 40b, 43, and 44 use the 150,453 vector-complete substrate, paper/v4/paper_a_methodology_v4_section_iii.md:31, and Script 44's report states n_big4_sources = 150,453. Correct the footnote to distinguish descriptor-complete sections from vector-complete §IV-M.2 / §IV-M.3 / §IV-M.5.
+
+3. **Public table cross-reference is stale.** §IV-I still says the consolidated v4-new ICCR calibration appears in "§IV-M Table XVI", paper/v4/paper_a_results_v4_section_iv.md:161. Current Table XVI is the K=3 firm cross-tab at :217; the ICCR calibration tables are XXI-XXVI at :280, :300, :317, :329, :340, :353.
+
+4. **Internal notes remain stale.** §IV's draft note still says v3.2 and Table XV-B, paper/v4/paper_a_results_v4_section_iv.md:3; the §IV close-out checklist repeats Table XV-B at :370. §III's internal cross-reference index still says within-firm collision concentration >=97% at paper/v4/paper_a_methodology_v4_section_iii.md:438 and "C1 hand-leaning" at :445. These are internal-only splice items, not empirical blockers, but they must be stripped or updated.
+
+5. **Minor terminology residue remains outside Opus's named M1 sites.** §III and §IV Table XI still call a K=3 / box-rule binary collapse "replicated vs not-replicated", paper/v4/paper_a_methodology_v4_section_iii.md:131 and paper/v4/paper_a_results_v4_section_iv.md:104. Because this is not the byte-identical positive-anchor ground-truth subset, a stricter v4 wording would be "high-cos / low-dHash vs other positions" or "replication-dominated vs less-replication-dominated."
+
+6. **"Less-replication-dominated" is long but not broken.** The phrase is readable in the public replacement sites. The only sentence I would smooth at copy-edit is paper/v4/paper_a_results_v4_section_iv.md:215 ("per-CPA less-replication-dominated ranking"), which could become "per-CPA ranking away from the replication-dominated corner."
+
+## Provenance spot-checks
+
+1. **Within-firm any-pair rates from §IV Table XXV.** From paper/v4/paper_a_results_v4_section_iv.md:340-349:
+   - Firm A: 14,447 / 14,622 = 98.8032% -> 98.8%.
+   - Firm B: 371 / 484 = 76.6529% -> 76.7%.
+   - Firm C: 149 / 178 = 83.7079% -> 83.7%.
+   - Firm D: 106 / 137 = 77.3723% -> 77.4%.
+   These match the corrected 76.7-98.8% any-pair range and the B/C/D 76.7-83.7% summary. Script 44's report gives the same matrix at /Volumes/NV2/PDF-Processing/signature-analysis/reports/v4_big4/firm_matched_pool/firm_matched_pool_report.md:36-43.
+
+2. **Same-pair joint range 97.0-99.96%.** §III-L.4 reports 11,314/11,319, 85/87, 54/55, and 64/66 at paper/v4/paper_a_methodology_v4_section_iii.md:281. The arithmetic is 99.9558%, 97.7011%, 98.1818%, and 96.9697%, matching the rounded 99.96% / 97.7% / 98.2% / 97.0%. §IV repeats the rates at paper/v4/paper_a_results_v4_section_iv.md:349.
+
+3. **Pooled Big-4 any-pair per-signature ICCR 0.1102.** Script 43's report gives 16,578 / 150,453 = 0.1102 at /Volumes/NV2/PDF-Processing/signature-analysis/reports/v4_big4/pool_normalized_far/pool_normalized_report.md:42-50. A normal approximation half-width is 1.96 * sqrt(0.1102 * 0.8898 / 150453) = 0.00158, consistent with the reported Wilson [0.1086, 0.1118] in §IV-M.3, paper/v4/paper_a_results_v4_section_iv.md:300-305.
+
+4. **Per-pair conditional ICCR 0.234.** Script 40b's report gives dHash <= 5 conditional on cos > 0.95 as 70 / 299 = 0.23411 at /Volumes/NV2/PDF-Processing/signature-analysis/reports/v4_big4/inter_cpa_far_sweep/far_sweep_report.md:87-99. This matches §III-L.1 at paper/v4/paper_a_methodology_v4_section_iii.md:208 and §IV-M.2 at paper/v4/paper_a_results_v4_section_iv.md:294.
+
+## Updated round-7 closure reassessment
+
+Opus M3 does not invalidate my round-7 closure of M2 (Big-4 scope language) or M11 (cross-scope pipeline reproducibility). Those findings were about overextending Big-4/full-dataset scope and overclaiming cross-scope reproducibility. The corrected current prose keeps the primary scope Big-4, treats full-dataset evidence as a narrow K=3 + Spearman robustness check, and now reports the within-firm collision pattern as any-pair 76.7-98.8% plus same-pair 97.0-99.96%.
+
+If I re-graded round 7 against the corrected current drafts, none of my major closures would move back to PARTIAL on empirical grounds. I would, however, reopen the abstract word-count minor item because the M3 repair pushed the abstract over the <=250-word target.
+
+## Phase 5 readiness
+
+Partial.
+
+The empirical core is ready: no script rerun or statistical redesign is needed. M2-M4 are closed in manuscript body text, and M3 is substantively corrected. Phase 5 is blocked only by splice/copy-edit/factual-reference hygiene:
+
+1. Trim the abstract from 261 to <=250 words.
+2. Correct §IV-J line 177's sample-size footnote so §IV-M.2 / M.3 / M.5 are identified as vector-complete / pair-recomputed analyses, not n=150,442 descriptor-complete analyses.
+3. Fix §IV-I's stale "§IV-M Table XVI" pointer.
+4. Strip or update internal draft notes, checklists, §III cross-reference index, and stale Table XV-B / >=97% / hand-leaning language.
+5. Optionally smooth the residual "replicated vs not-replicated" binary-collapse label and the long "less-replication-dominated" phrase in §IV-J.
+
+## Recommended next-step actions
+
+1. **Copy-edit blocker:** trim the abstract by at least 11 words while preserving the corrected any-pair headline. Do not add same-pair detail to the abstract unless other text is removed.
+
+2. **Factual cross-reference blocker:** replace the §IV-J Table XV footnote with a precise version: descriptor-complete analyses use 150,442; vector/pair-recomputed analyses use 150,453, including Scripts 40b, 43, and 44 (§IV-M.2, M.3, M.5).
+
+3. **Cross-reference blocker:** change §IV-I's "§IV-M Table XVI" to "§IV-M.2 Table XXI" or to "§IV-M Tables XXI-XXVI", depending on whether the intended pointer is per-comparison ICCR only or the whole calibration block.
+
+4. **Splice blocker:** remove all internal notes/checklists before manuscript assembly, especially the stale §IV v3.2 / Table XV-B note, §III's >=97% cross-reference-index shorthand, and the Phase 4 "§V-G Limitations / seven limitations" checklist item.
+
+5. **Terminology cleanup:** consider renaming "binary collapse, replicated vs not-replicated" to descriptor-position language in §III-K and §IV-F, while retaining "replicated class" only for the byte-identical positive-anchor ground truth.

paper/codex_review_gpt55_v4_round9.md

@@ -0,0 +1,96 @@
+# Paper A Round 29 Review — codex GPT-5.5 v4 round 9 (final cross-check)
+
+Reviewer: gpt-5.5
+Date: 2026-05-14
+Target: paper/v4/paper_a_prose_v4_phase4.md + paper/v4/paper_a_methodology_v4_section_iii.md + paper/v4/paper_a_results_v4_section_iv.md (post round-4)
+Prior reviewer artifacts: paper/codex_review_gpt55_v4_round{7,8}.md; paper/gemini_review_v4_round{1,2}.md; paper/opus_review_v4_round{1,2}.md
+Round-4 commit reviewed: d3ddf746f4555a68072ec2dacf5a455d6334033d
+
+## Verdict
+
+Minor Revision.
+
+Round-4 closes the intended structural shape of Opus N1-N4, but two provenance-sensitive wordings must be corrected before manuscript-splice assembly:
+
+1. The §IV-M.4 denominator reconciliation is arithmetically right but describes the 379 mixed-firm PDFs as Firm C "majority firm" cases. Direct verification against the database and Script 45 shows they are 1:1 Firm C/Firm D ties assigned to Firm C by the mode/tie-break implementation.
+2. The new §III-M Table XXVII composition-decomposition row is structurally right but its untested-assumption column over-attributes the within-firm corroboration to Script 39c. Script 39c's emitted raw dHash per-firm tests reject unimodality; the jittered non-Big-4 per-firm support is not emitted in the current Script 39c/39d reports.
+
+Phase 5 convergence by panel vote is still achieved: Gemini round-2 = Accept, Opus round-2 = Minor Revision, codex round-9 = Minor Revision. That is 3/3 reviewers in the Accept/Minor band. I would not splice the current text verbatim, but the remaining changes are small text/provenance patches, not new empirical work.
+
+## N1–N4 closure verification
+
+**N1. Firm C denominator reconciliation — partially closed, but not clean.**
+
+The reconciliation landed at paper/v4/paper_a_results_v4_section_iv.md:325. The arithmetic is correct: §IV-J Table XIX reports single-firm document rows with Firm C $n = 19{,}122$ and excludes 379 mixed-firm PDFs at paper/v4/paper_a_results_v4_section_iv.md:192-213; §IV-M.4 reports mode-assigned per-firm D2 denominators summing to $75{,}233$, with Firm C $n = 19{,}501$ at paper/v4/paper_a_results_v4_section_iv.md:317-325. Script 45's implementation maps each document to a firm mode via `np.unique(..., return_counts=True)` and `np.argmax(counts)` at signature_analysis/45_doc_level_far_full_5way.py:249-256, and the Script 45 report gives Firm C $n = 19{,}501$ in the per-firm doc-level table.
+
+The problem is the explanatory phrase "majority firm." A direct SQLite check of the Script 45 substrate returns exactly one mixed-document pattern: `Firm C:1+Firm D:1 | 379`; majority docs = 0 and tie docs = 379. Thus all 379 mixed-firm PDFs resolve to Firm C because of the sorted mode tie-break, not because Firm C is the empirical majority within those PDFs. Replace the current sentence with tie-break language, e.g. "The 379 mixed-firm PDFs are all 1:1 Firm C/Firm D mixed-firm documents; Script 45's mode-of-firms implementation assigns tied modes to the first sorted firm, so they are assigned to Firm C."
+
+**N2. Composition-decomposition added to §III-M ten-tool table — structurally closed, provenance wording needs correction.**
+
+The table now has ten rows, with composition decomposition inserted first at paper/v4/paper_a_methodology_v4_section_iii.md:318-331. §I contribution 8 now says "ten partial-evidence diagnostics (§III-M Table XXVII)" at paper/v4/paper_a_prose_v4_phase4.md:57, and §VI item 8 now says "ten-tool unsupervised-validation collection (§III-M Table XXVII)" at paper/v4/paper_a_prose_v4_phase4.md:147. This closes the count/cross-reference part of N2.
+
+The new composition row's assumption cell at paper/v4/paper_a_methodology_v4_section_iii.md:322 is not accurate as written. It says within-firm dip tests on every firm with $n \geq 500$ in Script 39c corroborate absence of within-population bimodality. Script 39c does run the eligible non-Big-4 per-firm tests at signature_analysis/39c_v4_midsmall_signature_diptest.py:147-158, but its emitted report shows raw dHash rejects in all ten eligible mid/small firms, while cosine fails to reject. The accurate decomposition is what §III-I.4 states more carefully: raw dHash rejects in all 14 firms, Big-4 per-firm dHash rejection disappears after jitter in Script 39d, and Big-4 pooled dHash needs both firm-mean centring and jitter in Script 39e (paper/v4/paper_a_methodology_v4_section_iii.md:57-68; paper/v4/paper_a_results_v4_section_iv.md:270-276).
+
+The non-Big-4 jittered per-firm claim is also not cleanly provenance-emitted: paper/v4/paper_a_methodology_v4_section_iii.md:382 cites Script 39d / 39c for a non-Big-4 jittered-dHash range, but the current Script 39d report emits Big-4 per-firm plus pooled non-Big-4, not the ten individual mid/small-firm jittered rows. My read-only rerun of the same jitter procedure did confirm 0/10 non-Big-4 firms reject after jitter, but it produced a median-$p$ range of $0.3755$-$1.0$, not the manuscript's $[0.71, 1.00]$. Either add the emitted table to Script 39c/39d, or narrow the Table XXVII assumption cell to the scripted evidence already visible in §IV-M.1.
+
+**N3. §III-M table numbering — closed.**
+
+The §III-M table is now explicitly introduced as Table XXVII at paper/v4/paper_a_methodology_v4_section_iii.md:316-318. The caption, "Ten-tool unsupervised-validation collection with disclosed untested assumptions," matches the table content: ten diagnostic rows, each with a measure and an untested-assumption column. The numbering also follows §IV-M.6 Table XXVI at paper/v4/paper_a_results_v4_section_iv.md:353.
+
+**N4. Cross-firm hit matrix assumption disclosure — closed, contingent on the N1 footnote fix.**
+
+The old "None — direct descriptive observation" assumption is gone. The current Table XXVII row at paper/v4/paper_a_methodology_v4_section_iii.md:327 discloses both deployed-rule dependence and same-pair vs any-pair semantics: same-pair joint event $97.0$-$99.96\%$ within-firm versus any-pair $76.7$-$98.8\%$. Those values match §IV-M.5 Table XXV and the following same-pair sentence at paper/v4/paper_a_results_v4_section_iv.md:340-349, and Script 44 computes the matrices at signature_analysis/44_firm_matched_pool_regression.py:274-327.
+
+The row's reference to Script 45 mode-of-firms assignment is appropriate, but it points to the §IV-M.4 footnote. Once N1's "majority firm" wording is corrected to "tie-break assignment," N4 reads cleanly.
+
+## Round-4 induced issues
+
+1. **N1 footnote overcorrects from "undisclosed denominator" to a false "majority firm" explanation.** The current prose at paper/v4/paper_a_results_v4_section_iv.md:325 should not say the 379 mixed-firm PDFs resolve to Firm C as majority firm. They are all Firm C/Firm D 1:1 ties.
+
+2. **The new Table XXVII composition row makes an existing provenance weakness load-bearing.** The row at paper/v4/paper_a_methodology_v4_section_iii.md:322 should not cite Script 39c as though Script 39c alone corroborates absence of within-population bimodality. Script 39c raw dHash rejects in all ten eligible mid/small firms; the no-rejection claim requires integer jitter. The current committed reports do not emit the ten non-Big-4 jittered per-firm values.
+
+3. **Ten-tool propagation is otherwise clean in public prose.** The public §I and §VI claims now say ten-tool / ten partial-evidence diagnostics at paper/v4/paper_a_prose_v4_phase4.md:57 and :147. I found no public leftover "nine-tool" validation claim except internal working material marked for removal: the Phase 4 draft note at paper/v4/paper_a_prose_v4_phase4.md:3 and the §III cross-reference checklist at paper/v4/paper_a_methodology_v4_section_iii.md:434-442. The separate "first nine limitations" statement at paper/v4/paper_a_prose_v4_phase4.md:111 is a limitations count, not a validation-tool count.
+
+4. **No new FAR/ICCR regression found.** The manuscript continues to avoid treating inter-CPA ICCR as true FAR in the public prose checked here. The remaining issues are denominator/tie-break wording and composition-diagnostic provenance.
+
+## Phase 5 round-3 convergence audit
+
+| Reviewer artifact | Verdict | Post-round-4 interpretation |
+|---|---|---|
+| Gemini round-2 | Accept | Accept remains within the convergence band; Gemini did not have round-4 in view. |
+| Opus round-2 | Minor Revision | N1-N4 were the requested round-4 targets. N3/N4 are closed; N1/N2 need wording/provenance cleanup. |
+| codex GPT-5.5 round-9 | Minor Revision | Current text is close, but not splice-ready verbatim because two new/retained provenance wordings are inaccurate. |
+
+Panel convergence on Accept/Minor consensus is **yes: 3 of 3 reviewers** are in the Accept/Minor band. The Phase 5 gate is therefore met by vote-count logic, but I recommend closing it only after the two "must do now" text patches below are applied and committed. No new empirical analysis or new full review round is required.
+
+## Splice readiness checklist
+
+**Must do now before splice assembly**
+
+1. Patch paper/v4/paper_a_results_v4_section_iv.md:325: replace "mode-of-firms (majority firm)" / "resolve to Firm C as the majority firm" with the actual 1:1 Firm C/Firm D tie-break explanation.
+
+2. Patch paper/v4/paper_a_methodology_v4_section_iii.md:322: revise the composition-decomposition row's untested-assumption cell so it does not imply Script 39c raw within-firm tests support the dHash no-bimodality claim. Either cite only the emitted Big-4 jittered evidence (Script 39d) plus Big-4 centred+jittered evidence (Script 39e), or emit/cite a proper ten-firm non-Big-4 jittered table.
+
+3. If retaining the non-Big-4 jittered per-firm claim, reconcile paper/v4/paper_a_methodology_v4_section_iii.md:59 and :382 plus paper/v4/paper_a_prose_v4_phase4.md:31 and :81 with a committed script/report. If not retaining it, narrow those sentences to the evidence already emitted in §IV-M.1.
+
+4. Re-run a targeted grep after patching: `rg -n "majority firm|9 tools|nine-tool|Script 39c|jittered-dHash" paper/v4`.
+
+**Splice-time mechanical strip**
+
+1. Remove the Phase 4 draft note at paper/v4/paper_a_prose_v4_phase4.md:3, which still contains the internal stale "nine-tool" wording.
+
+2. Remove the Phase 4 close-out notes at paper/v4/paper_a_prose_v4_phase4.md:153 onward before moving prose into the master manuscript.
+
+3. Remove the §III author cross-reference checklist at paper/v4/paper_a_methodology_v4_section_iii.md:434-450; it still says "9 tools" at line 442 and is explicitly marked "remove before submission."
+
+4. During master-file assembly, recheck table numbering after the actual splice, because Table XXVII currently lives in §III while Tables XX-XXVI are in §IV-M.
+
+## Recommended next-step actions
+
+1. Apply the N1 tie-break wording patch in §IV-M.4.
+
+2. Apply the N2 Table XXVII composition-row provenance patch; decide whether to emit the missing non-Big-4 jittered per-firm table or narrow the claim.
+
+3. Run the targeted grep in the checklist and commit the patch as the final Phase 5 text cleanup.
+
+4. Proceed to manuscript-master splice with the internal-note/checklist strip. Partner Jimmy review can then treat the manuscript as Phase 5-converged rather than re-litigating the empirical core.

paper/codex_review_gpt55_v4_round_3axis.md

@@ -0,0 +1,114 @@
+[軸 1]
+
+[MAJOR] §I Contributions, L42  
+原句：「resolves the ambiguity between *style consistency* and *image reproduction*」  
+問題：這比摘要語氣強。descriptor-only framework 不能真正「解開」style consistency 與 image reproduction 的機制歸因，§V-H 也說不能分離。  
+修改建議：改為「provides complementary evidence for screening cases where style consistency and image reproduction hypotheses diverge」。
+
+[MAJOR] §III-H.1, L314  
+原句：「High-confidence non-hand-signed (HC)」  
+問題：作為 rule label 可接受，但正文與表格反覆使用時，容易讀成已驗證分類結果，而非 screening label。  
+修改建議：改為「High-confidence image-reuse screening label (HC)」，並在首次定義處明說「label names are operational labels, not ground-truth classes」。
+
+[MAJOR] §III-H.1, L314  
+原句：「Both descriptors converge on strong replication evidence.」  
+問題：「strong replication evidence」過強；目前只保證兩個影像相似 descriptor 同時落入 rule box，不能保證 replication mechanism。  
+修改建議：改為「Both descriptors converge on image-similarity evidence consistent with replication」。
+
+[MAJOR] §III-H.1, L318  
+原句：「Likely hand-signed (LH): Cosine $\leq 0.837$.」  
+問題：沒有 hand-signed ground truth，不能把 low-similarity screening bin 命名成「likely hand-signed」而不冒認 ground-truth status。  
+修改建議：改為「Low-replication-similarity (LRS)」或「Low-alert similarity」，保留舊縮寫可在括號說明。
+
+[MAJOR] §V-C, L1060  
+原句：「similar, milder production-related reuse patterns at Firms B/C/D」  
+問題：這裡把 Firms B/C/D 的較溫和 within-firm collision 解讀為 production-related reuse，和 §V-H 的三機制不可分離聲明不一致。  
+修改建議：改為「similar, milder within-firm collision patterns, whose mechanisms may include template reuse, digitisation-pipeline homogeneity, or signing-style homogeneity」。
+
+[MINOR] §V-F, L1074  
+原句：「the deployed five-way classifier is characterised at three units」  
+問題：§V-H L1100 說 MC/HSC 與 document worst-case rule 未被本診斷組重新 characterise；這句像是整個 five-way classifier 都完成 ICCR calibration。  
+修改建議：改為「the HC sub-rule and document-level alarm definitions derived from the five-way output are characterised...」。
+
+[MINOR] §I Contributions, L44  
+原句：「Composition decomposition disproves the distributional-threshold path.」  
+問題：「disproves」語氣過硬；目前是對本資料與本診斷下不支持 natural-threshold reading。  
+修改建議：改為「does not support」或「rules out within the tested diagnostics」。
+
+[軸 2]
+
+[MAJOR] §III-G vs §III-L, L286 / L458  
+原句：「We analyse signatures at two units of resolution.」  
+問題：§III-G 說兩個 units（signature/accountant），§III-L 又說 calibration 有三個 units（per-comparison/per-signature/per-document）。讀者第一次讀會混淆「statistical summary unit」與「calibration/reporting unit」。  
+修改建議：在 §III-G 結尾加一個 bridge：accountant/signature 是 descriptor-summary units；§III-L 的 three units 是 ICCR reporting units。
+
+[MAJOR] §III-H.2, L326  
+原句：「The calibration distinguishes two reference populations」  
+問題：Firm A 後文反覆說不是 calibration anchor；這句仍像 v3 殘留，讓 Firm A 看起來參與 threshold calibration。  
+修改建議：改為「The supporting diagnostics use two reference populations」。
+
+[MAJOR] §III-H.1 / §III-L, L320 / L456  
+原句：「retain their prior calibration provenance」  
+問題：§III-L 說本分析不 re-derive thresholds，但標題仍叫 threshold calibration，且 §III-H.1 只在 L320 一句帶過。第一次閱讀時不夠清楚：deployed 5-way rule 是既有 rule，ICCR 是行為刻畫，不是重新最佳化。  
+修改建議：在 §III-H.1 後加一小段明確列出：「rule definition」「what §III-L calibrates」「what remains from supplement」。
+
+[MAJOR] §III-I.2 / §III-J, L342 / L369  
+原句：「K=2 / K=3 Gaussian mixture fits」  
+問題：K=2/K=3 數字、BIC、解讀在 §III-I.2 與 §III-J 重複，仍有 v3 splice 的疊床架屋感。  
+修改建議：§III-I.2 只保留「mixture path checked and demoted」摘要，完整模型細節集中到 §III-J。
+
+[MINOR] §III-K, L432  
+原句：「Leave-one-firm-out reproducibility ... Discussed in §III-J above.」  
+問題：LOOO 已在 §III-J 詳述，又在 §III-K 作為 internal-consistency check；分類上不自然，且增加重複。  
+修改建議：把 §III-K.3 改成單句 cross-reference，或移回 §III-J。
+
+[MINOR] §III-L.1, L489  
+原句：「dHash provides $\sim 4.3\times$ further per-comparison specificity」  
+問題：這裡漏了 proxy/disclaimer；全文已避免 FAR，但「specificity」單獨出現會弱化 ICCR 語氣。  
+修改建議：改為「specificity-proxy refinement」或「ICCR refinement」。
+
+[MINOR] §III-L.2, L513  
+原句：「consistent with the $1 - (1 - p_{\text{pair}})^{n_{\text{pool}}}$ form」  
+問題：這是有用直覺，但 independence limit 與 within-firm violation 的關係應在同段提醒，否則會像正式模型。  
+修改建議：補一句「This is an intuition, not an independence assumption used for estimation」。
+
+[軸 3]
+
+[MAJOR] §III-F, L277  
+原句：「Hand-signing, by contrast, often yields high dHash similarity」  
+問題：這句預設同一 CPA 多次親簽時「overall layout typically preserved」，接近不應預設的個別 CPA 跨文件一致性。雖然用 often，但仍在方法動機處承擔了未驗證手寫行為。  
+修改建議：改為「One working hypothesis is that some hand-signed repetitions may preserve coarse layout while varying in fine execution; the classifier does not require this to hold for all CPAs」。
+
+[MAJOR] §III-H.1, L316  
+原句：「consistent with a CPA who signs very consistently」  
+問題：HSC 被解讀成「同一 CPA 簽名很一致但非 reproduction」，這直接把高 cosine / 高 dHash 的 same-CPA pattern 歸因到個人書寫一致性。  
+修改建議：改為「high feature similarity without structural corroboration; mechanism unresolved」。
+
+[MAJOR] §III-H.1, L318  
+原句：「Likely hand-signed」  
+問題：低 max-cosine 並不等於親簽；也可能是跨年度書寫變化、掃描/PDF pipeline、裁切或多 template variant。這是對「沒有高 same-CPA match」的過度解讀。  
+修改建議：改成 descriptor-based label，例如「low-replication-similarity」。
+
+[MINOR] §III-G A1, L292  
+原句：「within the cross-year same-CPA pool」  
+問題：A1 本身不是年度一致性假設，但「cross-year」容易被讀成跨年度簽名應可比或應一致。  
+修改建議：改為「within the observed same-CPA candidate pool pooled over years; this does not assume temporal stability of handwriting or scanning」。
+
+[MINOR] §III-L.6, L587  
+原句：「same-CPA repeatability signal」  
+問題：已加 caveat，但「repeatability」仍可能被讀成個人簽名一致性訊號。  
+修改建議：改為「observed same-CPA-pool excess signal, whose sources are not identifiable」。
+
+[MINOR] §IV-M.6, L1043  
+原句：「interpreted as a same-CPA repeatability signal」  
+問題：同上，且出現在 results consolidation，容易被當成結果主張。  
+修改建議：改為「reported as same-CPA-pool excess under §III-M caveats, not attributed to handwriting repeatability」。
+
+總體判讀
+
+軸 1 verdict：大方向已和摘要一致，但仍有幾個「validated detector / mechanism attribution」味道偏重的句子，尤其是「resolves ambiguity」、HC/LH label、§V-C 對 B/C/D 的 production-related reuse。  
+軸 2 verdict：v3 殘留大多已被 demote，但 §III 的敘事仍偏重複；最大問題是 unit taxonomy 與 calibration/re-characterisation 範圍需要更早講清。  
+軸 3 verdict：沒有發現核心計算邏輯必然依賴「同一 CPA 或跨年度簽名必須一致」；但若干命名與動機句會讓讀者以為有這個假設。
+
+是否可送 partner 最終審查：可，但建議先做一輪小修，主要是改 label/語氣與 §III roadmap。  
+BLOCKER：無。

paper/export_v3.py

@@ -153,9 +153,19 @@ LATEX_TOKEN_REPLACEMENTS = [
     (r"\\leq(?![A-Za-z])", "≤"), (r"\\geq(?![A-Za-z])", "≥"),
     (r"\\neq(?![A-Za-z])", "≠"), (r"\\approx(?![A-Za-z])", "≈"),
     (r"\\equiv(?![A-Za-z])", "≡"), (r"\\sim(?![A-Za-z])", "~"),
+    (r"\\in(?![A-Za-z])", "∈"), (r"\\notin(?![A-Za-z])", "∉"),
     (r"\\to(?![A-Za-z])", "→"), (r"\\rightarrow(?![A-Za-z])", "→"),
     (r"\\leftarrow(?![A-Za-z])", "←"), (r"\\Rightarrow(?![A-Za-z])", "⇒"),
     (r"\\Leftarrow(?![A-Za-z])", "⇐"),
+    # Delimiters spelled as commands
+    (r"\\lvert(?![A-Za-z])", "|"), (r"\\rvert(?![A-Za-z])", "|"),
+    (r"\\lVert(?![A-Za-z])", "‖"), (r"\\rVert(?![A-Za-z])", "‖"),
+    # Symbols / operators that linearise to text
+    (r"\\checkmark(?![A-Za-z])", "✓"),
+    (r"\\bullet(?![A-Za-z])", "•"),
+    (r"\\max(?![A-Za-z])", "max"), (r"\\min(?![A-Za-z])", "min"),
+    (r"\\log(?![A-Za-z])", "log"), (r"\\ln(?![A-Za-z])", "ln"),
+    (r"\\exp(?![A-Za-z])", "exp"),
     # Binary operators
     (r"\\times(?![A-Za-z])", "×"), (r"\\cdot(?![A-Za-z])", "·"),
     (r"\\pm(?![A-Za-z])", "±"), (r"\\mp(?![A-Za-z])", "∓"),

paper/fusion_review_todo.md

@@ -0,0 +1,86 @@
+# 融合審查 — 修訂 TODO 清單
+
+**來源**：`Fusion_Review_Crosscheck_Table.docx`（Gemini 3.1 Pro + ChatGPT 5.5 + Opus 4.8 → Opus 4.8 融合，2026-06-19）
+**對象稿件**：`paper/v13_build/paper_v13_filled.md`（rev7 submission 單一真實來源）
+**建議結論**：Major Revision
+**統計**：致命共識 ×3 · 嚴重 ×4 · 融合新增 ×5 · 改向/不採納 ×1 · 結構/Minor ×15
+
+投票圖例：★ = 強烈標記 / ● = 提及 / ○ = 弱提及 / — = 未提。欄位 G=Gemini, C=ChatGPT, O=Opus。
+
+執行順序：① F1 → ② F2+G1 → ③ F3、F7 與宣稱降溫 → ④ F4–F6 與 F5 穩健性套組 → ⑤ 結構與 Minor。
+
+---
+
+## Tier 1 — 致命（三審高共識，不修則中心宣稱垮）
+
+- [x] **F1 — 校準 null 類別錯誤**（G★ C○ O★）｜偏誤方向 anti-conservative ✅ 文字完成（abstract/§I/§III-D/§III-E/§IV-C/§V/§VI）
+  - 已做：全文 ICCR→specificity proxy；撤回所有「139×／40–59×」式比較；§III-E 新增 between vs within 區分、明說 within-CPA 偽陽率不可估計、ICCR 連 bound 都不是、偏誤 anti-conservative。
+  - ✅ benchmark 問題已 §III-E 主動防守：無 OUR-population 親簽標記、公開集屬不同族群/字體/管線→借入會重引跨分布假設（違 label-free 初衷），故報告限制非誤導 proxy。不做（用戶/領域判斷確認做不出）。
+- [x] **F2 — Firm 為被混淆的 treatment**（G★ C★ O★）✅ 完成（PDF 解析具名證據）
+  - ✅ 880 份 PDF 管線審計：純掃描 2013 82% → 2021 崩到 1%；metadata 點名掃描機型（Fuji Xerox D125/ApeosPort）→ Table V 寫入 §V-B；量測基底本身在 2020/21 轉變 = 混淆具名鐵證。
+  - ✅ byte-identical era split：232 數位年代（偵測性放大，已 hedge）vs 30 掃描年代（管線無關鐵證，18 在 Firm A）→ §V-B + §IV-C 交叉註記。
+  - 腳本 pipeline_audit.py / pipeline_audit.csv 已存。
+- [x] **F3 — 吃重輸出零驗證**（G○ C★ O★）✅ 完成
+  - 已做：刪除 byte-identical「sanity check / recall 下界」修辭→改 prevalence signal；headline 全面 screening/triage/prevalence 口徑（abstract/貢獻/結論皆已轉），通讀確認無 detection-verdict 過度宣稱。
+
+## Tier 2 — 嚴重（雙審共識）
+
+- [x] **F4 — held-out ≠ blinded**（G○ C★ O★）｜Firm A 為已知 institutional positive case ✅ 完成
+  - 已做：§IV-C 標題改「Held-Out Benchmark: Firm A (a Known Positive)」+ 新增 quasi-positive institutional benchmark 段；abstract/§II 資料切分/§VI 全部改「known-positive benchmark / not a blinded test」。
+- [x] **F5 — 旗標率 pool-size / 極值依賴**（G● C★ O●）｜any-pair 爭議真正內核 ✅ 核心完成
+  - ✅ 已做：pool-size 分層（Firm A 每一層都壓制 BCD：<50 66 vs 20%、…、400+ 82 vs 29%）→ pool size 無法解釋 firm gap；accountant-clustered bootstrap gap 53.7pp CI[49.5,57.5]。皆寫入 §IV-C。
+  - ✅ 增補完成：firm+year FE logistic（控時間/管線後 B/C/D OR 0.116/0.061/0.070，仍低一個量級）；leave-one-year-out gap 53.1–54.9pp（任一年剔除皆穩，含 2022–23）。寫入 §IV-C「Four further checks」；腳本 f5_fe_loyo.py。
+- [x] **F6 — clean reference exogeneity** ✅ 完成（文字+既有證據）
+  - §III-E 新增：floor 為 conditional-on-correct-clean；污染只會抬高 floor → 對 Firm A 對比保守（known-safe 方向）；leave-one-baseline-firm-out 不動 floor；crossover scope 0.8547→0.8302（≤0.025）。ICCR 在不同 clean-group 下重算需 canonical sampler，未擅自重做。
+- [x] **F7 — 宣稱範圍過大**（G● C★ O●）｜detection / operational labels / tunable / 中文語料可直接採用 ✅ 完成
+  - 已做：貢獻條列 operational labels→risk strata；全文 specificity→specificity proxy；中文語料「adopt directly」→「starting reference for comparable Chinese-signature pipelines, subject to recalibration」；tunable 見 G3。
+
+## 仲裁（三審分歧，注意採納方向）
+
+- [x] **A — 不 fine-tune 是 label-free 的必然** ✅ 完成
+  - §II 末新增主動論證段：supervised metric learning 需 labelled pairs = 正是 archive 沒有的 ground truth；label-free 非弱化版而是唯一誠實選項；貢獻為方法論非架構；fine-tune 留待 protocol first-run 取得 labelled sample 後。
+- [x] **B — any-pair 嚴重性分歧** ✅ 併入 F5（已完成）
+
+## 融合新增（三審皆漏，全採納）
+
+- [x] **G1 — staggered e-signing adoption → event study** ｜**改為誠實描述+招認限制（用戶定案）** ✅ 完成
+  - 硬發現：資料無乾淨 staggered adoption（A 全程高、C 2022 跳、B 2023 跳、D 緩升），跳升跨整年且集中審計季 → 內部時序無法分離 adoption vs 管線變動，且反推導入點會循環。
+  - ✅ 已做：§V-B 升級為「Time Trend and the Firm–Pipeline Confound」，注入真實異質時序 + F2 指紋 + 明列 event study（需外部導入日期）為 future work；不杜撰日期。firm_year_hc_panel.csv 已存備圖。
+- [x] **G2 — 前處理壓縮 cosine 尺度** ✅ 完成（可驗證事實+construct，ablation 列 future）
+  - §V-A 新增：cosine 97.7% ≥0.90、median 0.969、僅 0.3% <0.85；0.95 cut 坐落飽和區（~76% 在其上）→ cosine 單獨幾乎不分辨、靠 dHash；padding/normalization 完整量化需重跑 CNN ablation（DB 做不到）列 future。注意：融合表「95.2%」我配對驗證對不上，未引用。
+- [x] **G3 — 「tunable operating point」單向空心**（recall 不可觀測 → 無 P–R trade-off）✅ 完成
+  - 已做：§III-D (i) + §V operating-point 改為「conservativeness dial, not a precision–recall control」；只能單向收緊、無可校準的 recall 取捨面；abstract 移除 operator-tunable 措辭。
+- [x] **G4 — byte-identical 跨案件/跨日期** ✅ 完成（DB 驗證）
+  - 驗證：262 筆 pixel-identical 全部 match 到**不同 source_pdf**（0 同檔），170/262 跨月 → 排除重複申報/同報表雙計；§IV-C 補述。
+- [x] **G5 — 低率 ≠ 少數** ✅ 完成
+  - ✅ 已做：§IV-A 補「888 期望巧合 HC flags（CI [677,1,098]）= 0.59%×150,442」+「low rate ≠ small number、單一 HC flag 不單獨解讀」。
+
+## 結構 / 格式 — Minor（含三審共識與細讀）
+
+- [x] **M1** — 摘要重構 ✅；problem→method→data→finding→limitation 弧線，刪「This is not forgery」口語句，併入 F1/F4/F5/F7 誠實框架。
+- [x] **M2** — §I 首次出現處新增 operational definition ✅；明確區分 handwritten/seal/overlay/e-sign/proxy + 排除 cryptographic digital signature，準則=image reuse 可見結果。
+- [x] **M3** — §III-A 強化免責 ✅；訪談 self-reported/anonymized/不可重製，吻合=consistency with domain knowledge，非 accuracy/recall 量測。
+- [x] **M4** — Figure 3 換真實 2D density ✅；make_fig3_density.py 產 n=150,441 log-density + 五區疊加 + 軸刻度；caption 改為描述真實分布。
+- [x] **M5** — §X→Section X ✅（114 處全替換，無 malformed）。
+- [x] **M6** — specificity→specificity proxy ✅（隨 F1/F7 完成）。
+- [x] **M7** — Table II-b 後新增 reconciliation 段 ✅；直接解釋 within(21–29%) vs between(0.59%) 不同量，「clean」=between 巧合罕見非 within 低。
+- [x] **M8** — 新增 §V-D Threats to Validity ✅；8 條集中列出（含 bias 方向與交叉引用）。
+- [x] **M9** — §IV-B 新增框架句 ✅；分清 empirical(比率/巧合率/byte-id) vs designed procedure(4 moves)；protocol first-run 明列 future work。
+- [x] **M10** — 對齊完成 ✅；168,755=matched、168,740=有測值（差 15=單簽會計師 pool=1，DB 驗證），§IV-A 補註；226=cell 全部、206=有足夠簽名子集，§V future-work 補「206 of 226」。
+- [x] **M11**（半-Major）— 語料範圍釐清 ✅
+  - §III-B 新增一句：primary sample=Big-4；150,442=valid∩有兩測值（60,448/38,993/34,248/16,752）；non-Big-4 僅入 §V-C crossover 穩健性、不入 calibration/headline。
+- [x] **M12** — 新增 Table II-c（A–D 2020–2023 五分類）✅；邏輯以 2013–2019 Firm B 重現 Table II-b 驗證通過。
+- [x] **M13** — 拼寫統一美式 ✅（behaviour/labelled/centring/colour→美式，30 處；references 內保留原拼寫）。⏳ placeholder 作者/機構/DOI/biography 投稿前補（double-blind，待你）。
+- [x] **M14** — 參考文獻體例 ✅（網路查證）
+  - 查證結論：[4]SigNet（dblp=CoRR）、[8]Brimoh、[9]Woodruff、[24]Qwen2.5-VL **皆無正式刊出版本，維持 arXiv 即正確**（reviewer 誤判有正式版）。[25] 補官方 docs URL；[27] 升級為精確永久連結 + 日期（Jan 21, 2013）。
+- [x] **M15** — 新增 Table I-a 縮寫/門檻對照表 ✅（HC/MC/HSC/UN/LH + cuts + ICCR/c/d 記號）。⏳ II-b vs IV 整併為主觀排版判斷，建議你決定。
+
+---
+
+## 分類：可立即文字改 vs 需新分析/資料 vs 需 co-author 決策
+
+**A. 純文字重構（無需新數據，可現在做）**：F1（術語/撤回 139×）、F4、F7、G3、G4（若資料已知）、M1、M2、M3、M5、M6、M8、M9、M13、M15、A（主動論證段落）
+
+**B. 需新分析 / 跑資料**：F2（firm-metadata 抽取）、F3（prevalence 數字）、F5（subsampling + bootstrap + FE + LOYO 套組）、F6（clean-group 敏感度）、G1（event study）、G2（前處理量化）、G5（絕對期望數）、M4（真實 density 圖）、M7（B/C/D HC 解釋需查數）、M11（語料分母核對）、M12（五分類表）
+
+**C. 需 co-author（Jimmy）決策 / 確認**：是否補 within-CPA 親簽 benchmark（F1 理想項）、G1 event study 範圍、最終宣稱降溫幅度

paper/gemini_review_v4_round1.md

@@ -0,0 +1,75 @@
+# Paper A Phase 5 Round 1 — Gemini 3.1 Pro independent review
+
+Reviewer: Gemini 3.1 Pro
+Date: 2026-05-14
+Target: paper/v4/paper_a_prose_v4_phase4.md + paper/v4/paper_a_methodology_v4_section_iii.md + paper/v4/paper_a_results_v4_section_iv.md
+Prior reviewer artifact: paper/codex_review_gpt55_v4_round7.md (codex GPT-5.5, Minor Revision)
+
+## Verdict
+Minor Revision. I corroborate codex's overall conclusion of Minor Revision, as the central empirical narrative (inter-CPA coincidence-rate calibration, K=3 descriptive demotion, and composition decomposition) is robust and strongly supported by the methodology and results sections. However, I explicitly dissent from codex on several critical findings. Most importantly, codex missed that references [42]-[44] *are* already present in the reference list, and codex did not flag the partner's dangerous suggestion to frame firm heterogeneity as "statistically insignificant." These gaps are addressed below.
+
+## Codex round-7 closure cross-check
+
+| # | Spot-checked finding | Verdict | Evidence |
+|---|---|---|---|
+| 1 | Replace "candidate classifiers" with "candidate checks" | CLOSED | `paper_a_prose_v4_phase4.md` explicitly uses "candidate checks" (e.g., "all three candidate checks — the inherited box rule..."); `paper_a_methodology_v4_section_iii.md` §III-K.4 uses "candidate check's positive-anchor miss rate". |
+| 2 | §II LOOO paragraph with refs [42]-[44] | CLOSED | `paper_a_prose_v4_phase4.md` §II contains a fully drafted addition citing Stone [42], Geisser [43], and Vehtari et al. [44]. |
+| 3 | Restore inherited v3.20.0 limitations in §V-G | CLOSED | `paper_a_prose_v4_phase4.md` §V-G lists "The last five are inherited from v3.20.0 §V-G..." and explicitly covers ImageNet features, red-stamp HSV, longitudinal confounds, source-exemplar misattribution, and legal interpretation. |
+| 4 | Limit full-dataset claims to K=3 + Spearman re-run | CLOSED | `paper_a_prose_v4_phase4.md` §V-G clarifies that full-dataset claims are limited: "We did not perform the full per-signature pool-normalised ICCR analysis at the full n = 686 scope; the §IV-K full-dataset Spearman re-run shows the K=3 + box-rule rank-convergence is preserved". |
+
+## Major findings
+
+1. **[Partner Query / Framing Risk] "Statistically insignificant" firm heterogeneity framing is unsupported.** (Codex missed) The partner queried whether firm heterogeneity could be framed as "statistically insignificant." This is completely unsupported and must be explicitly rejected. In `paper_a_results_v4_section_iv.md` §IV-M.5 (Table XXIII) and `paper_a_methodology_v4_section_iii.md` §III-L.4, logistic regression odds ratios for Firms B/C/D versus Firm A are 0.053, 0.010, 0.027. This indicates that Firms B/C/D have 19x to 100x *lower* odds than Firm A of firing the HC hit indicator even after controlling for pool size. This is an order-of-magnitude difference and highly statistically/practically significant.
+2. **[Codex Disagree] Codex falsely claimed refs [42]-[44] were absent.** (Codex disagree) Codex's round-7 review claimed that references [42]-[44] remained placeholders and were absent from `paper_a_references_v3.md`. This is incorrect; the reference file contains these three citations at the end of the list. The only residual issue is the draft note in the phase 4 close-out section (line ~145) stating `[add citation]`, which simply needs to be deleted.
+3. **[Disclaimer Adequacy] Unsupervised limits effectively disclosed.** The text properly disclaims the limits of its unsupervised setting. The "FAR" to "ICCR" terminology replacement reflects the structural fact that inter-CPA collision acts as a specificity proxy, fully acknowledging the "within-firm template-like collision" caveat in §III-L.4.
+4. **[K=3 Demotion Language] Consistent descriptive framing.** The language properly frames the K=2 and K=3 mixtures as firm-compositional partitions rather than inferential evidence for discrete mechanisms, correctly demoting K=3's operational standing based on the dip-test decomposition.
+5. **[Feature-Derived Scores] Caveat phrasing.** §III-K.1 and §V-E clearly caveat the high Spearman correlations ($\rho \ge 0.879$) as "not statistically independent measurements" since they are deterministic functions of the same descriptor pair, successfully framing it as internal consistency rather than external validation.
+
+## Minor findings
+
+1. **[m1] Stale internal draft notes and checklists.** The Phase 4 prose (`paper_a_prose_v4_phase4.md`) still contains internal draft notes at the top and the "Notes for Phase 4 close-out" at the bottom. Section III and IV files also contain similar `internal — remove before submission` blocks. These must be stripped.
+2. **[m2] Table numbering clash (XV-B vs XIX).** In `paper_a_results_v4_section_iv.md` §IV-J, a note acknowledges Table XV-B might need to be renumbered to Table XIX depending on journal style. This should be finalized to ensure sequential integer numbering (preferring Table XIX to avoid "B" suffixes).
+3. **[m3] Word count note.** The abstract word count note in the close-out checklist should be removed, as the abstract is independently verified at 243 words, satisfying the $\le 250$ requirement.
+
+## Provenance spot-checks
+
+1. **Spearman $\rho \ge 0.879$ floor.**
+   - *Claim Text:* "Three feature-derived scores agree on the per-CPA descriptor-position ranking at Spearman $\rho \ge 0.879$"
+   - *Location:* `paper_a_prose_v4_phase4.md` (Abstract & §V-E) and `paper_a_methodology_v4_section_iii.md` (§III-K.1).
+   - *Cited Script:* Script 38.
+   - *Verdict:* VERIFIED.
+2. **Firm A per-doc HC+MC alarm 0.62.**
+   - *Claim Text:* "Firm A's per-document HC+MC alarm rate is 0.62 versus 0.09–0.16 at Firms B/C/D"
+   - *Location:* `paper_a_prose_v4_phase4.md` (Abstract & §V-C).
+   - *Cited Script:* Script 45.
+   - *Verdict:* VERIFIED.
+3. **145/8/107/2 byte-identical split.**
+   - *Claim Text:* "262 byte-identical signatures in the Big-4 subset (Firm A 145, Firm B 8, Firm C 107, Firm D 2)"
+   - *Location:* `paper_a_methodology_v4_section_iii.md` (§III-K.4).
+   - *Cited Script:* Script 40.
+   - *Verdict:* VERIFIED.
+4. **Dip test $p_{\text{median}} = 0.35$ under joint centring + jitter.**
+   - *Claim Text:* "Once both confounds are removed (firm-mean centring plus uniform integer jitter), the Big-4 pooled dHash dip test yields $p_{\text{median}} = 0.35$"
+   - *Location:* `paper_a_methodology_v4_section_iii.md` (§III-I.4).
+   - *Cited Script:* Script 39e.
+   - *Verdict:* VERIFIED.
+5. **Logistic OR 0.053 / 0.010 / 0.027.**
+   - *Claim Text:* "logistic regression... yields odds ratios of 0.053 (Firm B), 0.010 (Firm C), and 0.027 (Firm D)"
+   - *Location:* `paper_a_methodology_v4_section_iii.md` (§III-L.4) and `paper_a_results_v4_section_iv.md` (Table XXIII).
+   - *Cited Script:* Script 44.
+   - *Verdict:* VERIFIED.
+
+## Newly introduced issues
+
+1. **Sample size nuance between text and tables:** §IV-J Table XV correctly sums to 150,442 signatures across Firms A-D, consistent with the descriptor-complete count. However, the vector-complete count of 150,453 used in the ICCR analyses (§III-L.2) could confuse readers comparing numbers across tables. A small footnote in Table XV directing readers back to the sample-size reconciliation in §III-G is recommended.
+
+## Phase 5 readiness
+
+Partial. The technical framing is solid and the methodological pivot is successfully integrated. Phase 5 readiness requires stripping the remaining internal draft notes and firmly rejecting any attempt to describe firm heterogeneity as "statistically insignificant."
+
+## Recommended next-step actions
+
+1. **[Empirical Blocker]** Explicitly reject the "statistically insignificant" framing of firm heterogeneity. The odds ratios derived from Script 44 confirm a massive, statistically significant difference between Firm A and Firms B/C/D.
+2. **[Copy-Edit Blocker]** Strip all internal draft notes, metadata tags, and the Phase 4 and Phase 3 close-out checklists from the prose, methodology, and results files.
+3. **[Copy-Edit Blocker]** Finalize Table XV-B's numbering (to Table XIX) to comply with sequential integer numbering formats typically preferred by IEEE Access.
+4. **[Copy-Edit Blocker]** Remove the `[add citation]` placeholder string in the notes; references [42]-[44] are fully integrated and listed in `paper_a_references_v3.md`.

paper/gemini_review_v4_round2.md

@@ -0,0 +1,42 @@
+# Paper A Phase 5 Round 2 — Gemini 3.1 Pro independent review
+
+Reviewer: Gemini 3.1 Pro
+Date: 2026-05-14
+Target: paper/v4/paper_a_prose_v4_phase4.md + paper/v4/paper_a_methodology_v4_section_iii.md + paper/v4/paper_a_results_v4_section_iv.md (post round-2 + round-3, commit 4a6f9c5)
+Prior reviewer artifacts: paper/codex_review_gpt55_v4_round7.md; paper/codex_review_gpt55_v4_round8.md; paper/gemini_review_v4_round1.md; paper/opus_review_v4_round1.md
+
+## Verdict
+Accept (Phase 5 Splice-Ready). The round-2 and round-3 changes have masterfully resolved the empirical and framing blockers surfaced by the multi-agent panel in round 1. No new empirical work is required. The manuscript is ready for Phase 5 master-file splice.
+
+## Round-1 / round-2 panel closure cross-check
+
+| Source | Finding | Current Status | Evidence / Note |
+|---|---|---|---|
+| Opus M1 | §IV K=3 mechanism-label reversion | CLOSED | Tables VIII, IX, XVI, XVII, and XVIII in `paper_a_results_v4_section_iv.md` now correctly use "low-cos / high-dHash" and "less-replication-dominated rate". The "hand-leaning" mechanistic framing has been successfully eradicated. |
+| Opus M3 | "98-100% within source firm" conflation | CLOSED | The Abstract in `paper_a_prose_v4_phase4.md` now accurately states "$77$–$99\%$ of inter-CPA collisions concentrate within the source firm" for the deployed any-pair rule, fixing the overclaim. |
+| Opus M4 | Duplicate §V-G heading | CLOSED | `paper_a_prose_v4_phase4.md` correctly sequence the sections as "G. Pixel-Identity..." and "H. Limitations". |
+| Codex r8 blocker | Abstract word count over 250 limit | CLOSED | The Abstract has been elegantly trimmed and now stands at approximately 235 words, well within the IEEE Access 250-word limit. |
+| Codex r8 blocker | §IV-I stale "Table XVI" cross-reference | CLOSED | The reference in `paper_a_results_v4_section_iv.md` now accurately points to "§IV-M Tables XXI–XXVI" for the ICCR calibration. |
+| Codex r8 blocker | §IV-J Table XV sample-size footnote | CLOSED | The footnote accurately reconciles the $150,442$ descriptor-complete versus $150,453$ vector-complete sub-samples in `paper_a_results_v4_section_iv.md`. |
+
+## Net-new findings
+1. **Abstract Trim:** The abstract trimming successfully reduced the word count without dropping any essential empirical substance. The retention of the $77$–$99\%$ any-pair collision stat over the $97$–$100\%$ same-pair stat is the right scientific choice, representing the actual deployed rule accurately.
+2. **"Replication-dominated" terminology:** The pivot to "less-replication-dominated" reads cleanly throughout §IV and maintains perfect consistency with the §III-J descriptive demotion.
+3. **Internal-note items:** The draft notes, close-out checklists, and the "Open questions remaining" in the files are tagged explicitly as `internal — remove before submission`. They are perfectly acceptable to defer to manuscript-splice time and are not empirical or structural blockers.
+
+## Provenance spot-checks
+I selected numerical claims not previously verified by Codex or Opus in their reviews:
+1. **Bootstrap CI half-width for marginal crossings:** Table VII in §IV-E reports a K=2 cosine crossing 95% CI of $[0.9742, 0.9772]$ and states a CI half-width of $0.0015$. $(0.9772 - 0.9742) / 2 = 0.0015$. The dHash CI of $[3.476, 3.969]$ yields a half-width of $(3.969 - 3.476) / 2 = 0.2465$, matching the reported $0.246$. VERIFIED.
+2. **Nine-tool validation table structure:** §III-M describes a "nine-tool unsupervised-validation collection." I verified the §III-M table counts exactly 9 diagnostics (from Per-comparison ICCR down to LOOO firm-level reproducibility) mapped to their untested assumptions. VERIFIED.
+3. **Table XVI K=3 Firm A Component Weights:** Table XVI in §IV-J reports Firm A has $0.00\%$ in C1 and $82.46\%$ in C3. This perfectly matches the prose claims in §V-C regarding Firm A's concentration in the templated end. VERIFIED.
+
+## Firm-heterogeneity framing audit
+The partner's suggestion to frame the firm heterogeneity as "statistically insignificant" remains correctly and decisively rejected in these post-round-3 drafts. The prose in §III-L.4 and the Abstract explicitly leverages the logistic regression odds ratios ($0.053, 0.010, 0.027$) to establish that Firms B/C/D have an order-of-magnitude lower HC alarm rate even after pool-size adjustment. Furthermore, the corrected any-pair $77$–$99\%$ / same-pair $97$–$100\%$ within-firm collision concentration explicitly *strengthens* the heterogeneity argument by showing that even false alarms cluster structurally within source firms. The framing is robust, decisive, and scientifically accurate.
+
+## Phase 5 readiness
+**Ready for Phase 5 Splice (Accept).** There are no remaining empirical, structural, or framing blockers.
+
+## Recommended next-step actions
+1. Execute the final master-file manuscript splice.
+2. During the splice, mechanically strip all markdown blocks tagged `> **Draft note... internal — remove before submission**`, as well as the close-out checklists and the open questions block at the end of §III.
+3. Finalize the `Table XV-B` versus `Table XIX` numbering decision based on the specific journal template requirements during typesetting.

paper/narrative_audit_v4.md

@@ -0,0 +1,220 @@
+# Paper A v4.0 — Narrative-Thread Audit
+
+Auditor: Claude Opus 4.7 (1M context)
+Date: 2026-05-14
+Target: paper/v4/paper_a_prose_v4_phase4.md + paper/v4/paper_a_methodology_v4_section_iii.md + paper/v4/paper_a_results_v4_section_iv.md (post round-5, commit 128a914)
+Purpose: Coherence check across Abstract / §I / §III / §IV / §V / §VI as a single argument, after Phase 5 AI peer-review panel convergence (3/3 in Accept/Minor band).
+
+## Headline assessment
+
+**Mostly Coherent — submission-ready after 2-3 small narrative-consistency patches.**
+
+The v4 story arc — *"v3.20.0 distributional path turns out to be composition + integer artefact; v4.0 replaces it with anchor-based ICCR + decisive firm heterogeneity; positioning is anchor-calibrated specificity-only screening, not validated detector"* — reads cleanly from Abstract through §VI. The 5 fix rounds + 3 reviewer panels have substantially closed the major framing, terminology, and provenance risks. What remains is narrow narrative-consistency residue between Phase 4 §I/§V prose and §III source-of-truth (3 specific items, all small), plus 1 interpretive caveat that would strengthen §V-H limitation 2.
+
+No empirical reruns required. No structural rewrites required. Submission-readiness gate: **conditionally pass** — recommend a 15-minute round-6 prose-consistency patch before manuscript-splice, then the manuscript is splice-ready.
+
+## 1. Abstract → body mirror audit
+
+The Abstract (Phase 4 line 11, 247 words) makes ~12 distinct claims. Each maps cleanly to a body location:
+
+| # | Abstract claim | §I location | §III/§IV body location | Status |
+|---:|---|---|---|---|
+| 1 | Non-hand-signed detection problem (regulation + digitization) | §I paras 1–3 (lines 19–21) | — (problem framing) | **Aligned** |
+| 2 | Pipeline: VLM + YOLOv11 + ResNet-50 + dual-descriptor | §I para 6 (line 29) + contribution 2 | §III-A..F (inherited) + §III-F dual-descriptor | **Aligned** |
+| 3 | 90,282 reports / 182,328 sigs / 758 CPAs | §I para 8 (line 39) | §IV-A..C (inherited) | **Aligned** |
+| 4 | Big-4 sub-corpus: 437 CPAs / 150,442 sigs | §I para 8 (line 39) | §III-G (line 19), §IV-D (line 9, line 15) | **Aligned** |
+| 5 | Composition decomposition $p_{\text{median}} = 0.35$ | §I para 5 (line 31) + contribution 4 | §III-I.4 (lines 55–73), §IV-M.1 Table XX (line 266) | **Aligned** |
+| 6 | Per-comparison ICCRs 0.0006 / 0.0013 / 0.00014 | §I para 6 (line 33) + contribution 5 | §III-L.1 (line 196), §IV-M.2 Table XXI (line 280) | **Aligned** |
+| 7 | Per-signature ICCR 0.11 | §I para 6 (line 33) | §III-L.2 (line 208), §IV-M.3 Table XXII (line 300) | **Aligned** |
+| 8 | Per-document ICCR 0.34 (HC+MC) | §I para 6 (line 33) | §III-L.3 (line 233), §IV-M.4 Table XXIII (line 317) | **Aligned** |
+| 9 | Firm heterogeneity: Firm A 0.62 vs B/C/D 0.09–0.16 | §I para 7 (line 35) + contribution 6 | §III-L.4 (line 259), §IV-M.4 (line 325) | **Aligned** |
+| 10 | Within-firm 77–99% (any-pair) | §I para 7 (line 35) + contribution 6 | §III-L.4 (line 283), §IV-M.5 Table XXV (line 340) | **Aligned** |
+| 11 | "Specificity-proxy-anchored screening + HITL, not validated detector" positioning | §I contribution 8 (line 57) | §III-M Table XXVII (line 316) + §V-G/H + §VI item 8 | **Aligned** |
+| 12 | "No calibrated error rates without ground truth" disclaimer | §I para 5 item (v) (line 25) | §III-M (line 312), §V-H limit 1 (line 113) | **Aligned** |
+
+Observation: Abstract does NOT mention the three-score Spearman convergence ($\rho \geq 0.879$). This is by design — the v4 pivot demoted three-score from a headline finding to "internal consistency" because the scores share inputs. §I contribution 7 and §V-E retain it with the demoted caveat. **No action needed.**
+
+## 2. §I contributions (8) → body implementation map
+
+| # | §I contribution | §III/§IV implementation | §V/§VI loop-back | Status |
+|---:|---|---|---|---|
+| 1 | Problem formulation | — (§V-A discusses) | §V-A (line 73), §VI implicit | **Aligned** |
+| 2 | End-to-end pipeline | §III-A..F (inherited) | §VI line 145 | **Aligned** |
+| 3 | Dual-descriptor verification | §III-F + §IV-L backbone ablation | §V-A/B implicit | **Aligned** |
+| 4 | Composition decomposition | §III-I.4 + §IV-M.1 Table XX | §V-B (line 81), §VI item 1 (line 147) | **Aligned** |
+| 5 | Anchor-based multi-level ICCR | §III-L + §IV-M.2/M.3/M.4 Tables XXI/XXII/XXIII | §V-F (line 99), §VI item 2 | **Aligned** |
+| 6 | Firm heterogeneity + within-firm collision | §III-L.4 + §IV-M.4/M.5 Tables XXIV/XXV | §V-C (line 83), §VI items 3+4 | **Aligned** |
+| 7 | K=3 descriptive + three-score convergence | §III-J + §III-K.1 + §IV-E/F/G | §V-D/E (lines 89–97), §VI items 5+6 | **Aligned** |
+| 8 | Annotation-free positive-anchor + ten-tool ceiling | §III-K.4 + §III-M Table XXVII + §IV-H Table XIV | §V-G (line 105), §VI items 7+8 | **Aligned** |
+
+All 8 contributions trace cleanly through §III/§IV implementation and §V/§VI loop-back. **No action needed.**
+
+## 3. v3→v4 pivot rhetoric thread
+
+The v4 pivot has four narrative nodes; each must reinforce the others:
+
+| Node | Location | Says |
+|---|---|---|
+| **Setup**: v3.x distributional path | §I para 5 line 31 (Phase 4 prose) | "Earlier work...adopted a distributional path...v4.0 reports a composition decomposition diagnostic that overturns this reading" |
+| **Proof**: 2×2 factorial composition decomposition | §III-I.4 Scripts 39b–39e (lines 55–73) | Joint firm-mean centring + integer-tie jitter eliminates rejection ($p_{\text{median}} = 0.35$) |
+| **Alternative**: anchor-based ICCR | §III-L (line 173+) | Replaces distributional thresholds with inter-CPA coincidence-rate calibration at 3 units |
+| **Discussion**: K=3 stays descriptive | §V-B + §V-D (lines 77–93) | Mixture fits are firm-compositional partitions, not mechanism modes |
+| **Conclusion**: pivot summary | §VI items 1+5 (line 147) | Demotes K=3 mechanism reading; positions ICCR as the operational calibration |
+
+All five nodes use consistent language ("composition + integer artefact"; "descriptive firm-compositional partition"; "no within-population bimodal antimode"). **No action needed.**
+
+## 4. K=3 demotion consistency
+
+Verified across 4 locations using consistent descriptor-position language (post round-2 M1 fix):
+
+- §III-J line 90 (source of truth): "The 'descriptive position' column replaces v3.x's 'hand-leaning / mixed / replicated' mechanism labels"
+- §I contribution 7 (line 55): "K=3 mixture demoted from 'three mechanism clusters' to a descriptive firm-compositional partition"
+- §V-D (line 93): "the K=3 stability supports a descriptive reading...*not* a three-mechanism latent-class structure"
+- §VI item 5 (line 147): same demotion language
+- §IV Tables XVI/XVII column headers: "C1 (low-cos / high-dHash) | C2 (central) | C3 (high-cos / low-dHash)" — descriptor-position labels throughout
+
+`grep -n "hand-leaning"` in v4 public prose: 0 hits (only internal-strip text). **Closed.**
+
+## 5. ICCR vs FAR terminology consistency
+
+Verified across all rate-reporting locations (post round-2 + round-5):
+
+- Abstract: "inter-CPA coincidence-rate (ICCR)" ✓
+- §I contribution 5 (line 51): explicit terminology adoption and FAR disclaimer ✓
+- §III-L.1 (line 185): "Terminological note on 'FAR'" with full disclaimer ✓
+- §IV-I (line 159): historical "FAR" cited only with the "v3.x terminology" caveat ✓
+- §V-G heading (line 105): "Inherited Inter-CPA Negative Anchor Reframed as Coincidence Rate" ✓
+- §V-H limitations: specificity-proxy framing under partially-violated assumption ✓
+- §VI item 2 (line 147): "explicit terminological replacement of 'FAR' by 'ICCR' given the unsupervised setting" ✓
+
+No public-prose "FAR" leak outside the historical-context caveats. **Closed.**
+
+## 6. Numbers consistency audit (cross-section)
+
+Headline numbers cross-referenced between Abstract / §I / §III / §IV:
+
+| Claim | Abstract | §I body | §III source | §IV table | Status |
+|---|---|---|---|---|---|
+| Per-comparison ICCR cos $>0.95$ | 0.0006 | 0.0006 | 0.00060 | 0.00060 Table XXI | **Match** (rounding consistent) |
+| Per-comparison ICCR dHash $\leq 5$ | 0.0013 | 0.0013 | 0.00129 | 0.00129 Table XXI | **Match** |
+| Per-comparison joint | 0.00014 | 0.00014 | 0.00014 | 0.00014 Table XXI | **Match** |
+| Per-signature ICCR | 0.11 | 0.11 | 0.1102 (Wilson) | 0.1102 Table XXII | **Match** |
+| Per-document ICCR (HC+MC) | 0.34 | 0.34 | 0.3375 | 0.3375 Table XXIII | **Match** |
+| Firm A doc HC+MC | 0.62 | 0.62 | 0.6201 | 0.6201 §IV-M.4 line 325 | **Match** |
+| Firms B/C/D doc HC+MC | 0.09–0.16 | 0.09–0.16 | 0.1600 / 0.1635 / 0.0863 | same | **Match** |
+| Within-firm any-pair | 77–99% (rounded) | 76.7–83.7% / 98.8% | same | Table XXV | **Match** |
+| Same-pair within-firm | — | 97.0–99.96% | 99.96 / 97.7 / 98.2 / 97.0 | line 349 | **Match** |
+| Composition $p_{\text{median}}$ | 0.35 | 0.35 | 0.35 | 0.35 Table XX | **Match** |
+| Logistic OR | — | 0.053 / 0.010 / 0.027 | same | Table XXIV | **Match** |
+| Spearman ρ floor | — | 0.879 | 0.879 | 0.8794 Table IX | **Match** (Spearman precision §III/§IV differ at 4th decimal — see §8) |
+
+All headline numbers reconcile across sections. **Closed.**
+
+## 7. Limitations vs §III-M Table XXVII coverage
+
+§V-H lists 14 limitations (9 v4-specific + 5 inherited from v3.20.0). Each Table XXVII assumption should be covered by an explicit §V-H item OR be self-evidently descriptive:
+
+| Table XXVII tool | Assumption | §V-H coverage |
+|---|---|---|
+| Composition decomposition | Jitter unbiased; Big-4 jittered + centred + jittered evidence | Implicit — covered by general "no signature-level ground truth" frame |
+| Per-comparison ICCR | Inter-CPA pairs are negative anchor (partially violated) | §V-H limit 2 (explicit) |
+| Per-signature ICCR | Same + pool replacement preserves negative-anchor property | §V-H limit 2 (implicit via "specificity-proxy rates under partially-violated assumption") |
+| Per-document ICCR | Same | Same |
+| Firm-heterogeneity logistic | Cluster-robust SE not run | **Gap** — no §V-H item explicitly flags the naive-SE caveat |
+| Cross-firm hit matrix | Deployed-rule semantics + mode-of-firms tie-break | §V-H limit 2 |
+| Alert-rate sensitivity | Descriptive gradient, not formal plateau | §V-H limit 5 (line 121, "alert-rate sensitivity analysis characterises only the HC threshold") |
+| Three-score Spearman | Scores share inputs | §V-H limit 6 (line 123, deployed-rate-excess interpretation) — partial; not the score-independence caveat directly |
+| Pixel-identical positive capture | Tautological (byte-identical ⇒ in HC region) | §V-H limit 4 (line 119, "pixel-identity is a conservative subset") |
+| LOOO firm-level reproducibility | Stability ≠ classification validity; K=3 membership ±12.8 pp | §V-H limit 8 (line 127, "K=3 hard-posterior membership is composition-sensitive") |
+
+**Gap 1**: §V-H does not explicitly flag the logistic-regression naive-SE caveat that Table XXVII row 5 discloses. Worth adding a half-sentence to §V-H or letting Table XXVII carry the disclosure since it's already in print.
+
+**Gap 2** (Opus N5 from round-2 audit): §V-H limit 2 discloses the firm-dependent within-firm violation numerically (98.8% at A; 76.7–83.7% at B/C/D) but does not interpret what this means for proxy reliability — namely that Firm A's per-firm ICCR is MORE contaminated by within-firm sharing than B/C/D's, so the per-firm B/C/D rates are closer to clean specificity. This nuance affects interpretation of the headline "firm heterogeneity is decisive" framing.
+
+## 8. Net-new narrative concerns (audit-surfaced)
+
+### Concern A — Phase 4 §I body line 31 cites "Script 39c" for jittered-dHash claim
+
+**Issue.** Phase 4 prose line 31 says:
+> "Within-firm signature-level cosine and jittered-dHash dip tests fail to reject in every individual Big-4 firm and in every individual mid/small firm with $\geq 500$ signatures (10 firms tested in Script 39c)."
+
+Codex round-9 verified that Script 39c on RAW dHash actually REJECTS unimodality in all 10 firms; only the JITTERED variant (codex's read-only spike on the Script 39c substrate) fails to reject. Round-5 corrected §III line 59 + provenance table line 382 to cite the spike attribution, but Phase 4 §I line 31's bare "Script 39c" citation now reads less precisely than the source-of-truth at §III line 59.
+
+**Severity.** Low. The qualitative claim ("fail to reject in 10 mid/small firms") is correct per codex's own rerun. The provenance attribution is what's slightly off.
+
+**Recommended fix.** Update Phase 4 line 31 to match §III line 59: "...in every individual mid/small firm with $\geq 500$ signatures (10 firms tested; cosine: Script 39c per-firm; jittered-dHash: codex-verified read-only spike on Script 39c substrate)." Or simpler: drop the parenthetical "in Script 39c" and let §III carry the precise provenance.
+
+### Concern B — Phase 4 §V-B line 81 carries the same jittered-dHash claim without provenance
+
+**Issue.** §V-B (line 81) says:
+> "Within-firm signature-level cosine and jittered-dHash dip tests fail to reject in every individual Big-4 firm and in every individual non-Big-4 firm with $\geq 500$ signatures (10 firms tested)."
+
+No script citation here; the bare "10 firms tested" is technically OK but less precise than §III line 59 after round-5.
+
+**Severity.** Low.
+
+**Recommended fix.** Add a §III cross-reference: "...10 firms tested; see §III-I.4 / §III provenance table line for the codex-verified read-only spike." Or leave as-is and let §III line 59 carry the detailed provenance.
+
+### Concern C — §III-K.4 line 149 stale cross-reference to v3.x §IV-I
+
+**Issue.** §III-K item 4 line 149 says:
+> "The corresponding signature-level inter-CPA negative-anchor ICCR evidence is developed in §III-L.1 (Big-4 sample) and the v3.x §IV-I corpus-wide version (reported under prior 'FAR' terminology)"
+
+After v4 §IV-I was shrunk to a 3-paragraph reframing stub (post round-3), the phrase "v3.x §IV-I corpus-wide version" is misleading — v4 §IV-I now exists, just as a pointer. Opus round-2 N6 flagged this; codex round-9 did not.
+
+**Severity.** Cosmetic.
+
+**Recommended fix.** Update line 149 to "§III-L.1 (Big-4 v4 sample) and the inherited corpus-wide v3.x version cited at §IV-I (reported under prior 'FAR' terminology)".
+
+### Concern D — Spearman precision mismatch §III vs §IV
+
+**Issue.** §III-K.1 line 123–127 reports Spearman ρ as 0.963 / 0.889 / 0.879 (3 decimal places); §IV-F Table IX line 81–87 reports 0.9627 / 0.8890 / 0.8794 (4 decimal places). Codex round-8 flagged this as OPEN / COPY-EDIT.
+
+**Severity.** Cosmetic.
+
+**Recommended fix.** Standardise on 4 decimal places (matches Script 38 reported precision) in §III + §IV + §V-E + §VI.
+
+## 9. Splice-readiness gate
+
+| Item | Status | Notes |
+|---|---|---|
+| Abstract word count | ✓ | 247 / 250 |
+| §I contributions count | ✓ | 8 contributions, all map to body |
+| §II LOOO addition with refs [42]-[44] | ✓ | Present (post round-1) |
+| §III sub-sections G..M complete | ✓ | Including Table XXVII numbered |
+| §IV table sequence V–XXVI sequential | ✓ | Post round-2 cascade |
+| §V sub-sections A..H complete | ✓ | Post round-2 M4 fix (G→H) |
+| §VI items 1..8 map to §I 1..8 | ✓ | 1:1 mapping verified |
+| References [1]–[44] present | ✓ | 44 entries; [42]-[44] = Stone 1974 / Geisser 1975 / Vehtari 2017 |
+| Internal draft notes stripped | ✗ | **Splice-time mechanical** — Phase 4 line 3 + lines 153-162; §III line 3 + lines 434-447; §IV line 3 + line 365+ |
+| "Nine-tool" / "Table XV-B" residue | ✗ | **Splice-time mechanical** — only in internal-strip text |
+| Cross-section number consistency | ✓ | All headline numbers match across Abstract / §I / §III / §IV |
+| Terminology consistency (ICCR / K=3 / less-replication-dominated) | ✓ | No public-prose leaks |
+| IEEE Access format | ✓ | Abstract single-paragraph ≤250 words; numbered references; numbered tables |
+
+## 10. Recommended round-6 narrative-consistency patch (15 min, optional)
+
+Before manuscript-splice, three small text patches would close the audit-surfaced concerns:
+
+1. **Concern A**: Phase 4 line 31 — narrow the "Script 39c" provenance attribution for the jittered-dHash claim to match §III line 59.
+2. **Concern C**: §III line 149 — update the "v3.x §IV-I corpus-wide version" wording to reflect v4 §IV-I's reframing-stub status.
+3. **Concern D**: Standardise Spearman precision to 4 decimal places across §III/§IV/§V.
+
+Optional:
+- **Gap 2 / Opus N5**: Add a half-sentence to §V-H limit 2 interpreting the firm-dependent within-firm violation as "Firm A's per-firm ICCR is more contaminated by within-firm sharing than Firms B/C/D's, so the per-firm B/C/D rates are closer to clean specificity than the pooled rate."
+
+None of these is empirical or structural. They are prose-level consistency polishings. **Submission can proceed without them**, but they would strengthen reviewer-pass robustness.
+
+## 11. Submission-readiness verdict
+
+**Conditionally ready.**
+
+The empirical core is sound and reproducible. The Phase 5 panel converged 3/3 in Accept/Minor band. Five fix rounds have closed every reviewer-flagged finding. Numbers are consistent across sections. Terminology is consistent. The v4 pivot's narrative thread reads as a coherent argument from Abstract through Conclusion.
+
+**Path A (ship now)**: proceed directly to manuscript-splice → DOCX export → partner Jimmy review → submission. Risk: the four audit-surfaced concerns above may be flagged by external reviewers as small cosmetic issues. Cost: zero pre-submission work.
+
+**Path B (15-min polish first)**: apply round-6 patches for the four concerns above → re-verify with `rg`-based grep → manuscript-splice → DOCX → partner. Risk: zero. Cost: 15 min.
+
+**Recommendation**: Path B. The four concerns are small, narrative-consistency only, and fixing them avoids putting cross-section attribution inconsistencies in front of partner Jimmy / IEEE Access reviewers.
+
+After Path B (or A): proceed to manuscript-splice as the next mechanical step.

paper/opus_review_v4_round1.md

@@ -0,0 +1,235 @@
+# Paper A Phase 5 Round 1 — Opus 4.7 max-effort independent review
+
+Reviewer: Claude Opus 4.7
+Date: 2026-05-14
+Target: paper/v4/paper_a_prose_v4_phase4.md + paper/v4/paper_a_methodology_v4_section_iii.md + paper/v4/paper_a_results_v4_section_iv.md
+Prior reviewer artifacts: paper/codex_review_gpt55_v4_round7.md (codex, Minor Revision); paper/gemini_review_v4_round1.md (Gemini 3.1 Pro, Minor Revision)
+
+## Verdict
+
+**Minor Revision (corroborates codex + Gemini on overall disposition), but I dissent on readiness.** The central empirical narrative — anchor-based multi-level ICCR calibration, the composition-decomposition demolition of distributional thresholds, the K=3-as-firm-compositional demotion, and the disclosed unsupervised-validation ceiling — is methodologically sound and survives my spot-checks. However, my independent pass surfaces three substantive blockers that both prior reviewers missed: (1) the headline "98–100% of inter-CPA collisions concentrated within the source firm" claim, repeated verbatim in the Abstract, §I contribution 6, §V-C, §V-G limitation 2, and §VI conclusion item 4, is **factually wrong for the deployed any-pair rule** at three of the four firms (Firms B/C/D within-firm rates are 77%/84%/77%, not 98–100%; the 98–100% range applies only to the stricter same-pair joint event); (2) §IV retains the demoted mechanism labels ("hand-leaning / mixed / replicated" in Table XVI columns, "hand-leaning rate" in Tables IX, X, XVII, XVIII, and prose at lines 102, 215, 226, 232, 254) that §III-J line 90 explicitly says are "replaced"; (3) §V has two sub-sections labelled "G" (line 105 "Pixel-Identity ..." and line 109 "Limitations"). I therefore corroborate Minor Revision on the empirical core but treat these three findings as additional copy-edit-or-rewrite blockers that must close before Phase 5 splice — finding (1) verges on empirical depending on whether the author chooses to rewrite or restrict scope.
+
+## Cross-reviewer agreement matrix
+
+| Theme | codex round-7 | Gemini round-1 | Opus round-1 |
+|---|---|---|---|
+| Overall disposition | Minor Revision | Minor Revision | Minor Revision |
+| Refs [42]-[44] present in `paper_a_references_v3.md` | Missed — claimed they were placeholders | Caught codex error | **Agree with Gemini.** Refs are at lines 87–91 of `paper/paper_a_references_v3.md` |
+| Partner "statistically insignificant" framing risk | Not flagged | Flagged as Major | **Agree with Gemini.** OR magnitudes 0.05/0.01/0.03 are decisive heterogeneity, not absence-of-effect |
+| Table XV-B → XIX renumbering | Not flagged | Flagged as Minor | **Agree with Gemini AND extend:** renumbering cascades — see Major M2 below |
+| Sample-size 150,442 vs 150,453 | Not flagged | Flagged as newly-introduced issue | **Partial agree:** the §IV-J line 177 footnote already reconciles this inline; the missing piece is a Table XV cross-pointer to §III-G |
+| Internal draft notes / checklists | Flagged | Flagged | **Agree.** Three draft-note blocks remain (Phase 4 prose, §III v7, §IV v3.3) plus three close-out checklists |
+| Three-score "feature-derived" caveat coverage | Closed | Closed | **Partial dissent.** Caveat is present at Abstract / §I / §III-K / §V-E, BUT §IV Tables IX/X/XVII/XVIII reintroduce mechanistic "hand-leaning" framing without the §III-K caveat |
+| K=3 demotion language consistency | Closed | Closed (line 25 says "Consistent descriptive framing") | **Dissent.** See Major M1 — §IV retains mechanistic labels throughout |
+| Cross-firm collision concentration "98–100%" | Closed implicitly | Verified VERIFIED in §IV-M.5 / §III-L.4 spot-check, but did not check whether prose claim matches the any-pair table | **Strong dissent.** See Major M3 |
+| Duplicate §V-G heading | Not flagged | Not flagged | **Both missed.** See Major M4 |
+
+## Major findings
+
+1. **§IV retains "hand-leaning / mixed / replicated" mechanism labels that §III-J line 90 explicitly demotes (both missed).**
+
+   - *Issue.* §III-I.4 establishes that the descriptor distributions contain no within-population bimodality; §III-J line 90 states: "The 'descriptive position' column **replaces** v3.x's 'hand-leaning / mixed / replicated' mechanism labels." §V-D, §I item 7, and §VI item 5 propagate the descriptive framing. **§IV does not.**
+   - *Where.*
+     - `paper_a_results_v4_section_iv.md` line 219 (Table XVI column headers): `C1 (hand-leaning) | C2 (mixed) | C3 (replicated)` — verbatim mechanism labels
+     - line 85 (Table IX): "K=3 P(C1) vs Paper A box-rule **hand-leaning** rate"
+     - line 86 (Table IX): "Reverse-anchor cosine percentile vs Paper A box-rule **hand-leaning** rate"
+     - line 93 (Table X): "mean Paper A **hand-leaning** rate"
+     - line 100: "more **hand-leaning** relative to the non-Big-4 reference"
+     - line 102: "the most-**hand-leaning** end of Big-4"; "more **hand-leaning**"; "ranks Firm D fractionally above Firm C"
+     - line 147 (Table XIV column): "Misclassified as **hand-leaning**"
+     - line 215 (§IV-J): "the per-CPA **hand-leaning** ranking"
+     - line 226 (Table XVI reading): "C3 **replicated** component"; "highest **hand-leaning** concentration of the Big-4"
+     - line 232 (§IV-K): "Paper A operational **hand-leaning** rate"
+     - line 238 (Table XVII): "C1 **hand-leaning**" / "C3 **replicated**"
+     - line 244 (Table XVIII title): "Paper A operational **hand-leaning** rate"
+     - line 246 (Table XVIII): "P(C1) vs Paper A **hand-leaning** rate"
+     - line 254 (§IV-K reading): "more non-templated CPAs"; "more mid/small-firm **hand-leaning** CPAs"
+   - *Reasoning.* §III-K explicitly renames Score 3 to "**inherited binary high-confidence box rule rate**" / "less-replication-dominated rate" (lines 119, 129), and §V-E uses "less-replication-dominated rate" (line 97). The §IV body silently retains v3.x mechanistic naming. A reader reaching §IV after §III-J's demotion will perceive that §IV has reverted to causal/mechanistic K=3 framing. This is the exact regression the v4 pivot is supposed to prevent.
+   - *Fix.* Global s/hand-leaning/less-replication-dominated/ in §IV; rename Table XVI columns to `C1 (low-cos / high-dHash)` / `C2 (central)` / `C3 (high-cos / low-dHash)` matching the §III-J Table 8 "descriptive position" column; rename Table XVII C1/C3 row labels and Table XVIII row labels likewise; rewrite the §IV-K Reading prose to use descriptor-position language ("CPAs whose descriptor mean sits further from the templated end of the descriptor plane").
+   - *Tag.* **Both missed.**
+
+2. **Table-numbering cascade if XV-B → XIX is finalised (Gemini missed the cascade; codex did not flag).**
+
+   - *Issue.* Gemini correctly recommended renumbering Table XV-B → Table XIX. But the §IV-M tables (§IV-M.1 through §IV-M.6) currently occupy Tables XIX, XX, XXI, XXII, XXIII, XXIV, XXV. Bumping XV-B → XIX forces a cascade renumber of all seven §IV-M tables to XX–XXVI. There is no acknowledgement of this cascade in either §IV-J close-out item 2 (line 370) or in Gemini's recommendation. Without addressing the cascade, fixing XV-B alone produces a duplicate Table XIX in the manuscript.
+   - *Where.* `paper_a_results_v4_section_iv.md` lines 192 (XV-B), 266 (XIX), 280 (XX), 300 (XXI), 317 (XXII), 329 (XXIII), 340 (XXIV), 353 (XXV).
+   - *Reasoning.* Sequential integer numbering is the IEEE Access norm. Either Table XV-B stays (and §IV-M tables stay XIX–XXV) or the rename cascades.
+   - *Fix.* If XV-B → XIX is preferred, also renumber §IV-M tables XIX→XX, XX→XXI, ..., XXV→XXVI; update all in-text references (§III provenance table at line 387 onward; §IV-J line 188 and 217 referencing "Table XVI"; §I and §V cross-references). Run a manuscript-wide grep on "Table XVI"…"Table XXVI" after the renumbering to catch stale internal pointers.
+   - *Tag.* **Gemini missed (cascade implication); codex missed entirely.**
+
+3. **The "98–100% of inter-CPA collisions concentrated within the source firm" claim is factually wrong for the deployed any-pair rule at three of four firms (both missed).**
+
+   - *Issue.* The Abstract (line 11), §I item 6 (line 53), §V-C (line 87), §V-G limitation 2 (line 115), and §VI conclusion item 4 (line 147) report "98–100% of inter-CPA collisions concentrated within the source firm". This range applies only to the **same-pair joint event** (a single inter-CPA candidate satisfying both cos>0.95 AND dHash≤5), which is the **stricter alternative** classifier explicitly contrasted to the deployed any-pair rule in §III-L.0 (line 183) and §III-L.4 (line 281).
+   - *Verification.* §IV-M Table XXIV (line 342) reports any-pair cross-firm hit counts. Computing within-firm fractions from Table XXIV:
+     - Firm A: 14,447 / 14,622 = 98.8%
+     - Firm B: 371 / 484 = 76.7%
+     - Firm C: 149 / 178 = 83.7%
+     - Firm D: 106 / 137 = 77.4%
+     - **Any-pair within-firm range: [76.7%, 98.8%]**, not [98%, 100%]. The 98–100% range corresponds to the same-pair rates (99.96/97.7/98.2/97.0%) stated separately at §III-L.4 line 281 and §IV-M.5 line 349.
+   - *Reasoning.* §III-L.0 makes a careful any-pair vs same-pair distinction and emphasises the deployed rule is **any-pair**. The Abstract / §I / §VI summarise the within-firm concentration using the same-pair number while attributing it to the deployed rule. §V-C line 87 is even worse: "98–100% of inter-CPA collisions originate from candidates within the source firm, **regardless of which Big-4 firm is the source**" — explicitly asserting the strong claim across all four firms, contradicted by Firms B/C/D any-pair rates of 77/84/77%.
+   - *Fix options.*
+     - **Option A (preferred):** Restate as "98% within-firm at Firm A, 77–84% at Firms B/C/D for the deployed any-pair rule; 97–100% within-firm under the stricter same-pair joint event across all four firms" in Abstract / §I / §V / §VI. This preserves the headline finding (within-firm dominance) while accurately characterising the gap between any-pair and same-pair.
+     - **Option B:** If the manuscript prefers the cleaner same-pair number, every occurrence must be re-attributed to "the same-pair joint event under the stricter alternative classifier (§III-L.4)", not to "the deployed rule" / "inter-CPA collisions". The current Abstract / §I framing ties the number to "inter-CPA collisions" without qualifier, which reads as applying to the deployed rule.
+   - *Severity.* This is the headline forensic finding in the Abstract and §I. Mis-stating it across five locations is more than a copy-edit — partners and reviewers form their understanding of "what the paper shows" from the Abstract.
+   - *Tag.* **Both missed.** Gemini's provenance spot-check #3 verified the byte-identical split (145/8/107/2) but did not verify whether the 98–100% claim in the Abstract matches the deployed-rule numbers in Table XXIV.
+
+4. **Duplicate §V-G heading (both missed).**
+
+   - *Issue.* `paper_a_prose_v4_phase4.md` has two §V-G headings: line 105 "G. Pixel-Identity as a Hard Positive Anchor; Inherited Inter-CPA Negative Anchor Reframed as Coincidence Rate" and line 109 "G. Limitations". The second should be "H. Limitations".
+   - *Where.* `paper_a_prose_v4_phase4.md` lines 105, 109. Also: line 37 (§I) cross-references "§V-G" for the conservative-subset caveat — ambiguous given two G sections; the content actually lives in BOTH (line 105 has "we caution that this result is necessary but not sufficient: for the box rule it is close to tautological", and line 119 has the "Pixel-identity is a conservative subset" bullet).
+   - *Reasoning.* §V originally went A through F in the v3 draft. The Phase 4 prose added a new §V-G ("Pixel-Identity ...") between §V-F and the existing §V-G Limitations without renaming the latter. Both prior reviewers spot-checked §V-G Limitations content for completeness (codex M9, Gemini codex-closure check #3) and confirmed the inherited v3.20.0 limitations are restored — but neither noticed the duplicate letter.
+   - *Fix.* Rename the second to §V-H ("H. Limitations"). Update every internal §V-G cross-reference: line 37, line 111 ("inherited from v3.20.0 §V-G"), line 160 (close-out checklist), and any §III or §IV pointer.
+   - *Tag.* **Both missed.**
+
+5. **Stale "seven limitations" assertion in close-out checklist (both missed).**
+
+   - *Issue.* Phase 4 close-out item 4 (line 160) says "The seven limitations are listed flat". Actual count in the §V-G Limitations section is **14** (9 v4.0-specific + 5 inherited from v3.20.0). §V-G line 111 correctly says "The first nine are v4.0-specific; the last five are inherited". The checklist item is stale from a previous version where the v4.0-specific count was 2.
+   - *Fix.* Either update to "The fourteen limitations are listed flat" or remove the checklist on Phase 5 splice (already on Gemini's list under m1 and codex's list under A8).
+   - *Tag.* **Both missed (a small instance of the broader internal-notes cleanup).**
+
+6. **§I contribution 4 and §IV-D both refer to a "2×2 factorial diagnostic" but only §IV-M Table XIX summarises it; §IV-D's body does not reproduce the 2×2 factorial table from §III-I.4 (codex implicitly covered; Gemini partly covered).**
+
+   - *Issue.* §IV-D line 23 says "the v4-new composition-decomposition diagnostics that establish this finding are tabulated in §IV-M below". The reader expects to find the §III-I.4 factorial table (4 rows: raw / centred-only / jittered-only / both) reproduced in §IV-M. But §IV-M Table XIX only summarises three of the four rows (collapses the "centred-only" and "jittered-only" cells into descriptive prose) and omits the "raw" baseline row's $p$-value comparison. Readers comparing §III-I.4 line 67 to §IV-M Table XIX will see a partial reproduction.
+   - *Severity.* Cosmetic — the same content is in §III-I.4 in full. But the §IV reader is implicitly told §IV-M is the destination for the §III-I.4 evidence, and §IV-M is not as complete as §III-I.4.
+   - *Fix.* Either reproduce the full 4-row 2×2 factorial table in §IV-M (currently Table XIX), or change §IV-D line 23 to "see §III-I.4 Table for the full factorial; §IV-M Table XIX summarises the key diagnostic outcomes".
+   - *Tag.* Both implicitly missed.
+
+7. **Inconsistent precision in three-score Spearman across §III-K and §IV-F (Gemini implicitly missed).**
+
+   - *Issue.* §III-K Table at line 125–127 reports three-score pairwise Spearman as +0.963, +0.889, +0.879 (3 decimal places). §IV-F Table IX at lines 85–87 reports the same correlations as +0.9627, +0.8890, +0.8794 (4 decimal places). §III provenance table line 365 uses 3 decimals.
+   - *Severity.* Low — the values are consistent at 3-dp rounding. But mixed precision across §III table / §IV table / Abstract is a copy-edit signal.
+   - *Fix.* Standardise on 4 decimal places everywhere (the §IV precision matches Script 38's output more faithfully) and update §III-K Table + §III provenance row + the Phase 4 abstract / §I body to match. Alternatively, standardise on the lower-bound floor "ρ ≥ 0.879" that the Abstract uses.
+   - *Tag.* Both missed (low priority).
+
+## Minor findings
+
+1. **Stale "approximately 235 words" / "243-244 words" abstract counts in two checklists.** Phase 4 close-out item 1 (line 157) says "Current draft is 243–244 words"; codex round-7 verified `wc -w` returns 243. Both are within the IEEE Access 250-word budget. Just remove the abstract-word-count notes before splice. (Gemini m3.)
+
+2. **Three "internal — remove before submission" draft notes still present.** Phase 4 line 3, §III v7 line 3 (lines 1–5), §IV v3.3 line 3. Plus two close-out checklists (§IV lines 365–375, Phase 4 lines 153–162) and §III's cross-reference index (lines 431–448). All flagged by codex (A8) and Gemini (m1). I corroborate.
+
+3. **§IV-J Table XV-B header pointer.** §IV-J line 228 "Document-level worst-case aggregation outputs are reported in Table XV-B above." — fine prose but does not specify which table number is the document-level table when XV-B → XIX renumber occurs. Once Major M2 is resolved, update accordingly.
+
+4. **Mixed pp vs decimal vs percentage notation across Abstract / §I / §III-L / §IV-M for the same numerical claims.** Examples:
+   - Abstract line 11 uses "0.34 for the operational HC+MC alarm"
+   - §I line 33 uses "33.75% of Big-4 documents"
+   - §III-L.3 Table reports "0.3375 [Wilson 95% [0.3342, 0.3409]]"
+   - §IV-M Table XXII reports "0.3375"
+   - Phase 4 conclusion line 147 uses "(0.34 for the operational HC+MC alarm)"
+   - Standardise on either decimal or percentage throughout numeric quotations; the lossy "0.34" in the Abstract drops the 95% CI which is reported elsewhere. The IEEE Access norm in this corpus is decimal-with-CI in tables, percentage-without-CI in prose summary.
+
+5. **"v3.x §IV-F.1" cross-reference is not validated against the v3 file's actual section letter.** §III-H line 37 and §V-C line 85 both cite "v3.x §IV-F.1" for the 145 pixel-identical signatures across ~50 distinct Firm A partners. If `paper_a_results_v3.md` uses a different sub-section letter (e.g., §IV-G or §IV-F-1), the citation will be stale at splice time. I did not verify this but flag it as a splice-time check.
+
+6. **"~50 distinct partners of 180" vs §III-H "50 distinct Firm A partners of 180 registered" — Firm A registered partner count of 180 is inherited from v3.20.0 / Script 28 with no in-paper provenance verification.** This is flagged in §III-H line 37 ("inherited from v3 §IV-F.1 / Script 28 / Appendix B byte-decomposition output and were not regenerated in v4.0 spike scripts") — adequately disclosed. Just confirm at splice that v3.20.0's count of 180 is reproducible.
+
+7. **§I.5 contribution 5 line 51 claim "We adopt 'inter-CPA coincidence rate' as the metric name throughout and reserve 'False Acceptance Rate' for terminology that requires ground-truth negative labels".** §IV-I line 159 prose uses "False Acceptance Rate (FAR)" in the historical-context phrase "previously reported as 'False Acceptance Rate' in v3.x" — this is intentional historical reference, but it does mean "FAR" appears once in §IV body, which contradicts the §I claim of "throughout". Either weaken §I.5 to "throughout v4.0 framing, with one historical-reference exception in §IV-I" or strip "FAR" from §IV-I line 159 and reword to "previously reported using biometric-verification 'False Acceptance Rate' terminology".
+
+8. **MC band per-firm proportions in §IV-J line 215.** "10.76% / 35.88% / 41.44% / 29.33% across Firms A through D". Cross-checking against Table XV per-firm breakdown line 183–186: Firm A MC = 10.76%, Firm B MC = 35.88%, Firm C MC = 41.44%, Firm D MC = 29.33%. Consistent ✓.
+
+9. **§V-G limitation 1 list says "first nine are v4.0-specific" but Phase 4 close-out item 4 calls them "seven limitations".** Already covered under Major M5.
+
+10. **Provenance-table cross-references inside the §III provenance table.** Line 364: "K=3 LOOO held-out C1 absolute differences 1.8–12.8 pp | direct | Script 37 held-out prediction check". Cross-checked against §IV-G Table XIII (line 134–137): the four held-out absolute differences are 4.68, 1.76, 12.77, 5.81 → range 1.76 to 12.77, which rounds to "1.8–12.8" ✓.
+
+11. **Abstract line 11 phrasing "98–100% of inter-CPA collisions concentrated within the source firm — consistent with firm-level template-like reuse"** —  this is the most-public statement of Major M3. The Abstract is the only thing many readers will read; the imprecision is more costly here than in §V-G.
+
+## Provenance spot-checks
+
+I deliberately chose 5 claims NOT covered by Gemini's spot-checks.
+
+1. **Within-source-firm any-pair collision rates (the "98–100%" claim).**
+   - *Claim text.* "with $98$–$100\%$ of inter-CPA collisions concentrated within the source firm" (Abstract line 11, §I line 53, §V-C line 87, §V-G line 115, §VI line 147).
+   - *Manuscript location of evidence.* Table XXIV in §IV-M.5 (line 342); §III-L.4 cross-firm hit matrix (line 274–281).
+   - *Cited script.* Script 44.
+   - *Verdict.* **INCONSISTENT.** Computed from Table XXIV: any-pair within-firm fractions are 98.8% / 76.7% / 83.7% / 77.4% (range 76.7–98.8%, not 98–100%). The 98–100% range corresponds to the same-pair joint event explicitly reported as the **stricter alternative** at §III-L.4 line 281 and Table XXIV line 349 (99.96% / 97.7% / 98.2% / 97.0%). The narrative-level claim conflates two distinct rule semantics. The script logic (44_firm_matched_pool_regression.py lines 274–327) does compute both matrices; the manuscript reports both correctly inside §III-L.4 / §IV-M.5; the failure is in the Abstract / §I / §V-C / §V-G / §VI narrative-summary layer. See Major M3.
+
+2. **Per-pair conditional ICCR dHash≤5 given cos>0.95 = 0.234 (Wilson [0.190, 0.285], 70 of 299 pairs).**
+   - *Claim text.* §III-L.1 line 208; Phase 4 §V-F line 103.
+   - *Cited script.* Script 40b.
+   - *Verdict.* **VERIFIED for logic; numerical value not independently verifiable from worktree.** Script 40b at lines 22–23 explicitly computes "Conditional FAR(dh<=k | cos>0.95)" for k∈[0, 20] and prints "P(dh<=k | cos>0.95)" for each k (lines 226–229 of the script). The 0.234 / 70 / 299 numbers cannot be checked without access to `/Volumes/NV2/PDF-Processing/signature-analysis/reports/v4_big4/inter_cpa_far_sweep/` (not in the worktree). Wilson [0.190, 0.285] is a plausible 95% CI for $\hat{p} = 70/299 = 0.2341$ (manual check: SE ≈ $\sqrt{0.234 \cdot 0.766/299} = 0.0245$, normal-approx 95% CI [0.186, 0.282] — Wilson CIs are slightly wider on each side, so [0.190, 0.285] is consistent). Logic + analytic plausibility verified.
+
+3. **Pooled Big-4 per-signature any-pair ICCR 0.1102 with Wilson [0.1086, 0.1118].**
+   - *Claim text.* §III-L.2 Table at line 220; Phase 4 §V-F line 101.
+   - *Cited script.* Script 43.
+   - *Verdict.* **VERIFIED analytically.** For $\hat{p} = 0.1102$ on $n_{\text{sig}} = 150{,}453$, the Wilson-approximate 95% CI half-width is $\approx 1.96 \cdot \sqrt{0.1102 \cdot 0.8898 / 150453} = 0.00159$, giving [0.1086, 0.1118] — exactly the reported interval. CPA-block bootstrap [0.0908, 0.1330] is much wider, as expected once CPA-level clustering is recognised (CIs widen by factor of $\sim 9\times$ here), consistent with the strong within-CPA correlation that the manuscript discusses.
+
+4. **Logistic OR pool-size effect = 4.01.**
+   - *Claim text.* §III-L.4 Table line 266; §IV-M.5 Table XXIII line 336.
+   - *Cited script.* Script 44.
+   - *Verdict.* **VERIFIED for logic; magnitude is intuitive.** Script 44 at lines 219–227 fits `logistic(hit) ~ intercept + FirmB + FirmC + FirmD + log(pool_size_centered)` and reports `OR = exp(beta)`. A 4× per-log-unit pool-size effect means each $e\times$ pool-size increase quadruples the per-signature odds — consistent with the §III-L.2 decile trend (decile 1 = 0.0249, decile 10 = 0.1905, ratio 7.65×, pool ratio approximately $1115/201 \approx 5.5$, log-ratio $\approx 1.71$, predicted OR $\approx 4.01^{1.71} = 10.7$ vs observed 0.1905/0.0249 = 7.65; the gap is small enough to be consistent with monotonicity + finite-sample). No red flag.
+
+5. **Alert-rate sensitivity local-gradient ratios (cos=0.95: ≈25×; dHash=5: ≈3.8×; dHash=15: ≈0.08).**
+   - *Claim text.* §III-L.5 line 291; §IV-M.6 Table XXV line 357–359.
+   - *Cited script.* Script 46.
+   - *Verdict.* **PARTIALLY VERIFIABLE.** Script 46 at lines 236–276 explicitly computes the local-to-median gradient ratio at the v3-inherited threshold (cos=0.95 and dh=5) and emits `ratio_local_to_median` for both. **However**, the script does NOT emit a dh=15 ratio: `DH_GRID = np.arange(0, 21, 1)` covers 15 (so the swept rates ARE available in the saved JSON), but the script only computes the ratio at dh=5. The ≈0.08 ratio at dh=15 in Table XXV must therefore be a **derived quantity** computed by the author from the JSON output post-hoc, not a direct script output. This is not a fabrication risk (the author has access to the swept rates), but the manuscript should either (a) add a brief in-paper note that the dh=15 ratio is computed identically to the dh=5 ratio but post-hoc on the JSON, or (b) extend Script 46's plateau-detection block to emit ratios at both dh=5 and dh=15.
+
+   *Secondary note.* The cosine sweep prose at §III-L.5 line 291 quotes "cosine sweep at dHash ≤ 5 yields rates of 0.5091 at cos > 0.945 vs 0.4789 at cos > 0.955". From Script 46 line 56–57: `COS_FOR_2D = np.arange(0.85, 1.00, 0.01)` only covers integer hundredths, so 0.945 and 0.955 are NOT on the 2D grid. The 1D `COS_GRID` covers the sweep at which 0.945 / 0.955 may be in the grid; I cannot fully verify without seeing the saved JSON. The numerical specificity is suspiciously precise (four decimals on a derived rate) — flag for partner spot-check at splice.
+
+## Newly introduced issues
+
+1. **The "specificity-proxy-anchored screening framework with human-in-the-loop review" positioning is consistent across Abstract / §I / §III-M / §V-G / §VI — except that the term "human-in-the-loop" appears NOWHERE in §III or §IV, only in §I line 30 (item 5), §III-L.3 line 255, §III-M line 334, and §VI line 147.** §I item 5 promises "anchor-based threshold calibration at three units of analysis ... against an inter-CPA negative-anchor coincidence-rate proxy", and §III-M closes with "positioning as a specificity-proxy-anchored screening framework with human-in-the-loop review". But §IV's results discussion never operationalises what "human-in-the-loop review" means: the per-document HC+MC alarm rate of 0.34 is reported as a number; whether the human reviewer triages all 34% of flagged docs or only a sub-set is not specified. Newly introduced v4-positional claim that v3.20.0 did not make, lacking a concrete operationalisation. Either explicitly punt to future work or describe the intended triage workflow briefly in §V or §VI.
+
+2. **The "feature-derived" qualifier coverage IS reliable across §III-K, §V-E, Abstract, §I, §VI — but breaks down in §IV.** Major M1 above. Newly introduced in v4 because v3.20.0 used "three independent scores" without the caveat; v4 added the caveat in §III/§V/§I/§VI but not §IV.
+
+3. **The "9-tool nine-diagnostic" validation collection (§III-M Table at lines 318–329) and §VI item 8 nine-tool claim — count is internally consistent (9 rows in the §III-M table), but the table is currently unnumbered.** All other tables in §III/§IV have a Table N: header. The §III-M validation table should be Table XXVI (or whatever number after the §IV-M cascade) for IEEE Access. Otherwise it is an unnumbered table inside a discussion of "Table-numbering scheme" elsewhere — inconsistent.
+
+4. **The §I cross-reference list at line 29 ("(1) signature page identification...; (2) signature region detection...; ... (8) a multi-tool unsupervised validation strategy") promises 8 pipeline steps in numbered order, but the actual pipeline order in §III is steps 1–5 are the embedding pipeline and 6–8 are validation framework / decomposition / framework positioning, not strict pipeline stages.** Cosmetic; the v3.20.0 contribution list had 7 sequential pipeline-stages; the v4.0 list collapses pipeline + validation into 8 items, blurring stage vs framework. Either say "Eight elements" rather than imply 8 pipeline steps, or split into "Pipeline (1–4): ...; Calibration and validation framework (5–8): ...".
+
+## Disagreements with prior reviewers
+
+1. **I disagree with codex round-7 M6 (closed) "§II citation-number gap and placeholder contradiction."** codex round-7 says "[42]-[44] remain placeholders and absent from the reference list." Gemini correctly identified that codex was wrong: [42]–[44] **are** present at lines 87–91 of `paper_a_references_v3.md`. I corroborate Gemini's correction. The only residual is the `[add citation]` placeholder string in Phase 4 close-out note line 159 (just a stale comment in a checklist that will be stripped at splice). codex's claim was based on the file ending at [41] in some earlier snapshot; the current file ends at [44].
+
+2. **I partially disagree with Gemini's finding #4 "K=3 Demotion Language Consistency" (verdict: "CLOSED. Consistent descriptive framing.").** §III, §I, §V, §VI properly demote K=3, but §IV (which Gemini did not separately enumerate for this check) retains mechanism labels throughout (Major M1 above). The K=3 demotion is **partially closed**, not fully closed.
+
+3. **I corroborate Gemini's finding #1 on "statistically insignificant" framing risk** (Major in Gemini's review, missed by codex). The OR magnitudes are 0.05/0.01/0.03 — these are 19×/100×/37× effects after pool-size adjustment, with standard errors that the manuscript flags as needing cluster-robust treatment but which are nonetheless an order of magnitude below 1. There is no defensible reading under which this is "statistically insignificant"; any such partner framing must be rejected.
+
+4. **Neither codex nor Gemini examined the cross-firm hit matrix arithmetic** to verify that the Abstract's "98–100%" claim is consistent with Table XXIV. Major M3 is my net-new finding.
+
+5. **Neither codex nor Gemini flagged the duplicate §V-G heading.** Major M4 is my net-new finding.
+
+6. **Neither codex nor Gemini flagged the §IV vs §III terminology drift around "hand-leaning / replicated".** Major M1 is my net-new finding.
+
+7. **codex M1 (Abstract independent-score correction closed) — I corroborate the closure for the Abstract** (line 11 says "Three feature-derived scores"), but extend the finding: the same correction is NOT propagated to §IV Tables IX/X/XVII/XVIII where the "Paper A box-rule hand-leaning rate" label retains mechanistic framing. This is Major M1 again.
+
+## Phase 5 readiness
+
+**Partial — closer to "blocked on copy-edit pass" than to "ready".** Three major issues require manuscript-text rewrites before Phase 5 splice:
+
+- **Empirical-language blocker:** Major M3 (98–100% within-firm claim) requires Abstract / §I / §V-C / §V-G / §VI prose rewrite. This is more than copy-edit — the headline finding's numerical scope must be reconciled with Table XXIV.
+- **Terminology blocker:** Major M1 (§IV mechanism labels) requires global s/hand-leaning/less-replication-dominated/ across §IV tables and prose plus Table XVI column-header rename.
+- **Structural blocker:** Major M4 (duplicate §V-G heading) requires §V renumber.
+- **Numbering blocker:** Major M2 (table-numbering cascade) requires either keeping XV-B suffix or cascading XIX→XX→...→XXVI across §IV-M and updating every cross-reference.
+
+The empirical core (scripts 32–46, ICCR multi-level calibration, composition decomposition, three-score convergence, byte-identical anchor, logistic regression) is sound and reproducible from script inspection. No new statistical work is required for Phase 5.
+
+## Recommended next-step actions
+
+Ranked by severity, distinguishing prose-rewrite blockers from copy-edit-only items.
+
+1. **[Manuscript rewrite — prose blocker]** Reconcile the "98–100% within-firm" claim with Table XXIV. Adopt Major M3 Option A or Option B in Abstract / §I item 6 / §V-C / §V-G limitation 2 / §VI item 4. Preferred phrasing (Option A): "98% of inter-CPA collisions at Firm A and 77–84% at Firms B/C/D under the deployed any-pair rule, rising to 97–100% across all four firms under the stricter same-pair joint event".
+
+2. **[Manuscript rewrite — terminology blocker]** Global s/hand-leaning/less-replication-dominated/ across §IV (Table IX/X column labels, Table XIV column, Table XVI columns, Table XVII rows, Table XVIII title, §IV-J/K/M prose). Rename Table XVI columns to match §IV-E Table VIII's "descriptive position" column. Keep §I, §V, §VI as currently worded.
+
+3. **[Manuscript rewrite — structural blocker]** Rename §V's second "G. Limitations" to "H. Limitations". Update every §V-G cross-reference.
+
+4. **[Manuscript rewrite — partner framing]** Explicitly state at §V-C or §V-G that firm heterogeneity is highly statistically significant (OR magnitudes 19×, 100×, 37× after pool-size adjustment with $z$-equivalent ≥ 10 in absolute value on standard SE; cluster-robust SE is flagged as a robustness check, not as the reason heterogeneity might be insignificant). Reject any reframing as "statistically insignificant". (Gemini Major #1.)
+
+5. **[Copy-edit blocker — table renumbering]** Decide on the XV-B suffix. If renamed to XIX, cascade through §IV-M XIX→XX, XX→XXI, …, XXV→XXVI. Update §III provenance table cross-references. Update §IV-J line 188 / 217 / 228 references. Update §IV-L line 258 to disambiguate from v3.20.0 Table XVIII.
+
+6. **[Copy-edit blocker — strip internal artefacts]** Remove:
+   - Phase 4 prose draft note (line 3) and close-out checklist (lines 153–162)
+   - §III v7 draft note (lines 1–5) and cross-reference index + open-questions block (lines 431–448)
+   - §IV v3.3 draft note (line 3) and close-out checklist (lines 365–375)
+
+7. **[Copy-edit — terminology consistency]** Decide on decimal vs percentage notation for the headline ICCRs (0.34 vs 33.75% etc.) and apply uniformly across Abstract / §I / §V / §VI.
+
+8. **[Copy-edit — Spearman precision]** Standardise three-score correlations on either 3-dp (0.963/0.889/0.879) or 4-dp (0.9627/0.8890/0.8794) across §III-K / §IV-F / §III provenance.
+
+9. **[Copy-edit — §III-M validation table]** Assign a table number (Table XXVI after the §IV-M cascade) to the nine-tool validation table at §III-M lines 318–329.
+
+10. **[Copy-edit — §V-G item count]** Update Phase 4 close-out item 4 "seven limitations" to "fourteen limitations" (or strip the checklist).
+
+11. **[Splice-time check]** Verify v3.20.0 §IV-F.1 cross-reference letters resolve correctly when §III-H / §V-C are spliced into the master file. Verify v3.20.0 Table XVIII (backbone ablation) does not collide with v4 Table XVIII (Spearman drift) once the final renumbering is applied. Verify all "v3.x" pointers point to actual sections that survive in the master manuscript.
+
+12. **[Minor — script 46 dh=15 ratio]** Either add a brief note in §IV-M that the dh=15 plateau ratio is computed post-hoc from the swept JSON (the swept rates themselves are emitted by Script 46), or extend Script 46 to compute the ratio at dh=15 directly.
+
+13. **[Minor — pipeline-step vs framework-element framing]** Reword §I line 29's "(1)…(8)" enumeration to distinguish pipeline stages (1–4) from validation/framework elements (5–8).
+
+14. **[Minor — human-in-the-loop operationalisation]** Add one sentence in §V or §VI describing what "human-in-the-loop review" means operationally (e.g., manual inspection of the 0.34 flagged-document fraction; sampling strategy; reviewer-effort estimate).

paper/opus_review_v4_round2.md

@@ -0,0 +1,233 @@
+# Paper A Phase 5 Round 2 — Opus 4.7 max-effort independent review
+
+Reviewer: Claude Opus 4.7
+Date: 2026-05-14
+Target: paper/v4/paper_a_prose_v4_phase4.md + paper/v4/paper_a_methodology_v4_section_iii.md + paper/v4/paper_a_results_v4_section_iv.md (post round-2 + round-3, commit 4a6f9c5)
+Prior reviewer artifacts: paper/codex_review_gpt55_v4_round7.md; paper/codex_review_gpt55_v4_round8.md; paper/gemini_review_v4_round1.md; paper/opus_review_v4_round1.md
+
+## Verdict
+
+**Minor Revision (corroborates codex round-8).** The empirical core is sound and reproducible. Round-2 (b884d39) closed my round-1 M1–M4 cleanly, and round-3 (4a6f9c5) closed codex round-8's three concrete splice blockers (abstract trim 247 w; §IV-J n=150,442 vs 150,453 footnote correctly distinguishing descriptor-complete vs vector-complete; §IV-I "§IV-M Tables XXI-XXVI" replacing the stale "Table XVI"). I dissent on **readiness for splice**: a fresh round-2 pass surfaces three net-new findings that the panel collectively missed — a denominator inconsistency between §IV-J Table XIX and §IV-M.4 per-firm doc counts (379-doc mixed-firm-PDF mode-of-firms tie-break), the absence of the composition-decomposition diagnostic from the §III-M nine-tool validation table that anchors the v4 narrative, and the unnumbered status of the §III-M table itself. None of the three is empirical-blocker grade; they are substantive copy-edit / structural fixes that should be patched in a single round-4 pass before splice.
+
+I align with codex round-8 on the disposition and on M1-M4 closure judgments. I do NOT corroborate Gemini round-2 directly (parallel; not read).
+
+## Cross-reviewer convergence summary
+
+Post-round-3 (4a6f9c5):
+
+| Theme | codex r8 | Opus r2 (this) |
+|---|---|---|
+| Overall disposition | Minor Revision | Minor Revision (corroborate) |
+| Opus M1 K=3 mechanism-label reversion | CLOSED (Table XI residue → fixed in r3 to "replication-dominated vs less-replication-dominated") | **CLOSED.** Verified by grep: `hand-leaning` returns 0 matches in §IV body; the only 2 §III matches are internal-checklist open-question text (line 445) and an unrelated $\Delta$BIC line, both stripped at splice. |
+| Opus M2 Table XV-B cascade | CLOSED in public body | **CLOSED.** Tables XV→XIX, §IV-M cascades XX-XXVI correctly. Only "XV-B" residue is in internal draft notes (lines 3, 370) that strip at splice. |
+| Opus M3 within-firm any-pair vs same-pair | CLOSED in body; abstract uses rounded any-pair 77-99% | **CLOSED.** Abstract line 11 "77-99%" rounds the deployed-rule any-pair range (76.7-98.8%). §I item 6 (line 53), §V-C (line 87), §V-H limitation 2 (line 115), §VI item 4 (line 147), §VI future work (line 149) all give the correct any-pair 76.7-83.7% / 98.8% split plus same-pair 97.0-99.96% subrange. |
+| Opus M4 duplicate §V-G | CLOSED | **CLOSED.** §V headings now run A-H sequentially (lines 73/77/83/89/95/99/105/109). |
+| Gemini Table XV sample-size footnote | CLOSED in r3 | **CLOSED.** §IV-J line 177 footnote now correctly groups §IV-M.2/M.3/M.5 (Scripts 40b/43/44) as vector-complete 150,453. |
+| Codex r8 splice blockers | r8 status open; r3 fixed | **CLOSED.** Abstract 247 w (verified `wc -w` on line 11); §IV-I (line 161) now points to "§IV-M Tables XXI-XXVI"; binary-collapse label is "replication-dominated vs less-replication-dominated" (§III line 131, §IV Table XI line 104). |
+| Internal draft notes | OPEN (splice-strip pending) | **OPEN.** Three draft-note blocks + three close-out checklists + §III cross-reference index + §III open-questions block still present and must strip at splice. |
+
+## M1–M4 closure verification (full audit)
+
+### M1 — §IV K=3 mechanism-label reversion: CLOSED
+
+Provenance grep `hand-leaning` returns only 2 matches in §III, both non-substantive:
+- `paper_a_methodology_v4_section_iii.md:90` — false positive ("lower than $K{=}2$ by $3.48$", the term "leaning" never appears).
+- `paper_a_methodology_v4_section_iii.md:445` — Open-question item 2 in the §III internal checklist ("Firm C is the firm most concentrated in C1 hand-leaning at 23.5%"). This is in the **author working notes** at lines 441-447, scheduled for splice-strip per Gemini m1 / codex Opus minor 2.
+
+§IV body is entirely cleansed: Table IX (line 85-87) "less-replication-dominated rate"; Table X (line 93) "mean Paper A less-replication-dominated rate"; Table XI (line 104) "binary collapse, replication-dominated vs less-replication-dominated" (round-3 fix); Table XIV (line 147) "Misclassified as less-replication-dominated"; Table XVI (line 219) "C1 (low-cos / high-dHash) | C2 (central) | C3 (high-cos / low-dHash)"; Table XVII (line 238) "C1 (low-cos / high-dHash)" / "C3 (high-cos / low-dHash)"; Table XVIII (line 244-246) "Paper A operational less-replication-dominated rate"; §IV-F prose (line 102) "the most replication-dominated"; §IV-K reading (line 254) "non-templated CPAs". All match §III-J line 90's intent.
+
+**One residual concern (low priority):** the K=3 LOOO upstream Script 37 report file at `/Volumes/NV2/PDF-Processing/signature-analysis/reports/v4_big4/k3_loo_check/k3_loo_report.md` still uses "C1 hand-leaning / C2 mixed / C3 replicated" labels (lines 7-9). This is in the empirical pipeline output, not the manuscript; it doesn't affect the published paper but does mean a reviewer following the data trail will see legacy labels. Worth noting in author working notes that the script report file naming convention has not been updated.
+
+### M2 — Table-numbering cascade: CLOSED
+
+Verified the cascade: §IV-J Table XV (line 167, five-way per-sig), Table XVI (line 217, K=3 cross-tab), Table XVII (line 234, full-vs-Big-4 K=3 drift), Table XVIII (line 244, Spearman full-vs-Big-4), Table XIX (line 192, document-level worst-case), Table XX (line 266, composition decomposition), Table XXI (line 280, per-comparison ICCR), Table XXII (line 300, pool-normalised per-sig ICCR), Table XXIII (line 317, document-level ICCR), Table XXIV (line 329, logistic regression), Table XXV (line 340, cross-firm hit matrix), Table XXVI (line 353, alert-rate sensitivity).
+
+In-text §IV cross-references all consistent:
+- §IV-D line 23 "tabulated in §IV-M below" — non-specific, OK.
+- §IV-I line 161 (round-3 fix) "§IV-M Tables XXI-XXVI" — correct.
+- §IV-J line 188 "qualitatively aligns with the K=3 cluster cross-tab of Table XVI" — correct.
+- §IV-J line 228 "Document-level worst-case aggregation outputs are reported in Table XIX above" — correct.
+
+§III provenance table (lines 386-427) does not by-number reference §IV tables, only by §III subsection / Script — robust to the cascade.
+
+Phase 4 prose: no Table-XV-X references in §V or §VI; safe.
+
+Only stale "Table XV-B" residue lives in the §IV draft-note (line 3) and close-out checklist (line 370), strip-at-splice items.
+
+### M3 — Within-firm collision semantic conflation: CLOSED
+
+The Abstract (line 11) uses the rounded any-pair-only range "77-99% of inter-CPA collisions concentrate within the source firm — consistent with firm-level template-like reuse". 76.7-98.8% rounds to 77-99% (lossy at the boundary but defensible as a 2-significant-figure summary).
+
+I verified by grepping "98-100\\|98–100" — zero matches in §III/§IV/Phase 4. The legacy framing is fully removed. The corrected pattern (any-pair + same-pair subrange disclosure) appears at:
+- §III-J line 99: "within-firm collision concentration is $98.8\\%$ at Firm A and $76.7$–$83.7\\%$ at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–$99.96\\%$ within-firm across all four firms)"
+- §III-L.4 line 283: same body text in primary methodology location.
+- §V-C line 87 (Phase 4): same pattern + the v3.x byte-level evidence cross-reference.
+- §V-H limitation 2 line 115 (Phase 4): correctly attributes the partial-violation framing.
+- §VI item 4 line 147 (Phase 4): same pattern.
+- §VI future work line 149: "(any-pair $76.7$–$98.8\\%$ across Big-4; same-pair joint $97.0$–$99.96\\%$)" — full precision.
+
+§I contribution 6 (line 53) also gives the any-pair 98.8% / 76.7-83.7% + same-pair 97.0-99.96% split.
+
+### M4 — Duplicate §V-G heading: CLOSED
+
+`grep -n '^## [A-Z]\\.'` on Phase 4 prose shows §V sub-sections A-H in correct order: A (line 73), B (77), C (83), D (89), E (95), F (99), G (105 "Pixel-Identity..."), H (109 "Limitations"). One cross-reference to §V-G in the close-out checklist (line 160) is internal-strip-at-splice; §V-H line 111 "inherited from v3.20.0 §V-G" correctly cites the original v3.x letter.
+
+## Net-new findings (fresh look post-round-3)
+
+### N1. Denominator inconsistency between §IV-J Table XIX per-firm document counts and §IV-M.4 per-firm D2 counts — **discoverable in 30s by a careful reader**.
+
+*Issue.* §IV-J Table XIX (lines 207-211) reports per-firm document-level breakdown **"single-firm PDFs only; mixed-firm PDFs $n = 379$ excluded"**: Firm A 30,226 / Firm B 17,127 / Firm C **19,122** / Firm D 8,379. §IV-M.4 line 325 reports per-firm D2 document-level ICCR with denominators: Firm A 30,226 / Firm B 17,127 / Firm C **19,501** / Firm D 8,379.
+
+The Firm A/B/D denominators are identical between the two tables; only Firm C differs by exactly 379, the mixed-firm count. The full sum 75,233 reconciles in §IV-M.4 but not in §IV-J (74,854 = 75,233 - 379).
+
+*Root cause (verified against Script 45 source).* Script 45 (`signature_analysis/45_doc_level_far_full_5way.py`) lines 250-260 assigns each PDF to its firms-mode via `np.unique(firms[idxs], return_counts=True); doc_firm[pdf] = str(vals[np.argmax(counts)])`. For mixed-firm PDFs with a 1:1 firm tie, `np.argmax` returns the first index, which under `np.unique`'s alphabetical sort means tie-break order A < B < C < D. So if a PDF has signatures from {Firm A, Firm C}, the mode-of-firms is Firm A by tie-break, not Firm C. Yet all 379 mixed-firm PDFs land in Firm C in the output — meaning Firm C is the genuine majority firm in every mixed-firm PDF, not a tie-break artefact. This is empirically plausible (Firm C may dominate joint audits) but the manuscript does not disclose it.
+
+*Severity.* Medium. The §IV-M.4 Firm C ICCR of 0.1635 is computed on 19,501 docs (including 379 mixed-firm), while the §IV-J Table XIX Firm C 5-way distribution is on 19,122 docs (single-firm only). A reader who tries to compose §IV-J + §IV-M.4 to reason about Firm C will see two different denominators with no inline reconciliation.
+
+*Fix.* Add a one-line footnote to §IV-M.4 Table XXIII clarifying that "per-firm document counts use Script 45's mode-of-firms assignment, which assigns mixed-firm PDFs to their majority firm; the 379 mixed-firm PDFs all resolve to Firm C and are excluded from §IV-J Table XIX's single-firm-only breakdown." Alternatively, harmonise on a single rule (mode-of-firms throughout, or single-firm-only throughout).
+
+*Tag.* **Both prior rounds + codex r8 missed this.**
+
+### N2. The §III-M nine-tool validation table **omits the composition-decomposition diagnostic** that anchors the v4 narrative.
+
+*Issue.* The §III-M table (lines 318-329) lists 9 tools: per-comparison ICCR, per-signature ICCR, per-document ICCR, firm-heterogeneity logistic regression, cross-firm hit matrix, alert-rate sensitivity sweep, convergent score Spearman ranking, pixel-identical conservative positive capture, LOOO firm-level reproducibility. The §III-I.4 composition-decomposition diagnostic (Scripts 39b-39e), which is the most novel v4 contribution and is foregrounded in:
+- Abstract line 11 ("dissolves under joint firm-mean centring and integer-tie jitter")
+- §I contribution 4 ("2×2 factorial diagnostic ... fully attributable to between-firm location shifts and integer mass-point artefacts")
+- §VI conclusion item 1 ("composition decomposition (Scripts 39b-39e) that establishes the absence of a within-population bimodal antimode")
+- §V-B (Phase 4 line 81)
+
+...is **not in the nine-tool table**. The reader is told the system has a "multi-tool collection of partial-evidence diagnostics" and the table claims to enumerate them, but the most distinctive v4 diagnostic is absent. The omission is structurally awkward because the composition decomposition is what justifies the entire anchor-based-rather-than-distributional pivot.
+
+*Reasoning.* This is not just nomenclature — §III-M is the manuscript's explicit answer to "what validates this in the unsupervised setting?" Omitting the composition decomposition from §III-M Table reads as if the v4 authors do not consider it part of the validation collection. But §I item 4 + §VI item 1 + Abstract all rely on it as the foundation for the anchor-based pivot.
+
+*Fix.* Add a row: "Composition decomposition (§III-I.4; Scripts 39b-39e) | Demonstrates that Big-4 dip-test rejection is attributable to between-firm location shift + integer-tie artefact, not within-population bimodality | Assumes within-firm signature-level distribution is the appropriate unit; bootstrap resolution $n_{\\text{boot}} = 2000$ bounds $p$-value precision at $5 \\times 10^{-4}$".
+
+This converts the framing from "nine-tool" to "ten-tool". The "nine" appears in §I contribution 8 (line 57), §VI item 8 (line 147), and §III-M's framing. All three would need to update to "ten-tool". Alternatively, fold the composition decomposition into an existing row (e.g., merge with "Convergent score Spearman" as a "Distributional / convergent diagnostic" row) — but that obscures the v4 pivot.
+
+*Tag.* **All three prior reviewers missed this. Highest-priority net-new finding.**
+
+### N3. The §III-M nine-tool table is structurally **unnumbered**.
+
+This was flagged in my round-1 as new-issue #3 (low priority) and codex r8 reflagged it as Opus-new-issue-3. It remains unfixed in round-3. The other §III tables (lines 60-66 factorial; the K=3 component table in §III-J at line 83-87) are inline tables also unnumbered. But §III-M's table is referenced from §I (line 57 "a multi-tool unsupervised validation strategy") and §VI (line 147 "nine-tool unsupervised-validation collection (§III-M)") as a load-bearing artefact. If the journal style requires every numbered display object to have a Table N: header, this needs Table XXVII (the next number after §IV-M.6's Table XXVI).
+
+*Fix.* Either assign "Table XXVII" to the §III-M table, or restate §I item 8 / §VI item 8 to "a nine-tool collection (see §III-M)" without table-numbered cross-reference.
+
+### N4. §III-M row 5 ("Cross-firm hit matrix") **understates the untested assumption** as "None — direct descriptive observation".
+
+*Issue.* Line 324: "Cross-firm hit matrix (§III-L.4; Script 44) | Concentration of inter-CPA collisions within source firm | None — direct descriptive observation".
+
+This is too strong. The cross-firm hit matrix is computed under the deployed any-pair rule, against an inter-CPA candidate pool drawn from non-same-CPA signatures. The "concentration within source firm" depends on (a) the deployed-rule semantics (any-pair vs same-pair: §V-H limitation 2's whole point is that the rate differs across the two semantics), (b) the candidate-pool construction (all non-same-CPA across all firms; the report doc shows this draws on 168,755 total signatures), and (c) for the per-document version, the mode-of-firms tie-breaking surfaced in N1.
+
+*Fix.* Replace "None" with something like: "Reflects deployed any-pair rule semantics (the stricter same-pair joint event yields 97-99.96% within-firm across all four firms; §III-L.4); per-document per-firm assignment uses Script 45's mode-of-firms rule (§IV-M.4 N1)."
+
+*Tag.* Net-new.
+
+### N5. The §V-H limitations list (14 items) does **not include a limitation about within-firm collision firm-dependence**.
+
+*Issue.* The §V-H list at line 113-139 covers 9 v4-specific limitations and 5 inherited from v3.20.0. Item 2 (line 115) covers the assumption-violation but not the firm-dependent fact: under the deployed any-pair rule, Firm A is 98.8% within-firm but Firms B/C/D are only 76.7-83.7% within-firm. This means the inter-CPA-as-negative assumption is more violated at Firm A than at Firms B/C/D — so per-firm ICCRs at Firm A are most contaminated by within-firm sharing, while per-firm ICCRs at B/C/D are closer to clean specificity. The implication is that the headline pooled rate (per-document HC+MC 0.34) is over-influenced by Firm A's higher within-firm contamination, and the per-firm B/C/D rates of 0.09-0.16 are more nearly a clean specificity estimate.
+
+This nuance is not in §V-H but matters for a reader interpreting "per-firm rates differ by an order of magnitude" as evidence of differential template-sharing rates rather than as a confound on the inter-CPA proxy itself.
+
+*Severity.* Low. §V-H item 2 implicitly covers it; the question is whether to spell it out as a separate "interpretive caveat" item.
+
+*Fix.* Optional. Add to limitation 2 a sentence: "The within-firm violation is firm-dependent (Firm A 98.8%, Firms B/C/D 76.7-83.7% any-pair), so per-firm ICCRs at Firm A are more contaminated by within-firm sharing than at Firms B/C/D."
+
+*Tag.* Net-new (low priority).
+
+### N6. §III-K item 4 line 149 cross-references "§III-L.1 (Big-4 sample) and the v3.x §IV-I corpus-wide version" but §IV-I has been substantially shrunk in v4.
+
+*Issue.* §III-K.4 line 149 says "The corresponding signature-level inter-CPA negative-anchor ICCR evidence is developed in §III-L.1 (Big-4 sample) and the v3.x §IV-I corpus-wide version (reported under prior 'FAR' terminology)". §IV-I in v4 (lines 157-161) is a 3-paragraph stub that mostly redirects to §IV-M Tables XXI-XXVI. It is no longer a "corpus-wide v3.x version" but a v4 reframing pointer.
+
+*Severity.* Cosmetic. The cross-reference still works for an informed reader.
+
+*Fix.* Update §III-K.4 line 149 to "§III-L.1 (Big-4 v4 sample) and the inherited corpus-wide v3.x version cited at §IV-I (reported under prior 'FAR' terminology)".
+
+## Provenance spot-checks (three fresh)
+
+I selected three claims not previously verified by codex r7/8 or my round-1.
+
+### S1. §IV-F line 112 per-signature K=3 C1 cosine drift = 0.018; C3 drift = 0.006. — **VERIFIED**
+
+Per-CPA fit C1 = 0.9457 (§III-J Table line 86 / §IV-E Table VIII line 71); per-signature fit C1 = 0.928 (manuscript at line 112).
+Manual: |0.9457 − 0.928| = 0.01770 ≈ 0.018 ✓.
+Per-CPA fit C3 = 0.9826; per-signature C3 = 0.989. |0.9826 − 0.989| = 0.00640 ≈ 0.006 ✓.
+
+### S2. §IV-G Table XIII C1 component shape stability (max deviations: cosine 0.005, dHash 0.96, weight 0.023). — **VERIFIED against upstream Script 37 report**
+
+Script 37 report at `/Volumes/NV2/PDF-Processing/signature-analysis/reports/v4_big4/k3_loo_check/k3_loo_report.md` (lines 30-35) gives:
+- Fold C1 cos means: [0.9425, 0.9441, 0.9504, 0.9439]
+- Baseline: 0.9457
+- Max |dev| vs baseline: 0.0047 — rounded to 0.005 in manuscript ✓ (slightly liberal rounding from 0.0047 → 0.005, conventional 1-sf rounding).
+- Max |dh dev|: 0.955 — manuscript reports 0.96 ✓.
+- Max |weight dev|: 0.023 ✓ exact.
+
+Held-out C1 rates: 4.68 / 7.14 / 36.27 / 17.31% — manuscript Table XIII (lines 134-137) values 4.68 / 7.14 / 36.27 / 17.31% match exactly ✓.
+
+### S3. §IV-M.4 Table XXIII D1 rate 0.1797, Wilson 95% CI [0.1770, 0.1825]. — **VERIFIED, with N1 caveat**
+
+Script 45 report at `/.../doc_level_far_full/doc_far_full_report.md`: "D1 | any sig HC | 0.1797 | 13,519 / 75,233". 13,519/75,233 = 0.17968 → rounds to 0.1797 ✓.
+
+Wilson 95% CI on $\\hat{p} = 0.1797$ at $n = 75{,}233$:
+- Half-width ≈ 1.96·√(0.18·0.82/75233) = 1.96·0.001405 = 0.002753
+- [0.17968 − 0.002753, 0.17968 + 0.002753] = [0.17693, 0.18243]
+- Manuscript reports [0.1770, 0.1825] ✓ (rounds to within reporting precision).
+
+The per-firm D2 rates (line 325) also verify exactly:
+- Firm A 18,743/30,226 = 0.6201 ✓ (Script 45 reports 0.6201)
+- Firm B 2,740/17,127 = 0.1600 ✓
+- Firm C ? / 19,501 = 0.1635 → 19,501 · 0.1635 = 3,189 ✓
+- Firm D 723/8,379 = 0.0863 ✓
+
+**N1 caveat:** the Firm C denominator 19,501 differs from §IV-J Table XIX's 19,122 by exactly 379 mixed-firm PDFs that Script 45 mode-of-firms assigns to Firm C (verified against Script 45 line 256). Not a bug; not disclosed in §IV-M.4.
+
+## Phase 5 splice readiness
+
+**Partial.** Empirical core is splice-ready: no script reruns required; M1–M4 closed; codex r8 splice blockers fixed in round-3; all spot-checked numbers reconcile against upstream report files.
+
+Outstanding for round-4 (the merge-to-master copy-edit pass):
+
+1. **[Substantive — recommended before splice]** Patch N1: add inline footnote to §IV-M.4 Table XXIII reconciling the 19,122 vs 19,501 Firm C denominator with the §IV-J Table XIX 379-mixed-PDF exclusion.
+
+2. **[Substantive — recommended before splice]** Patch N2: add the composition-decomposition row to the §III-M validation table and update "nine-tool" → "ten-tool" in §I item 8, §VI item 8, and §III-M framing. **OR** weaken §III-M to "primary-tool collection (the composition decomposition that anchors the framework is treated as foundational rather than as a validation tool)" — but this re-introduces the framing problem the round-3 reframe was supposed to solve.
+
+3. **[Structural — splice-time]** Patch N3: assign Table XXVII to the §III-M validation table.
+
+4. **[Editorial — splice-time]** Patch N4: rewrite the §III-M cross-firm hit matrix row's "None" assumption to disclose any-pair-vs-same-pair semantics + mode-of-firms tie-breaking.
+
+5. **[Editorial — optional]** Patch N5: add the firm-dependent within-firm violation nuance to §V-H limitation 2.
+
+6. **[Splice-strip — required]** Remove all internal draft notes + close-out checklists + §III cross-reference index + §III open-questions block (already on codex r8 + Gemini m1 list).
+
+7. **[Splice-time]** Verify v3.x §IV-F.1 letter resolves correctly in master manuscript.
+
+## Recommended next-step actions
+
+Numbered, prioritised, distinguishing empirical from copy-edit.
+
+**Empirical / substantive:**
+
+1. **N2 (§III-M omits composition decomposition).** Add the composition-decomposition row to the validation table or restate the framing. This affects the §I / §VI claims of "nine-tool" / "multi-tool unsupervised validation framework" — if the v4 pivot's foundational diagnostic isn't in the table, the framework label reads as incomplete. **Highest priority.**
+
+2. **N1 (denominator inconsistency).** Add the Firm C 19,122 vs 19,501 reconciliation footnote to §IV-M.4 Table XXIII. Half a sentence; high payoff for reviewer trust.
+
+3. **N4 (cross-firm hit matrix "None" assumption).** Replace with the actual mode-of-firms + any-pair-vs-same-pair assumption disclosure.
+
+**Copy-edit / structural:**
+
+4. **N3 (unnumbered Table).** Assign Table XXVII to §III-M validation table.
+
+5. **N5 (within-firm violation firm-dependence).** Add half-sentence to §V-H limitation 2.
+
+6. **N6 (§IV-I reduced to a stub).** Update §III-K.4 line 149 cross-reference wording.
+
+7. **Splice-strip pass.** Remove all internal draft notes + checklists per the pre-existing list (codex r8 + Gemini m1 + Opus r1 minor 2).
+
+8. **Spearman precision (Opus M7 from r1; codex r8 OPEN COPY-EDIT).** Standardise 4-dp across §III-K Table / §IV-F Table IX / §III provenance.
+
+9. **Decimal vs percentage notation.** Standardise the 0.34 / 33.75% / 0.3375 mix across Abstract / §I / §III-L / §IV-M.
+
+**Splice-time checks:**
+
+10. v3.x §IV-F.1 cross-reference letter resolution in master manuscript.
+11. v3.x Table XVIII (backbone ablation) vs v4 Table XVIII (Spearman drift) collision avoidance in the final manuscript table sequence.
+12. Confirm the upstream Script 37 LOOO report file's legacy "C1 hand-leaning / C2 mixed / C3 replicated" labels do not propagate to any supplementary material exported with the manuscript.

paper/paper_a_abstract_v3.md

@@ -2,6 +2,6 @@
 
 <!-- IEEE Access target: <= 250 words, single paragraph -->
 
-Regulations require Certified Public Accountants (CPAs) to attest to each audit report by affixing a signature, but digitization makes reusing a stored signature image across reports---through administrative stamping or firm-level electronic signing---technically trivial and visually invisible to report users, undermining individualized attestation. We build an end-to-end pipeline that detects such *non-hand-signed* signatures at scale: a Vision-Language Model identifies signature pages, a YOLOv11 detector localizes signature regions, ResNet-50 supplies deep features, and a dual-descriptor verification layer combines deep-feature cosine similarity with perceptual hashing (difference hash, dHash) to separate *style consistency* (high cosine, divergent dHash) from *image reproduction* (high cosine, low dHash). The operational classifier outputs a five-way verdict per signature with a worst-case document-level aggregation; the cosine cut is anchored on a transparent whole-sample Firm A P7.5 percentile (cos $> 0.95$), and the dHash cuts on the same reference. Applied to 90,282 audit reports filed in Taiwan over 2013-2023 (182,328 signatures from 758 CPAs), the operational dual rule cos $> 0.95$ AND $\text{dHash}_\text{indep} \leq 15$ captures 92.46\% of Firm A and yields FAR = 0.0005 against a $\sim$50,000-pair inter-CPA negative anchor; intra-report agreement is 89.9\% at Firm A versus 62-67\% at the other Big-4 firms (a 23-28 percentage-point cross-firm gap). Validation uses three annotation-free anchors (310 byte-identical positives, $\sim$50,000 inter-CPA negatives, and a 70/30 held-out Firm A fold) reported with Wilson 95\% intervals. Three statistical diagnostics applied to the per-signature similarity distribution (Hartigan dip test, EM-fitted Beta mixture with logit-Gaussian robustness check, Burgstahler-Dichev / McCrary density-smoothness procedure) jointly characterise the distribution as a continuous quality spectrum, which motivates the percentile-based anchor and is itself a substantive finding for similarity-threshold selection in document forensics.
+Regulations require Certified Public Accountants (CPAs) to attest each audit report with a signature, but digitization makes it feasible to reuse a stored signature image across reports — through administrative stamping or firm-level electronic signing — thereby undermining individualized attestation. We build an end-to-end pipeline for screening such *non-hand-signed* signatures at scale: a Vision-Language Model identifies signature pages, YOLOv11 localizes signatures, ResNet-50 supplies deep features, and a dual-descriptor layer combines cosine similarity with an independent-minimum perceptual hash (dHash) to separate *style consistency* from *image reproduction*. Applied to 90,282 Taiwan audit reports (2013–2023), the pipeline yields 182,328 signatures from 758 CPAs; primary analyses are scoped to the Big-4 sub-corpus (437 CPAs; 150,442 signatures). Distributional diagnostics show that the apparent multimodality of the descriptor distribution dissolves under joint firm-mean centring and integer-tie jitter ($p$ rises to $0.35$), so no within-population bimodal antimode anchors the operational thresholds. We instead adopt an anchor-based inter-CPA coincidence-rate (ICCR) calibration at three units: per-comparison ($0.0006$ at cos$>0.95$; $0.0013$ at dHash$\leq 5$; $0.00014$ jointly), pool-normalised per-signature ($0.11$ under the deployed any-pair high-confidence rule), and per-document ($0.34$ for the operational HC+MC alarm). The framework surfaces pronounced firm-level heterogeneity: Firm A's per-document HC+MC ICCR is $0.62$ versus $0.09$–$0.16$ at Firms B/C/D (gap persists after pool-size adjustment), and $77$–$99\%$ of inter-CPA collisions concentrate within the source firm. This contrast is consistent with firm-level template-like reuse but not independently diagnostic, since descriptor-only data cannot separate reuse from digitisation-pipeline or signing-style homogeneity within a firm; we report it as a scope limitation rather than a mechanism finding. We position the system as a specificity-proxy-anchored screening framework with human-in-the-loop review, not as a validated forensic detector; no calibrated error rates are reportable without signature-level ground truth.
 
-<!-- Target word count: 240 -->
+<!-- Word count: 281 -->

paper/paper_a_appendix_v3.md

@@ -1,9 +1,12 @@
-# Appendix A. BD/McCrary Bin-Width Sensitivity (Signature Level)
+# Appendix A. Supplementary Diagnostic Detail
 
-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.
+## 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 $\alpha = 0.05$, $|Z| > 1.96$).
 
-<!-- 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   |
@@ -20,45 +23,38 @@ This appendix documents the empirical basis for that framing by sweeping the bin
 | 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 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.
+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 per-signature similarity does not form a clean two-mechanism mixture.
+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.
 
-# Appendix B. Table-to-Script Provenance
+## A.2. Diagnostic Summary
 
-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.
+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 B.I: Manuscript table → reproduction artifact
-| Manuscript table | Generating script | Report artifact |
-|------------------|-------------------|-----------------|
-| Table III (extraction results) | `02_extract_features.py`; `09_pdf_signature_verdict.py` | `reports/extraction_methodology.md`; `reports/pdf_signature_verdicts.json` |
-| Table IV (intra/inter all-pairs cosine statistics) | `10_formal_statistical_analysis.py` | `reports/formal_statistical_data.json`; `reports/formal_statistical_report.md` |
-| Table V (Hartigan dip test) | `15_hartigan_dip_test.py` | `reports/dip_test/dip_test_results.json` |
-| Table VI (signature-level threshold-estimator summary) | `17_beta_mixture_em.py`; `25_bd_mccrary_sensitivity.py` | `reports/beta_mixture/beta_mixture_results.json`; `reports/bd_sensitivity/bd_sensitivity.json` |
-| Table IX (Firm A whole-sample capture rates) | `19_pixel_identity_validation.py`; `24_validation_recalibration.py` | `reports/pixel_validation/pixel_validation_results.json`; `reports/validation_recalibration/validation_recalibration.json` |
-| Table X (cosine threshold sweep, FAR vs inter-CPA negatives) | `21_expanded_validation.py` | `reports/expanded_validation/expanded_validation_results.json` |
-| Table XI (held-out vs calibration Firm A capture rates) | `24_validation_recalibration.py` | `reports/validation_recalibration/validation_recalibration.json` |
-| Table XII (operational-cut sensitivity 0.95 vs 0.945) | `24_validation_recalibration.py` | `reports/validation_recalibration/validation_recalibration.json` |
-| Table XII-B (cosine-threshold tradeoff: capture vs inter-CPA FAR) | `21_expanded_validation.py` (FAR column; canonical 50k-pair anchor); inline computation in revision (Firm A and non-Firm-A capture columns) | `reports/expanded_validation/expanded_validation_results.json` |
-| Table XIII (Firm A per-year cosine distribution) | `29_firm_a_yearly_distribution.py` | `reports/firm_a_yearly/firm_a_yearly_distribution.json` |
-| Fig. 4 (per-firm yearly best-match cosine, 2013-2023) | `30_yearly_big4_comparison.py` | `reports/figures/fig_yearly_big4_comparison.{png,pdf}`; `reports/firm_yearly_comparison/firm_yearly_comparison.{json,md}` |
-| Tables XIV / XV (partner-level similarity ranking) | `22_partner_ranking.py` | `reports/partner_ranking/partner_ranking_results.json` |
-| Table XVI (intra-report classification agreement) | `23_intra_report_consistency.py` | `reports/intra_report/intra_report_results.json` |
-| Table XVII (document-level five-way classification) | `09_pdf_signature_verdict.py`; `12_generate_pdf_level_report.py` | `reports/pdf_signature_verdicts.json`; `reports/pdf_signature_verdict_report.md` (CSV / XLSX bulk reports also at `reports/`) |
-| Table XVIII (backbone ablation) | `paper/ablation_backbone_comparison.py` | `ablation/ablation_results.json` (sibling of `reports/`) |
-| Table A.I (BD/McCrary bin-width sensitivity) | `25_bd_mccrary_sensitivity.py` | `reports/bd_sensitivity/bd_sensitivity.json` |
-| Byte-identity decomposition (145 / 50 / 180 / 35; Section IV-F.1) | `28_byte_identity_decomposition.py` | `reports/byte_identity_decomp/byte_identity_decomposition.json` |
-| Cross-firm dual-descriptor convergence (Section IV-H.2) | `28_byte_identity_decomposition.py` | `reports/byte_identity_decomp/byte_identity_decomposition.json` |
--->
+**Table A.II.** Diagnostics, failure mode addressed, and disclosed untested assumption.
 
-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.
+| 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.

paper/paper_a_conclusion_v3.md

@@ -1,30 +1,7 @@
 # VI. Conclusion and Future Work
 
-## Conclusion
+We present a fully automated pipeline for screening non-hand-signed CPA signatures in Taiwan-listed financial audit reports, together with an anchor-calibrated screening framework that characterises the pipeline's operational behaviour at the Big-4 sub-corpus scope under explicit unsupervised assumptions. The pipeline processes raw PDFs through VLM-based page identification, YOLO-based signature detection, ResNet-50 feature extraction, and dual-descriptor (cosine + independent-minimum dHash) similarity computation. The operational output is the deployed five-way per-signature classifier with worst-case document-level aggregation (§III-H.1; calibrated in §III-L). Applied to 90,282 audit reports filed between 2013 and 2023, the pipeline extracts 182,328 signatures from 758 CPAs, with the Big-4 sub-corpus (437 CPAs at accountant level; 150,442–150,453 signatures at signature level) as the primary analytical population.
 
-We have presented an end-to-end AI pipeline for detecting non-hand-signed auditor signatures in financial audit reports at scale.
-Applied to 90,282 audit reports from Taiwanese publicly listed companies spanning 2013--2023, our system extracted and analyzed 182,328 CPA signatures using a combination of VLM-based page identification, YOLO-based signature detection, deep feature extraction, and dual-descriptor similarity verification, with the operational classifier's cosine cut anchored on a whole-sample Firm A percentile heuristic and the per-signature similarity distribution characterised through two threshold estimators and a density-smoothness diagnostic.
+Our central methodological contributions are: (1) a composition decomposition that establishes the absence of a within-population bimodal antimode in the Big-4 descriptor distribution: the apparent multimodality dissolves under joint firm-mean centring and integer-tie jitter ($p_{\text{median}} = 0.35$), so distributional "natural-threshold" framings of the deployed operating points are not empirically supported; (2) an anchor-based inter-CPA coincidence-rate (ICCR) calibration at three units of analysis — per-comparison ($0.0006$ at cos$>0.95$; $0.0013$ at dHash$\leq 5$; $0.00014$ jointly), pool-normalised per-signature ($0.11$ for the deployed any-pair HC rule), and per-document ($0.34$ for the operational HC$+$MC alarm) — with explicit terminological replacement of "FAR" by "ICCR" given the unsupervised setting; (3) firm-level heterogeneity surfaced by the framework: logistic regression with pool-size adjustment gives odds ratios $0.053$, $0.010$, $0.027$ for Firms B/C/D relative to Firm A reference, a contrast that pool-size differences do not explain and that we report as a framework-discriminative observation rather than a mechanism finding (§V-H); (4) cross-firm hit matrix evidence that under the deployed any-pair rule, within-firm collision concentration is $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms), consistent with — but not independently establishing — firm-level template-like reuse, digitisation-pipeline homogeneity, or signing-style similarity, which descriptor-only data cannot separate (§V-H); (5) K=3 mixture demoted from "three mechanism clusters" to a descriptive firm-compositional partition; (6) three feature-derived scores converging on the per-CPA descriptor-position ranking at Spearman $\rho \geq 0.879$, reported as internal consistency rather than external validation; (7) $0\%$ positive-anchor miss rate on 262 byte-identical Big-4 signatures with the conservative-subset caveat; and (8) explicit disclosure of each diagnostic's untested assumption (Appendix A Table A.II), positioning the system as an anchor-calibrated screening framework with human-in-the-loop review rather than as a validated forensic detector.
 
-The seven numbered contributions listed in Section I can be grouped into four broader methodological themes, summarized below.
-
-First, we argued that non-hand-signing detection is a distinct problem from signature forgery detection, requiring analytical tools focused on the upper tail of intra-signer similarity rather than inter-signer discriminability.
-
-Second, we showed that combining cosine similarity of deep embeddings with difference hashing is essential for meaningful classification---among 71,656 documents with high feature-level similarity, the dual-descriptor framework revealed that only 41% exhibit converging structural evidence of non-hand-signing while 7% show no structural corroboration despite near-identical feature-level appearance, demonstrating that a single-descriptor approach conflates style consistency with image reproduction.
-
-Third, we characterised the per-signature similarity distribution using three diagnostics---a Hartigan dip test, an EM-fitted Beta mixture (with logit-Gaussian robustness check), and a Burgstahler-Dichev / McCrary density-smoothness procedure---and showed that no two-mechanism mixture cleanly explains it: the dip test fails to reject unimodality for Firm A ($p = 0.17$), BIC strongly prefers a 3-component over a 2-component Beta fit ($\Delta\text{BIC} = 381$ for Firm A), and the BD/McCrary candidate transition lies inside the non-hand-signed mode rather than between modes (and is not bin-width-stable; Appendix A).
-The substantive reading is that *pixel-level output quality* is a continuous spectrum produced by firm-specific reproduction technologies (administrative stamping in early years, firm-level e-signing later) and scan conditions, rather than a discrete class cleanly separated from hand-signing.
-This reading motivates anchoring the operational classifier's cosine cut on a whole-sample Firm A P7.5 percentile heuristic (cos $> 0.95$) rather than on a mixture-fit crossing.
-
-Fourth, we introduced a *replication-dominated* calibration methodology---explicitly distinguishing replication-dominated from replication-pure calibration anchors and validating classification against a byte-level pixel-identity anchor (310 byte-identical signatures) paired with a $\sim$50,000-pair inter-CPA negative anchor.
-To document the within-firm sampling variance of using the calibration firm as its own validation reference, we split the firm's CPAs 70/30 at the CPA level and report capture rates on both folds with Wilson 95% confidence intervals; extreme rules agree across folds while rules in the operational 85--95% capture band differ by 1--5 percentage points, reflecting within-firm heterogeneity in replication intensity rather than generalization failure.
-This framing is internally consistent with the available evidence: the byte-level pair analysis finding of 145 pixel-identical calibration-firm signatures across 50 distinct partners of 180 registered (Section IV-F.1); the 92.5% / 7.5% split in signature-level cosine thresholds and the dip-test-confirmed unimodal-long-tail shape of Firm A's per-signature cosine distribution (Section IV-D.1); and the 95.9% top-decile concentration of Firm A auditor-years in the threshold-independent partner-ranking analysis (Section IV-G.2).
-
-An ablation study comparing ResNet-50, VGG-16 and EfficientNet-B0 confirmed that ResNet-50 offers the best balance of discriminative power, classification stability, and computational efficiency for this task.
-
-## Future Work
-
-Several directions merit further investigation.
-Domain-adapted feature extractors, trained or fine-tuned on signature-specific datasets, may improve discriminative performance beyond the transferred ImageNet features used in this study.
-The pipeline's applicability to other jurisdictions and document types (e.g., corporate filings in other countries, legal documents, medical records) warrants exploration.
-The replication-dominated calibration strategy and the pixel-identity anchor technique are both generalizable to settings in which (i) a reference subpopulation has a known dominant mechanism and (ii) the target mechanism leaves a byte-level signature in the artifact itself, conditional on the availability of analogous anchors in the new domain and on artifact-generation physics that preserve the byte-level trace.
-Finally, integration with regulatory monitoring systems and a larger negative-anchor study---for example drawing from inter-CPA pairs under explicit accountant-level blocking---would strengthen the practical deployment potential of this approach.
+Future work falls in four directions. *First*, a small-scale human-rated labelled set would enable direct ROC optimisation and provide the signature-level ground truth that the present analysis fundamentally lacks; without such ground truth, no true error rates can be reported. *Second*, the within-firm collision concentration documented in §III-L.4 (any-pair $76.7$–$98.8\%$ across Big-4; same-pair joint $97.0$–$99.96\%$) invites a separate study to distinguish deliberate template sharing from passive firm-level production artefacts (shared scanners, common form templates, identical report-generation infrastructure) — a question the inter-CPA-anchor analysis alone cannot resolve. *Third*, the descriptive Firm A versus Firms B/C/D contrast (per-document HC$+$MC alarm $0.62$ vs $0.09$–$0.16$) — together with the byte-level evidence of 145 pixel-identical signatures across $\sim 50$ distinct Firm A partners — invites a companion analysis examining whether such firm-level signing patterns correlate with established audit-quality measures. *Fourth*, generalisation to mid- and small-firm contexts requires extending the anchor-based ICCR framework to scopes where firm-level LOOO folds are not available; the §III-I.4 composition diagnostics already document that the absence of within-population bimodality holds across the tested eligible scopes, so the calibration approach in principle generalises, but a full extension with cluster-robust uncertainty quantification is left as future work.

paper/paper_a_declarations_v3.md

@@ -4,4 +4,6 @@
 
 **Data availability.** All audit reports analysed in this study were obtained from the Market Observation Post System (MOPS) operated by the Taiwan Stock Exchange Corporation, a publicly accessible regulatory disclosure platform. The CPA registry used to map signatures to certifying CPAs is publicly available. Signature images, model weights, and reproducibility scripts are available in the supplementary materials.
 
-**Funding.** [To be filled in before submission.]
+<!-- Funding statement to be inserted before submission:
+**Funding.** [acknowledge any grants, awards, or institutional support here]
+-->

paper/paper_a_discussion_v3.md

@@ -2,108 +2,76 @@
 
 ## A. Non-Hand-Signing Detection as a Distinct Problem
 
-Our results highlight the importance of distinguishing *non-hand-signing detection* from the well-studied *signature forgery detection* problem.
-In forgery detection, the challenge lies in modeling the variability of skilled forgers who produce plausible imitations of a target signature.
-In non-hand-signing detection the signer's identity is not in question; the challenge is distinguishing between legitimate intra-signer consistency (a CPA who signs similarly each time) and image-level reproduction of a stored signature (a CPA whose signature on each report is a byte-level or near-byte-level copy of a single source image).
+Non-hand-signing differs from forgery in that the questioned signature is produced by its legitimate signer's own stored image rather than by an impostor. The detection problem is therefore framed around *intra-signer image reproduction* rather than *inter-signer imitation*. This framing has analytical consequences. The within-CPA signature distribution is the analytical population of interest; the cross-CPA inter-class distribution is a *reference* against which intra-CPA similarity is interpreted, not the population to be modelled. This contrasts with most prior offline signature verification work, which treats genuine-versus-forged as the central two-class problem.
 
-This distinction has direct methodological consequences.
-Forgery detection systems optimize for inter-class discriminability---maximizing the gap between genuine and forged signatures.
-Non-hand-signing detection, by contrast, requires sensitivity to the *upper tail* of the intra-class similarity distribution, where the boundary between consistent handwriting and image reproduction becomes ambiguous.
-The dual-descriptor framework we propose---combining semantic-level features (cosine similarity) with structural-level features (dHash)---addresses this ambiguity in a way that single-descriptor approaches cannot.
+## B. Per-Signature Similarity is a Continuous Quality Spectrum; the Accountant-Level Multimodality is Composition-Driven
 
-## B. Per-Signature Similarity is a Continuous Quality Spectrum
+The Big-4 accountant-level descriptor distribution rejects unimodality on both marginals at $p < 5 \times 10^{-4}$ (§IV-D Table V). The composition decomposition of §III-I.4 shows that this rejection is fully attributable to two non-mechanistic sources: (a) between-firm location-shift effects on both axes — Firm A's mean dHash of $2.73$ versus Firms B/C/D's $6.46$, $7.39$, $7.21$ creates a multi-peaked pooled distribution that any single firm's distribution lacks — and (b) integer mass-point artefacts on the integer-valued dHash axis, which inflate the dip statistic against a continuous-density null. A 2×2 factorial diagnostic applied to the Big-4 pooled dHash (firm-mean centring × uniform integer jitter $[-0.5, +0.5]$, 5 jitter seeds) shows that the dip test fails to reject ($p_{\text{median}} = 0.35$, 0/5 seeds reject) when *both* corrections are applied; either correction alone leaves the rejection in place. Within the Big-4 firms, the descriptor marginals at the signature level are unimodal once integer ties are broken (Scripts 39b, 39d); eligible non-Big-4 firms provide corroborating raw-axis evidence on the cosine dimension (Script 39c) but are not used as calibration evidence (§III-I.4). The descriptor distributions therefore lack a within-population bimodal antimode that could anchor an operational threshold. The K=2 / K=3 mixture fits are retained in §III-J as descriptive partitions of the joint Big-4 distribution that reflect firm-compositional structure, not as inferential evidence for two or three latent mechanism modes.
 
-A central empirical finding of this study is that per-signature similarity does not form a clean two-mechanism mixture (Section IV-D).
-Firm A's signature-level cosine is formally unimodal (Hartigan dip test $p = 0.17$) with a long left tail.
-The all-CPA signature-level cosine rejects unimodality ($p < 0.001$), reflecting the heterogeneity of signing practices across firms, but its structure is not well approximated by a two-component Beta mixture: BIC clearly prefers a three-component fit ($\Delta\text{BIC} = 381$ for Firm A; $10{,}175$ for the full sample), and the forced 2-component Beta crossing and its logit-GMM robustness counterpart disagree sharply on the candidate threshold (0.977 vs. 0.999 for Firm A).
-The BD/McCrary discontinuity test locates its transition at cosine 0.985---*inside* the non-hand-signed mode rather than at a boundary between two mechanisms---and the transition is not bin-width-stable (Appendix A).
+## C. Firm A as the Templated End of Big-4 (Case Study, Not Calibration Anchor)
 
-Taken together, these results indicate that non-hand-signed signatures form a continuous quality spectrum rather than a discrete class cleanly separated from hand-signing.
-Replication quality varies continuously with scan equipment, PDF compression, stamp pressure, and firm-level e-signing system generation, producing a heavy-tailed distribution that no two-mechanism mixture explains at the signature level.
+Firm A is empirically the firm whose CPAs are most concentrated in the high-cosine, low-dHash corner of the Big-4 descriptor plane. In the Big-4 K=3 hard-posterior assignment (now interpreted as a firm-compositional position assignment; §III-J), Firm A accounts for $0\%$ of C1 (low-cos / high-dHash position) and $82.5\%$ of C3 (high-cos / low-dHash position); the opposite pattern holds at Firm C, which has the highest C1 concentration at $23.5\%$. Firm A also accounts for 145 of the 262 byte-identical signatures in the Big-4 byte-identical anchor of §IV-H (with Firm B 8, Firm C 107, Firm D 2). Byte-level decomposition of the 145 Firm A pixel-identical signatures (see supplementary materials) shows they span 50 distinct Firm A partners (of 180 registered), with 35 byte-identical matches occurring across different fiscal years.
 
-The methodological implication is that the operational classifier's cosine cut should not be derived from a mixture-fit crossing.
-We accordingly anchor the operational cosine cut on the whole-sample Firm A P7.5 percentile (Section III-K), and treat the signature-level threshold-estimator outputs (KDE antimode, Beta and logit-Gaussian crossings) as descriptive characterisation of the similarity distribution rather than as the source of operational thresholds.
-The BD/McCrary procedure plays a *density-smoothness diagnostic* role in this framing rather than that of an independent threshold estimator.
+We treat Firm A as a *templated-end case study within the Big-4 sub-corpus* rather than as the calibration anchor for the operational threshold. Firm A enters the Big-4 anchor-based ICCR calibration on equal footing with the other three Big-4 firms (§III-L). The cross-firm hit matrix of §III-L.4 strengthens this framing: under the deployed any-pair rule, within-firm collision concentration is $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms). Firm A's per-document D2 inter-CPA proxy ICCR of $0.6201$ (versus Firms B/C/D's $0.09$–$0.16$) — the counterfactual rate at which Firm A documents would fire HC$+$MC if same-CPA pools were replaced by random inter-CPA candidates — reflects high inter-CPA collision concentration under the deployed rule, consistent with firm-specific template, stamp, or document-production reuse. (The corresponding observed rate on real same-CPA pools, from Table XVI, is substantially higher: $97.5\%$ HC$+$MC for Firm A; the proxy and observed rates measure different quantities and are not directly comparable.) The inter-CPA-anchor analysis alone is not diagnostic of deliberate template sharing. The byte-level evidence above (Firm A's 145 pixel-identical signatures across $\sim 50$ distinct partners) provides direct evidence of image-level reuse among Firm A signatures; the distribution across many partners is consistent with a firm-level template or production workflow, and the within-firm collision pattern at all four Big-4 firms is consistent with similar, milder within-firm collision patterns at Firms B/C/D, whose mechanisms may include template-like reuse, digitisation-pipeline homogeneity, or signing-style homogeneity (§V-H).
 
-This continuous-spectrum finding also has substantive implications for downstream interpretation.
-Because pixel-level output quality varies continuously, *signature-level rates* (such as the 92.5% / 7.5% Firm A split) reflect the share of signatures whose similarity falls above or below a chosen threshold rather than the share that came from a "non-hand-signing mechanism" versus a "hand-signing mechanism."
-We accordingly report all rates as signature-level quantities and abstain from partner-level frequency claims (Section III-G).
+## D. K=2 / K=3 as Descriptive Firm-Compositional Partitions
 
-## C. Firm A as a Replication-Dominated, Not Pure, Population
+Leave-one-firm-out cross-validation of the Big-4 mixture fit reveals a sharp contrast between K=2 and K=3 behaviour. K=2 is unstable: across-fold cosine-crossing deviation is $0.028$, and holding Firm A out gives a fold rule (cos $> 0.938$, dHash $\leq 8.79$) that classifies $100\%$ of held-out Firm A in the upper component, while holding any non-Firm-A Big-4 firm out gives a fold rule near (cos $> 0.975$, dHash $\leq 3.76$) that classifies $0\%$ of the held-out firm in the upper component. The K=2 boundary is essentially a Firm-A-vs-others separator — direct evidence that the K=2 partition reflects firm-compositional rather than mechanistic structure.
 
-A recurring theme in prior work that treats Firm A or an analogous reference group as a calibration anchor is the implicit assumption that the anchor is a pure positive class.
-Our evidence across multiple analyses rules out that assumption for Firm A while affirming its utility as a calibration reference.
+K=3 in contrast has a *reproducible component shape* at the descriptor-position level: across the four folds the C1 (low-cos / high-dHash) component cosine mean varies by at most $0.005$, the dHash mean by at most $0.96$, and the weight by at most $0.023$. Hard-posterior membership for the held-out firm is composition-sensitive (absolute differences $1.8$–$12.8$ pp across folds). Together with the §III-I.4 composition decomposition (no within-population bimodal antimode), the K=3 stability supports a descriptive reading: the Big-4 descriptor plane has a reproducible three-region partition that reflects how firm-compositional weight is distributed across the descriptor space, *not* a three-mechanism latent-class structure. We accordingly do not use K=3 hard-posterior membership as an operational classifier; we use it as the accountant-level descriptive summary that complements the deployed signature-level five-way classifier of §III-H.1.
 
-Two convergent strands of evidence support the replication-dominated framing.
-First, the byte-level pair evidence: 145 Firm A signatures (from 50 distinct partners of 180 registered) have a byte-identical same-CPA match in a different audit report, with 35 of these matches spanning different fiscal years.
-Independent hand-signing cannot produce byte-identical images across distinct reports, so these pairs directly establish image reuse within Firm A as a concrete, threshold-free phenomenon, and the 50/180 partner spread shows that replication is widespread rather than confined to a handful of CPAs.
-Second, the signature-level distributional evidence: Firm A's per-signature cosine distribution is unimodal long-tail (Hartigan dip test $p = 0.17$) rather than a tight single peak; 92.5% of Firm A signatures exceed cosine 0.95, with the remaining 7.5% forming the left tail.
-The unimodal-long-tail *shape*, not the precise 92.5 / 7.5 split, is the structural evidence: it is consistent with a dominant high-similarity regime plus residual within-firm heterogeneity, and a noise-only explanation of the left tail would predict a shrinking share as scan/PDF technology matured over 2013--2023, which is not what we observe (Section IV-G.1).
+## E. Three-Score Convergent Internal-Consistency
 
-Two additional checks, reported in Section IV-G, are robust to threshold choice and complement the two primary strands:
-the held-out Firm A 70/30 validation (Section IV-F.2) gives capture rates on a non-calibration Firm A subset that sit in the same replication-dominated regime as the calibration fold across the full range of operating rules (extreme rules are statistically indistinguishable; operational rules in the 85--95% band differ between folds by 1--5 percentage points, reflecting within-Firm-A heterogeneity in replication intensity rather than a generalization failure), and the threshold-independent partner-ranking analysis (Section IV-G.2) shows that Firm A auditor-years occupy 95.9% of the top decile of similarity-ranked auditor-years against a 27.8% baseline share---a 3.5$\times$ concentration ratio that uses only ordinal ranking and is independent of any absolute cutoff.
+Three feature-derived scores agree on the per-CPA descriptor-position ranking at Spearman $\rho \geq 0.879$: the K=3 mixture posterior (a firm-compositional position score, not a mechanism cluster posterior); the reverse-anchor cosine percentile under a non-Big-4 reference distribution; and the deployed box-rule less-replication-dominated rate. The three scores are *not* statistically independent measurements — they are deterministic functions of the same per-CPA descriptor pair — so the convergence is documented as internal consistency rather than external validation against an independent ground truth (which the corpus does not provide for the hand-signed class). The strength of the convergence (all pairwise $|\rho| > 0.87$) and its persistence at the signature level (Cohen $\kappa = 0.87$ between per-CPA-fit and per-signature-fit K=3 binary labels) are nevertheless informative: per-CPA aggregation does not collapse the broad three-region ordering, and three different summarisations of the descriptor space produce broadly concordant per-CPA rankings, with a residual non-Firm-A disagreement (the reverse-anchor cosine percentile ranks Firm D fractionally above Firm C, while the mixture posterior and the deployed box-rule rate rank Firm C highest among non-Firm-A firms).
 
-The replication-dominated framing is internally coherent with both pieces of evidence, and it predicts and explains the residuals that a "near-universal" framing would be forced to treat as noise.
-We therefore recommend that future work building on this calibration strategy should explicitly distinguish replication-dominated from replication-pure calibration anchors.
+## F. Anchor-Based Multi-Level Calibration
 
-## D. The Style-Replication Gap
+The operational specificity-proxy behaviour of the deployed HC sub-rule and document-level HC$+$MC alarm derived from the five-way classifier is characterised at three units of analysis (§III-L), all against the same inter-CPA negative-anchor coincidence-rate proxy. The per-comparison ICCR is consistent with the corpus-wide rate reported in §IV-I (cos$>0.95 \to 0.00060$) and extends it to the structural dimension (dHash$\leq 5 \to 0.00129$; joint $\to 0.00014$). The pool-normalised per-signature ICCR captures the deployed rule's effective per-signature rate under inter-CPA candidate-pool replacement ($0.1102$ pooled Big-4 any-pair HC), exposing that the per-comparison rate is not the deployed-rule rate at the per-signature classifier level: the deployed classifier takes max-cosine and min-dHash over a same-CPA pool of size $n_{\text{pool}}$, so the inter-CPA-equivalent rate scales approximately as $1 - (1 - p_{\text{pair}})^{n_{\text{pool}}}$ in the independence limit. The per-document ICCR aggregates to operational alarm-rate units: HC alone $0.18$, the operational HC$+$MC alarm $0.34$.
 
-Within the 71,656 documents exceeding cosine $0.95$, the dHash descriptor partitions them into three distinct populations: 29,529 (41.2%) with high-confidence structural evidence of non-hand-signing, 36,994 (51.7%) with moderate structural similarity, and 5,133 (7.2%) with no structural corroboration despite near-identical feature-level appearance.
-A cosine-only classifier would treat all 71,656 documents identically; the dual-descriptor framework separates them into populations with fundamentally different interpretations.
+Two additional findings refine the calibration story. First, the per-pair conditional ICCR for dHash$\leq 5$ given cos$>0.95$ is $0.234$ (Wilson 95% $[0.190, 0.285]$): given the cosine gate, the structural dimension provides further per-comparison specificity at $\sim 4.3\times$ refinement. Second, the alert-rate sensitivity analysis (§III-L.5) shows the deployed HC threshold is locally sensitive rather than plateau-stable (local gradient $\approx 25\times$ the median for cosine, $\approx 3.8\times$ for dHash); alternative operating points can be characterised by inverting the ICCR curves (e.g., a tighter rule cos$>0.95$ AND dHash$\leq 3$ on the same-pair joint corresponds to per-signature ICCR $\approx 0.045$). The MC/HSC sub-band boundary at dHash$=15$, by contrast, *is* plateau-like (local-to-median ratio $\approx 0.08$), consistent with high-dHash-tail saturation.
 
-The 7.2% classified as "high style consistency" (cosine $> 0.95$ but dHash $> 15$) are particularly informative.
-Several plausible explanations may account for their high feature similarity without structural identity, though we lack direct evidence to confirm their relative contributions.
-Many accountants may develop highly consistent signing habits---using similar pen pressure, stroke order, and spatial layout---resulting in signatures that appear nearly identical at the semantic feature level while retaining the microscopic variations inherent to handwriting.
-Some may use signing pads or templates that further constrain variability without constituting image-level reproduction.
-The dual-descriptor framework correctly identifies these cases as distinct from non-hand-signed signatures by detecting the absence of structural-level convergence.
+## G. Pixel-Identity Positive Anchor and Inter-CPA Coincidence-Rate Negative Anchor
 
-## E. Value of a Replication-Dominated Calibration Group
+The only conservative hard-positive subset in the corpus is pixel-identical signatures: those whose nearest same-CPA match is byte-identical after crop and normalisation. Independent hand-signing cannot produce byte-identical images, so these signatures are a conservative hard-positive subset for image replication. On the Big-4 subset ($n = 262$ pixel-identical signatures), all three candidate checks — the deployed box rule, the K=3 hard label, and the reverse-anchor metric with a prevalence-calibrated cut — achieve $0\%$ positive-anchor miss rate (Wilson 95% upper bound $1.45\%$). We caution that this result is necessary but not sufficient: for the deployed box rule it is close to tautological, because byte-identical neighbours have cosine $\approx 1$ and dHash $\approx 0$, well inside the rule's high-confidence region. The corresponding signature-level *negative* anchor evidence is developed in §III-L.1 above (per-comparison ICCR $= 0.00060$ at cos$>0.95$, consistent with the corpus-wide rate of $0.0005$ reported in §IV-I). We frame the per-comparison rate as a specificity proxy under the assumption that inter-CPA pairs constitute a clean negative anchor, and we document in §III-L.4 that this assumption is partially violated by within-firm cross-CPA template-like collision structures.
 
-The use of Firm A as a calibration reference addresses a fundamental challenge in document forensics: the scarcity of ground truth labels.
-In most forensic applications, establishing ground truth requires expensive manual verification or access to privileged information about document provenance.
-Our approach uses practitioner background---one Big-4 firm reportedly relies predominantly on stamping or e-signing workflows---only as a *motivation* for selecting that firm as a candidate reference population; the calibration role is then established from the audit-report images themselves (byte-identical same-CPA pairs, the Firm A per-signature similarity distribution, partner-ranking concentration, and intra-report consistency), so the calibration does not depend on the practitioner-background claim being externally verified (Section III-H).
+## H. Limitations
 
-This calibration strategy has broader applicability beyond signature analysis.
-Any forensic detection system operating on real-world corpora can benefit from identifying subpopulations with known dominant characteristics (positive or negative) to anchor threshold selection, particularly when the distributions of interest are non-normal and non-parametric or mixture-based thresholds are preferred over parametric alternatives.
-The framing we adopt---replication-dominated rather than replication-pure---is an important refinement of this strategy: it prevents overclaim, accommodates the within-firm heterogeneity visible in the unimodal-long-tail shape of Firm A's per-signature cosine distribution, and yields classification rates that are internally consistent with the data.
+Several limitations should be transparent. We group them into primary methodological limitations, secondary scope and validation caveats, documented design features, and engineering-level caveats of the pipeline.
 
-## F. Pixel-Identity and Inter-CPA Anchors as Annotation-Free Validation
+**Primary methodological limitations.**
 
-A further methodological contribution is the combination of byte-level pixel identity as an annotation-free *conservative* gold positive and a large random-inter-CPA negative anchor.
-Handwriting physics makes byte-identity impossible under independent signing events, so any pair of same-CPA signatures that are byte-identical after crop and normalization is pair-level proof of image reuse and, modulo the narrow source-template edge case discussed in the seventh limitation below, a conservative positive for non-hand-signing without requiring human review.
-In our corpus 310 signatures satisfied this condition.
-We emphasize that byte-identical pairs are a *subset* of the true non-hand-signed positive class---they capture only those whose nearest same-CPA match happens to be bytewise identical, excluding replications that are pixel-near-identical but not byte-identical (for example, under different scan or compression pathways).
-Perfect recall against this subset therefore does not generalize to perfect recall against the full non-hand-signed population; it is a lower-bound calibration check on the classifier's ability to catch the clearest positives rather than a generalizable recall estimate.
+*No signature-level ground truth; no true error rates reportable.* The corpus does not contain labelled hand-signed or replicated classes at the signature level. We therefore cannot report False Rejection Rate, sensitivity, recall, Equal Error Rate, ROC-AUC, precision, or positive predictive value against ground truth. All quantitative rates reported in §III-L are inter-CPA negative-anchor coincidence rates (ICCRs) under the assumption that inter-CPA pairs constitute a clean negative anchor; this is a specificity proxy, not a calibrated specificity (§III-M).
 
-Paired with the $\sim$50,000-pair inter-CPA negative anchor, the byte-identical positives yield FAR estimates with tight Wilson 95% confidence intervals (Table X), which is a substantive improvement over the low-similarity same-CPA negative ($n = 35$) we originally considered.
-The combination is a reusable pattern for other document-forensics settings in which the target mechanism leaves a byte-level physical signature in the artifact itself, provided that its generalization limits are acknowledged: FAR is informative, whereas recall is valid only for the conservative subset.
+*Inter-CPA negative-anchor assumption is partially violated and the violation is firm-dependent.* The cross-firm hit matrix of §III-L.4 shows that under the deployed any-pair rule, within-firm collision concentration is $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms), consistent with firm-specific template, stamp, or document-production reuse. The inter-CPA-as-negative assumption is therefore not exactly satisfied — some inter-CPA pairs may share firm-level templates rather than being independent random matches. Our reported per-comparison ICCRs are best read as specificity-proxy rates under a partially-violated assumption, not as calibrated FARs. Because the violation is firm-dependent, Firm A's per-firm ICCR is more contaminated by within-firm sharing than Firms B/C/D's; the per-firm B/C/D rates of $0.09$–$0.16$ may therefore be less contaminated than the pooled rate, and the Firm A vs Firms B/C/D contrast reflects both genuine firm heterogeneity and a firm-dependent proxy-contamination gradient.
 
-## G. Limitations
+*Mechanism attribution for the firm-level heterogeneity is not identifiable from descriptor-only data.* The observed firm-level contrast (Firm A's per-document HC$+$MC ICCR of $0.62$ versus $0.09$–$0.16$ at Firms B/C/D; within-firm collision concentration $77$–$99\%$ under the deployed any-pair rule; byte-identical evidence of §IV-H) is consistent with at least three non-mutually-exclusive firm-level mechanisms: (i) template, stamp, or e-signature production reuse; (ii) digitisation-pipeline homogeneity — shared scanners, common PDF generation infrastructure, identical compression and form-template settings — that systematically inflates image-descriptor similarity without signature replication; and (iii) signing-style or training homogeneity that produces correlated handwritten signatures within a firm. The descriptor pair (cosine, dHash) operates at the image-similarity level and is, by construction, indifferent to which mechanism generated a given near-identical pair. We therefore report the firm contrast as a methodological observation — the framework discriminates at firm-level resolution — rather than as a mechanism finding. The byte-identical Firm A signatures across $\sim 50$ distinct partners (§IV-H, §V-C) provide direct evidence for (i) at Firm A specifically, but do not exclude additive contribution from (ii) or (iii); the milder within-firm collision patterns at Firms B/C/D are individually consistent with all three mechanisms. Image-acquisition metadata (scanner identifiers, PDF generator fingerprints, compression-codec markers), partner-level intent records, or controlled hand-signed baselines would be needed to attribute the contrast across (i), (ii), and (iii).
 
-Several limitations should be acknowledged.
+*Scope.* The primary analyses are scoped to the Big-4 sub-corpus. We did not perform the full per-signature pool-normalised ICCR analysis at the full $n = 686$ scope; the §IV-K full-dataset Spearman re-run shows the K=3 $+$ deployed box-rule rank-convergence is preserved at $n = 686$ but does not establish portability of the Big-4 operational ICCRs, the LOOO firm-fold structure, or the five-way operational classifier at the broader scope.
 
-First, comprehensive per-document ground truth labels are not available.
-The pixel-identity anchor is a strict *subset* of the true non-hand-signing positives (only those whose nearest same-CPA match happens to be byte-identical), so perfect recall against this anchor does not establish the classifier's recall on the broader positive class.
-The low-similarity same-CPA anchor ($n = 35$) is small because intra-CPA pairs rarely fall below cosine 0.70; we use the $\sim$50,000-pair inter-CPA negative anchor as the primary negative reference, which yields tight Wilson 95% CIs on FAR (Table X), but it too does not exhaust the set of true negatives (in particular, same-CPA hand-signed pairs with moderate cosine similarity are not sampled).
-A manual-adjudication study concentrated at the decision boundary---for example 100--300 auditor-years stratified by cosine band---would further strengthen the recall estimate against the full positive class.
+**Secondary scope and validation caveats.**
 
-Second, the ResNet-50 feature extractor was used with pre-trained ImageNet weights without domain-specific fine-tuning.
-While our ablation study and prior literature [20]--[22] support the effectiveness of transferred ImageNet features for signature comparison, a signature-specific feature extractor could improve discriminative performance.
+*Pixel-identity is a conservative subset.* Byte-identical pairs are the easiest replicated cases, and for the deployed box rule the positive-anchor miss rate against byte-identical pairs is close to tautological (byte-identical $\Rightarrow$ cosine $\approx 1$, dHash $\approx 0$, well inside the high-confidence box). A score that fails the pixel-identity check would be disqualified, but passing the check does not guarantee correct behaviour on the broader replicated population (e.g., re-stamped or noisy-template-variant signatures).
 
-Third, the red stamp removal preprocessing uses simple HSV color-space filtering, which may introduce artifacts where handwritten strokes overlap with red seal impressions.
-In these overlap regions, blended pixels are replaced with white, potentially creating small gaps in the signature strokes that could reduce dHash similarity.
-This effect would bias classification toward false negatives rather than false positives, but the magnitude has not been quantified.
+*Rule components not separately re-characterised by the present diagnostic battery.* The five-way classifier's moderate-confidence band (cos $> 0.95$ AND $5 < \text{dHash} \leq 15$), the style-consistency band ($\text{dHash} > 15$), and the document-level worst-case aggregation rule retain their prior calibration and capture-rate evidence (supplementary materials); the anchor-based ICCR calibration covers the binary high-confidence sub-rule (and its tightening alternatives such as dHash$\leq 3$), and the alert-rate sensitivity analysis (§III-L.5) characterises only the HC threshold. The MC and HSC sub-band boundaries are not separately re-characterised by the present diagnostic battery.
 
-Fourth, scanning equipment, PDF generation software, and compression algorithms may have changed over the 10-year study period (2013--2023), potentially affecting similarity measurements.
-While cosine similarity and dHash are designed to be robust to such variations, longitudinal confounds cannot be entirely excluded.
+*Deployed-rate excess is not a presumed true-positive rate.* The $\sim 44$-pp per-document gap between the observed deployed alert rate (HC: $0.62$ on real same-CPA pools) and the inter-CPA proxy rate (HC: $0.18$) cannot be interpreted as a presumed true-positive rate without additional assumptions that §III-M shows are unsafe (consistent within-CPA signing can exceed inter-CPA similarity at the cosine axis; within-firm template sharing inflates the inter-CPA proxy baseline). The gap is best read as a same-CPA repeatability signal.
 
-Fifth, the max/min detection logic treats both ends of a near-identical same-CPA pair as non-hand-signed.
-In the rare case that one of the two documents contains a genuinely hand-signed exemplar that was subsequently reused as the stamping or e-signature template, the pair correctly identifies image reuse but misattributes the non-hand-signed status to the source exemplar.
-This misattribution affects at most one source document per template variant per CPA (the exemplar from which the template was produced), is not expected to be common given that stored signature templates are typically generated in a separate acquisition step rather than extracted from submitted audit reports, and does not materially affect aggregate capture rates at the firm level.
+*A1 pair-detectability stipulation.* The per-signature detector requires at least one same-CPA pair to be near-identical when a CPA uses image replication. A1 is plausible for high-volume stamping or firm-level electronic signing but not guaranteed when a corpus contains only one observed replicated report for a CPA, multiple template variants used in parallel, or scan-stage noise that pushes a replicated pair outside the detection regime.
 
-Sixth, our analyses remain at the signature level; we abstain from partner-level frequency inferences such as "X% of CPAs hand-sign in a given year."
-Per-signature labels in this paper are not translated to per-report or per-partner mechanism assignments (Section III-G).
-The signature-level rates we report, including the 92.5% / 7.5% Firm A split and the year-by-year left-tail share of Section IV-G.1, should accordingly be read as signature-level quantities rather than partner-level frequencies.
+**Documented design features.**
 
-Finally, the legal and regulatory implications of our findings depend on jurisdictional definitions of "signature" and "signing."
-Whether non-hand-signing of a CPA's own stored signature constitutes a violation of signing requirements is a legal question that our technical analysis can inform but cannot resolve.
+*K=3 hard-posterior membership is composition-sensitive.* The K=3 hard-posterior membership for any single firm varies by up to $12.8$ pp across LOOO folds. This is documented as a composition-sensitivity band rather than failure, but it means K=3 hard labels are not used as operational classifier output; they are reported only as accountant-level descriptive characterisation.
+
+*No partner-level mechanism attribution.* The analysis reports population-level patterns; it does not perform partner-level mechanism attribution or report-level claims of intent. The signature-level outputs are signature-level quantities throughout. The within-firm cross-CPA collision concentration of §III-L.4 is consistent with template-like reuse but is not by itself diagnostic of deliberate sharing.
+
+**Engineering-level caveats of the pipeline.**
+
+*Transferred ImageNet features.* The ResNet-50 feature extractor uses pre-trained ImageNet weights without signature-domain fine-tuning. While our backbone-ablation study (§IV-L) and prior literature support the effectiveness of transferred ImageNet features for signature comparison, a signature-domain fine-tuned feature extractor could improve discriminative performance.
+
+*Red-stamp HSV preprocessing artifacts.* The red stamp removal preprocessing uses simple HSV color-space filtering, which may introduce artifacts where handwritten strokes overlap with red seal impressions. Blended pixels are replaced with white, potentially creating small gaps in signature strokes that could reduce dHash similarity. This bias would push classifications toward false negatives rather than false positives.
+
+*Longitudinal scan / PDF / compression confounds.* Scanning equipment, PDF generation software, and compression algorithms may have changed over the 2013–2023 study period, potentially affecting similarity measurements. While cosine similarity and dHash are designed to be robust to such variations, longitudinal confounds cannot be entirely excluded.
+
+*Source-exemplar misattribution in max/min pair logic.* The max-cosine / min-dHash detection logic treats both ends of a near-identical same-CPA pair as non-hand-signed. In the rare case where one of the two documents contains a genuinely hand-signed exemplar that was subsequently reused as a stamping or e-signature template, the pair correctly identifies image reuse but misattributes non-hand-signed status to the source exemplar. This affects at most one source document per template variant per CPA and is not expected to be common.
+
+*Legal and regulatory interpretation.* Whether non-hand-signing of a CPA's own stored signature constitutes a violation of signing requirements is a jurisdiction-specific legal question. Our technical analysis can inform such determinations but cannot resolve them.

paper/paper_a_impact_statement_v3.md

@@ -1,21 +1,17 @@
 <!--
-ARCHIVED. Not part of the IEEE Access submission.
-
-IEEE Access Regular Papers do not include a separate Impact Statement
-section. The text below is retained for possible reuse in a cover
-letter, grant report, or non-IEEE venue. It is excluded from the
-assembled paper by export_v3.py.
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-If reused, note that the wording "distinguishes genuinely hand-signed
+ARCHIVED. Not part of the IEEE Access submission. The block below is wrapped
+in an HTML comment so it does not render in the assembled paper. It is
+retained for possible reuse in a cover letter, grant report, or non-IEEE
+venue. If reused, note that the wording "distinguishes genuinely hand-signed
 signatures from reproduced ones" overstates what a five-way confidence
 classifier without a fully labeled test set establishes; soften before
 external use.
--->
 
 # Impact Statement (archived; not in IEEE Access submission)
 
 Auditor signatures on financial reports are a key safeguard of corporate accountability.
 When the signature on an audit report is produced by reproducing a stored image instead of by the partner's own hand---whether through an administrative stamping workflow or a firm-level electronic signing system---this safeguard is weakened, yet detecting the practice through manual inspection is infeasible at the scale of modern financial markets.
 We developed a pipeline that automatically extracts and analyzes signatures from over 90,000 audit reports spanning a decade of filings by publicly listed companies in Taiwan.
-Combining deep-learning visual features with perceptual hashing and two methodologically distinct threshold estimators (plus a density-smoothness diagnostic), the system stratifies signatures into a five-way confidence-graded classification and quantifies how the practice varies across firms and over time.
-After further validation, the technology could support financial regulators in screening signature authenticity at national scale.
+Combining deep-learning visual features with perceptual hashing, distributional diagnostics, and anchor-based inter-CPA coincidence-rate calibration, the system stratifies signatures into a five-way confidence-graded classification and quantifies how the practice varies across firms and over time.
+With a future labelled evaluation set, the technology could support financial regulators in screening candidate non-hand-signed signatures at national scale.
+-->

paper/paper_a_introduction_v3.md

@@ -2,85 +2,44 @@
 
 <!-- Target: ~1.5 pages double-column IEEE format. Double-blind: no author/institution info. -->
 
-Financial audit reports serve as a critical mechanism for ensuring corporate accountability and investor protection.
-In Taiwan, the Certified Public Accountant Act (會計師法 §4) and the Financial Supervisory Commission's attestation regulations (查核簽證核准準則 §6) require that certifying CPAs affix their signature or seal (簽名或蓋章) to each audit report [1].
-While the law permits either a handwritten signature or a seal, the CPA's attestation on each report is intended to represent a deliberate, individual act of professional endorsement for that specific audit engagement [2].
+Financial audit reports serve as a critical mechanism for ensuring corporate accountability and investor protection. In Taiwan, the Certified Public Accountant Act (會計師法 §4) and the Financial Supervisory Commission's attestation regulations (查核簽證核准準則 §6) require certifying CPAs to affix their signature or seal (簽名或蓋章) to each audit report [1]. While the law permits either a handwritten signature or a seal, the CPA's attestation on each report is intended to represent a deliberate, individual act of professional endorsement for that specific audit engagement [2].
 
-The digitization of financial reporting has introduced a practice that complicates this intent.
-As audit reports are now routinely generated, transmitted, and archived as PDF documents, it is technically and operationally straightforward to reproduce a CPA's stored signature image across many reports rather than re-executing the signing act for each one.
-This reproduction can occur either through an administrative stamping workflow---in which scanned signature images are affixed by staff as part of the report-assembly process---or through a firm-level electronic signing system that automates the same step.
-From the perspective of the output image the two workflows are equivalent: both can reproduce one or more stored signature images, producing same-CPA signatures that are identical or near-identical up to reproduction, scanning, compression, and template-variant noise.
-We refer to signatures produced by either workflow collectively as *non-hand-signed*.
-Although this practice may fall within the literal statutory requirement of "signature or seal," it raises substantive concerns about audit quality, as an identically reproduced signature applied across hundreds of reports may not represent meaningful individual attestation for each engagement.
-The accounting literature has long examined the audit-quality consequences of partner-level engagement transparency: studies of partner-signature mandates in the United Kingdom find measurable downstream effects [31], cross-jurisdictional evidence on individual partner signature requirements highlights similar quality channels [32], and Taiwan-specific evidence on mandatory partner rotation documents how individual-partner identification interacts with audit-quality outcomes [33].
-Unlike traditional signature forgery, where a third party attempts to imitate another person's handwriting, non-hand-signing involves the legitimate signer's own stored signature being reused.
-This practice, while potentially widespread, is visually invisible to report users and virtually undetectable through manual inspection at scale: regulatory agencies overseeing thousands of publicly listed companies cannot feasibly examine each signature for evidence of image reproduction.
+The digitization of financial reporting has introduced a practice that complicates this intent. As audit reports are now routinely generated, transmitted, and archived as PDF documents, it is technically and operationally straightforward to reproduce a CPA's stored signature image across many reports rather than re-executing the signing act for each one. This reproduction can occur either through an administrative stamping workflow — in which scanned signature images are affixed by staff as part of the report-assembly process — or through a firm-level electronic signing system that automates the same step. We refer to signatures produced by either workflow collectively as *non-hand-signed*. Although this practice may fall within the literal statutory requirement of "signature or seal," it raises substantive concerns about audit quality, as an identically reproduced signature applied across hundreds of reports may not represent meaningful individual attestation for each engagement. The accounting literature has examined the audit-quality consequences of partner-level engagement transparency: studies of partner-signature mandates in the United Kingdom find measurable downstream effects [31], cross-jurisdictional evidence on individual partner signature requirements highlights similar quality channels [32], and Taiwan-specific evidence on mandatory partner rotation documents how individual-partner identification interacts with audit-quality outcomes [33]. Unlike traditional signature forgery, where a third party attempts to imitate another person's handwriting, non-hand-signing involves the legitimate signer's own stored signature being reused, and is visually invisible to report users at scale.
 
-The distinction between *non-hand-signing detection* and *signature forgery detection* is both conceptually and technically important.
-The extensive body of research on offline signature verification [3]--[8] has focused almost exclusively on forgery detection---determining whether a questioned signature was produced by its purported author or by an impostor.
-This framing presupposes that the central threat is identity fraud.
-In our context, identity is not in question; the CPA is indeed the legitimate signer.
-The question is whether the physical act of signing occurred for each individual report, or whether a single signing event was reproduced as an image across many reports.
-This detection problem differs fundamentally from forgery detection: while it does not require modeling skilled-forger variability, it introduces the distinct challenge of separating legitimate intra-signer consistency from image-level reproduction, requiring an analytical framework focused on detecting abnormally high similarity across documents.
+The distinction between *non-hand-signing detection* and *signature forgery detection* is conceptually and technically important. The extensive body of research on offline signature verification [3]–[8] focuses almost exclusively on forgery detection — determining whether a questioned signature was produced by its purported author. In our context, identity is not in question; the CPA is indeed the legitimate signer. The question is whether the physical act of signing occurred for each individual report, or whether a single signing event was reproduced as an image across many reports. This detection problem differs fundamentally from forgery detection: while it does not require modeling skilled-forger variability, it introduces the distinct challenge of separating legitimate intra-signer consistency from image-level reproduction.
 
-A secondary methodological concern shapes the research design.
-Many prior similarity-based classification studies rely on ad-hoc thresholds---declaring two images equivalent above a hand-picked cosine cutoff, for example---without principled statistical justification.
-Such thresholds are fragile in an archival-data setting where the cost of misclassification propagates into downstream inference.
-A defensible approach requires (i) a transparent threshold anchored to an empirical reference population drawn from the target corpus; (ii) statistical diagnostics that characterise the *shape* of the underlying similarity distribution and so motivate the choice of anchor; and (iii) external validation against naturally-occurring anchor populations---byte-level identical pairs as a conservative gold positive subset and large random inter-CPA pairs as a gold negative population---reported with Wilson 95% confidence intervals on per-rule capture / FAR rates, since precision and $F_1$ are not meaningful when the positive and negative anchor populations are sampled from different units.
+A methodological concern shapes the research design. Many prior similarity-based classification studies rely on ad-hoc thresholds — declaring two images equivalent above a hand-picked cosine cutoff, for example — without principled statistical justification. Such thresholds are fragile in an archival-data setting. A defensible approach requires (i) explicit calibration of the operational thresholds against measurable negative-anchor evidence; (ii) diagnostic procedures that test whether the descriptor distribution itself supports a within-population threshold, including formal decomposition of apparent multimodality into between-group composition and integer-tie artefacts; (iii) annotation-free reporting of operational alarm rates at multiple analysis units (per-comparison, per-signature pool, per-document) with Wilson 95% confidence intervals; (iv) per-firm stratification of the reported rates to surface heterogeneity that aggregate metrics conceal; and (v) explicit disclosure of the unsupervised setting's limits — in particular, the inability to estimate true error rates without signature-level ground-truth labels.
 
-Despite the significance of the problem for audit quality and regulatory oversight, no prior work has specifically addressed non-hand-signing detection in financial audit documents at scale with these methodological safeguards.
-Woodruff et al. [9] developed an automated pipeline for signature analysis in corporate filings for anti-money-laundering investigations, but their work focused on author clustering (grouping signatures by signer identity) rather than detecting reuse of a stored image.
-Copy-move forgery detection methods [10], [11] address duplicated regions within or across images but are designed for natural images and do not account for the specific characteristics of scanned document signatures, where legitimate visual similarity between a signer's authentic signatures is expected and must be distinguished from image reproduction.
-Research on near-duplicate image detection using perceptual hashing combined with deep learning [12], [13] provides relevant methodological foundations but has not been applied to document forensics or signature analysis.
-From the statistical side, the methods we adopt for distributional characterisation---the Hartigan dip test [37] and finite mixture modelling via the EM algorithm [40], [41], complemented by a Burgstahler-Dichev / McCrary density-smoothness diagnostic [38], [39]---have been developed in statistics and accounting-econometrics but have not, to our knowledge, been combined as a joint diagnostic toolkit for document-forensics threshold selection.
+Despite the significance of the problem for audit quality and regulatory oversight, to our knowledge no prior work has specifically addressed non-hand-signing detection in financial audit documents at scale with these methodological safeguards. Woodruff et al. [9] developed an automated pipeline for signature analysis in corporate filings for anti-money-laundering investigations, but their work focused on author clustering rather than detecting image reuse. Copy-move forgery detection methods [10], [11] address duplicated regions within or across images but are designed for natural images and do not account for the specific characteristics of scanned document signatures. Research on near-duplicate image detection using perceptual hashing combined with deep learning [12], [13] provides relevant methodological foundations but has not been applied to document forensics or signature analysis. From the statistical side, the methods we adopt for distributional characterisation — the Hartigan dip test [37] and finite mixture modelling via the EM algorithm [40], [41], complemented by a Burgstahler-Dichev / McCrary density-smoothness diagnostic [38], [39] — have been developed in statistics and accounting-econometrics but have not been combined as a joint diagnostic toolkit for document-forensics threshold characterisation.
 
-In this paper, we present a fully automated, end-to-end pipeline for detecting non-hand-signed CPA signatures in audit reports at scale.
-Our approach processes raw PDF documents through the following stages:
-(1) signature page identification using a Vision-Language Model (VLM);
-(2) signature region detection using a trained YOLOv11 object detector;
-(3) deep feature extraction via a pre-trained ResNet-50 convolutional neural network;
-(4) dual-descriptor similarity computation combining cosine similarity on deep embeddings with difference hash (dHash) distance;
-(5) signature-level distributional characterisation using two threshold estimators---KDE antimode with a Hartigan unimodality test and finite Beta mixture via EM with a logit-Gaussian robustness check---complemented by a Burgstahler-Dichev / McCrary density-smoothness diagnostic, used to read the structure of the per-signature similarity distribution and to motivate a percentile-based operational anchor rather than a mixture-fit crossing; and
-(6) validation against a pixel-identical anchor, a low-similarity anchor, and a replication-dominated Big-4 calibration firm.
+In this paper we present a fully automated, end-to-end pipeline for screening non-hand-signed CPA signatures in audit reports at scale, together with an anchor-calibrated screening framework that characterises the pipeline's operational behaviour under explicit unsupervised assumptions. The pipeline processes raw PDF documents through (1) signature page identification with a Vision-Language Model; (2) signature region detection with a trained YOLOv11 object detector; (3) deep feature extraction via a pre-trained ResNet-50; (4) dual-descriptor similarity (cosine + independent-minimum dHash); (5) anchor-based threshold calibration at three units of analysis (per-comparison, pool-normalised per-signature, per-document) against an inter-CPA negative-anchor coincidence-rate proxy (§III-L); (6) firm-stratified per-rule reporting and a within-firm cross-CPA hit-matrix analysis (§III-L.4); (7) a composition decomposition that establishes the absence of a within-population bimodal antimode in the descriptor distributions (§III-I.4); and (8) disclosure of each diagnostic's untested assumption (§III-M).
 
-The dual-descriptor verification is central to our contribution.
-Cosine similarity of deep feature embeddings captures high-level visual style similarity---it can identify signatures that share similar stroke patterns and spatial layouts---but cannot distinguish between a CPA who signs consistently and one whose signature is reproduced from a stored image.
-Perceptual hashing (specifically, difference hashing) encodes structural-level image gradients into compact binary fingerprints that are robust to scan noise but sensitive to substantive content differences.
-By requiring convergent evidence from both descriptors, we can differentiate *style consistency* (high cosine but divergent dHash) from *image reproduction* (high cosine with low dHash), resolving an ambiguity that neither descriptor can address alone.
+A key empirical finding is that the descriptor distributions do not support a within-population natural threshold. The apparent multimodality in the Big-4 accountant-level distribution is explained by between-firm location-shift effects (Firm A's mean dHash of $2.73$ versus Firms B/C/D's $6.46$, $7.39$, $7.21$) and integer mass-point artefacts on the integer-valued dHash axis. After joint firm-mean centring and uniform integer-tie jitter, the pooled dHash dip-test rejection disappears ($p_{\text{median}} = 0.35$ across five seeds). Within-firm diagnostics in every Big-4 firm fail to reveal stable bimodal structure after accounting for integer ties; eligible non-Big-4 firms provide corroborating raw-axis evidence on the cosine dimension (§III-I.4). We therefore treat mixture fits as descriptive summaries of firm-compositional structure rather than threshold-generating mechanisms, and calibrate the deployed operating rules using inter-CPA coincidence-rate anchors.
 
-A second distinctive feature is our framing of the calibration reference.
-One major Big-4 accounting firm in Taiwan (hereafter "Firm A") was selected as a candidate calibration reference based on practitioner-knowledge motivation; its benchmark status is then evaluated using the image evidence reported in this paper, not asserted by the practitioner-knowledge motivation itself.
-We therefore treat Firm A as a *replication-dominated* calibration reference rather than a pure positive class.
-This framing is important because the statistical signature of a replication-dominated population is visible in our data: Firm A's per-signature cosine distribution is unimodal with a long left tail (Hartigan dip $p = 0.17$), 92.5% of Firm A signatures exceed cosine 0.95 with the remaining 7.5% forming the left tail, and 145 Firm A signatures across 50 distinct partners are byte-identical to a same-CPA match in a different audit report (35 spanning different fiscal years).
-Adopting the replication-dominated framing---rather than a near-universal framing that would have to absorb the 7.5% residual as noise---ensures internal coherence between the byte-level pixel-identity evidence and the signature-level distributional shape.
+In place of distributional anchoring, we adopt an anchor-based inter-CPA coincidence-rate (ICCR) calibration. At the per-comparison unit, the cos$>0.95$ operating point yields ICCR $= 0.00060$ on a $5 \times 10^5$-pair Big-4 sample; the dHash$\leq 5$ structural cutoff yields ICCR $= 0.00129$; the joint rule cos$>0.95$ AND dHash$\leq 5$ yields joint ICCR $= 0.00014$ (any-pair semantics, matching the deployed extrema rule). At the pool-normalised per-signature unit, the same rule's effective coincidence rate is materially higher because the deployed classifier takes max-cosine and min-dHash over a same-CPA pool: pooled Big-4 any-pair ICCR is $0.1102$ (Wilson 95% CI $[0.1086, 0.1118]$; CPA-block bootstrap 95% $[0.0908, 0.1330]$). At the per-document unit, the operational HC$+$MC alarm fires on $33.75\%$ of Big-4 documents under the inter-CPA candidate-pool counterfactual.
 
-A third distinctive feature is the empirical reading we take from the per-signature distributional analysis.
-Three diagnostics applied to the per-signature similarity distribution---the Hartigan dip test, an EM-fitted Beta mixture (with logit-Gaussian robustness check), and the Burgstahler-Dichev / McCrary density-smoothness procedure---jointly indicate that no two-mechanism mixture cleanly explains per-signature similarity: the dip test fails to reject unimodality for Firm A, BIC strongly prefers a 3-component over a 2-component Beta fit, and the BD/McCrary candidate transition lies *inside* the non-hand-signed mode rather than between modes (and is not bin-width-stable; Appendix A).
-The substantive reading is that *pixel-level output quality* is a continuous spectrum shaped by firm-specific reproduction technologies (administrative stamping in early years, firm-level e-signing later) and scan conditions, rather than a discrete class cleanly separated from hand-signing.
-This reading motivates anchoring the operational classifier on a percentile heuristic over the Firm A reference distribution rather than on a mixture-fit crossing, and it motivates the byte-level pixel-identity anchor (Section IV-F.1) as a threshold-free positive reference that does not depend on resolving signature-level mixture structure.
+The pooled per-signature and per-document rates conceal striking firm heterogeneity. A logistic regression of the per-signature hit indicator on firm dummies (Firm A reference) and centred log pool size yields odds ratios of $0.053$ (Firm B), $0.010$ (Firm C), and $0.027$ (Firm D) — Firms B/C/D are an order of magnitude below Firm A even after controlling for the pool-size confound. Cross-firm hit matrix analysis under the deployed any-pair rule shows within-firm collision concentrations of $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D (Table XXV; the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms). The pattern is consistent with firm-specific template, stamp, or document-production reuse mechanisms — though not by itself diagnostic of deliberate sharing. The deployed five-way box rule defines a reproducible screening classifier; the calibration contribution is to characterise its multi-level inter-CPA coincidence behaviour rather than to derive new thresholds. The high-confidence sub-rule (cos $> 0.95$ AND dHash $\leq 5$) and moderate-confidence sub-rule (cos $> 0.95$ AND $5 < \text{dHash} \leq 15$) are explicit decision rules whose calibrated false-positive and false-negative error rates remain unknown in the absence of signature-level labels.
 
-We apply this pipeline to 90,282 audit reports filed by publicly listed companies in Taiwan between 2013 and 2023, extracting and analyzing 182,328 individual CPA signatures from 758 unique accountants.
-To our knowledge, this represents the largest-scale forensic analysis of signature authenticity in financial documents reported in the literature.
+Three feature-derived scores converge on the per-CPA descriptor-position ranking with Spearman $\rho \geq 0.879$: the K=3 mixture posterior (a firm-compositional position score under §III-J's reading, not a mechanism cluster posterior), a reverse-anchor cosine percentile relative to a strictly-out-of-target non-Big-4 reference, and the box-rule less-replication-dominated rate. The three scores are deterministic functions of the same per-CPA descriptor pair, so the convergence is documented as internal consistency among feature-derived ranks rather than external validation. A conservative hard-positive subset for image replication is provided by 262 byte-identical signatures in the Big-4 subset (Firm A 145, Firm B 8, Firm C 107, Firm D 2), against which all three candidate checks achieve $0\%$ positive-anchor miss rate (Wilson 95% upper bound $1.45\%$). For the box rule this result is close to tautological at byte-identity; we discuss the conservative-subset caveat in §V-G.
 
-The contributions of this paper are summarized as follows:
+We apply this pipeline to 90,282 audit reports filed by publicly listed companies in Taiwan between 2013 and 2023, extracting and analyzing 182,328 individual CPA signatures from 758 unique accountants. The Big-4 sub-corpus comprises 437 CPAs and 150,442 signatures with both descriptors available.
 
-1. **Problem formulation.** We formally define non-hand-signing detection as distinct from signature forgery detection and argue that it requires an analytical framework focused on intra-signer similarity distributions rather than genuine-versus-forged classification.
+The contributions of this paper are:
 
-2. **End-to-end pipeline.** We present a pipeline that processes raw PDF audit reports through VLM-based page identification, YOLO-based signature detection, deep feature extraction, and dual-descriptor similarity computation, with automated inference requiring no manual intervention after initial training and annotation.
+1. **Problem formulation.** We define non-hand-signing detection as distinct from signature forgery detection and frame it as a detection problem on intra-signer similarity distributions.
 
-3. **Dual-descriptor verification.** We demonstrate that combining deep-feature cosine similarity with perceptual hashing resolves the fundamental ambiguity between style consistency and image reproduction, and we validate the backbone choice through an ablation study comparing three feature-extraction architectures.
+2. **End-to-end pipeline.** We present a pipeline that processes raw PDF audit reports through VLM-based page identification, YOLO-based signature detection, ResNet-50 feature extraction, and dual-descriptor similarity computation, with automated inference and no manual intervention after initial training.
 
-4. **Percentile-anchored operational threshold.** We anchor the operational classifier's cosine cut on the whole-sample Firm A P7.5 percentile (cos $> 0.95$), a transparent and reproducible reference drawn from a replication-dominated reference population, and complement it with dHash structural cuts derived from the same reference distribution. Operational thresholds are therefore explained by an empirical reference rather than asserted.
+3. **Dual-descriptor similarity.** We demonstrate that combining deep-feature cosine similarity with independent-minimum dHash provides complementary evidence for screening cases where *style consistency* and *image reproduction* hypotheses diverge, and we support the backbone choice through a feature-backbone ablation.
 
-5. **Distributional characterisation of per-signature similarity.** We apply three statistical diagnostics---a Hartigan dip test, an EM-fitted Beta mixture with logit-Gaussian robustness check, and a Burgstahler-Dichev / McCrary density-smoothness procedure---to characterise the shape of the per-signature similarity distribution. The three diagnostics jointly find that per-signature similarity forms a continuous quality spectrum, which both motivates the percentile-based operational anchor over a mixture-fit crossing and is itself a substantive finding for the document-forensics literature on similarity-threshold selection.
+4. **Composition decomposition does not support the distributional-threshold path.** We show via a 2×2 factorial diagnostic (firm-mean centring × integer-tie jitter) that the apparent multimodality of the Big-4 accountant-level descriptor distribution is fully attributable to between-firm location shifts and integer mass-point artefacts. The descriptor distributions contain no within-population bimodal antimode; a distributional "natural threshold" reading of the operating points is not empirically supported.
 
-6. **Replication-dominated calibration methodology.** We introduce a calibration strategy using a replication-dominated reference group, distinguishing *replication-dominated* from *replication-pure* anchors; and we validate classification using byte-level pixel identity as an annotation-free gold positive, requiring no manual labeling.
+5. **Anchor-based multi-level inter-CPA coincidence-rate calibration.** We characterise the deployed high-confidence (HC) sub-rule and document-level HC$+$MC alarm derived from the five-way classifier at three units of analysis: per-comparison ICCR (cos$>0.95$: $0.0006$; dHash$\leq 5$: $0.0013$; joint: $0.00014$), pool-normalised per-signature ICCR ($0.11$ for the deployed any-pair high-confidence rule), and per-document ICCR ($0.34$ for the operational HC$+$MC alarm). We adopt "inter-CPA coincidence rate" as the metric name throughout and reserve "False Acceptance Rate" for terminology that requires ground-truth negative labels, which the corpus does not provide.
 
-7. **Large-scale empirical analysis.** We report findings from the analysis of over 90,000 audit reports spanning a decade, providing the first large-scale empirical evidence on non-hand-signing practices in financial reporting under a methodology designed for peer-review defensibility.
+6. **Firm heterogeneity quantification and within-firm cross-CPA collision concentration.** Per-document D2 inter-CPA proxy ICCRs differ by an order of magnitude across firms (Firm A: $0.62$ versus Firms B/C/D: $0.09$–$0.16$); a per-signature logistic regression of the any-pair HC hit indicator on firm dummies and centred log pool size confirms the firm gap persists after pool-size control. Cross-firm hit matrix analysis shows within-firm collision concentrations of $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D under the deployed any-pair rule (the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms); the pattern is consistent with — but does not independently establish — firm-level template-like reuse, digitisation-pipeline homogeneity, or signing-style homogeneity, which descriptor-only data cannot separate (§V-H).
 
-The remainder of this paper is organized as follows.
-Section II reviews related work on signature verification, document forensics, perceptual hashing, and the statistical methods we adopt for distributional characterisation.
-Section III describes the proposed methodology.
-Section IV presents experimental results including the signature-level distributional characterisation, pixel-identity validation, and backbone ablation study.
-Section V discusses the implications and limitations of our findings.
-Section VI concludes with directions for future work.
+7. **K=3 as descriptive firm-compositional partition; three-score convergent internal consistency.** We fit a K=3 Gaussian mixture as a descriptive partition of the Big-4 accountant-level distribution (interpreted as firm-compositional structure, not as three mechanism clusters). Three feature-derived scores agree on the per-CPA descriptor-position ranking at Spearman $\rho \geq 0.879$; we report this as internal consistency rather than external validation, given that the scores share the underlying descriptor pair.
+
+8. **Annotation-free positive-anchor capture check and unsupervised-setting disclosure.** We achieve $0\%$ positive-anchor miss rate (Wilson 95% upper bound $1.45\%$) on 262 byte-identical Big-4 signatures, with the conservative-subset caveat that byte-identical pairs are by construction near cos$=1$ and dHash$=0$. Each supporting diagnostic in §III-M addresses one specific failure mode of an unsupervised screening classifier — composition artefacts, inter-CPA coincidence, pool-size confounding, firm heterogeneity, threshold sensitivity, or positive-anchor capture — with an explicitly disclosed untested assumption. We do not claim a validated forensic detector; we position the system as a specificity-proxy-anchored screening framework with human-in-the-loop review.
+
+The remainder of the paper is organized as follows. Section II reviews related work on signature verification, document forensics, perceptual hashing, and the statistical methods used. Section III describes the proposed methodology. Section IV presents the experimental results — distributional characterisation, mixture fits, convergent internal-consistency checks, leave-one-firm-out reproducibility, pixel-identity positive-anchor check, and full-dataset robustness. Section V discusses the implications and limitations. Section VI concludes with directions for future work.

paper/paper_a_methodology_v3.md

@@ -4,19 +4,19 @@
 
 We propose a six-stage pipeline for large-scale non-hand-signed auditor signature detection in scanned financial documents.
 Fig. 1 illustrates the overall architecture.
-The pipeline takes as input a corpus of PDF audit reports and produces, for each document, a classification of its CPA signatures along a confidence continuum anchored on whole-sample Firm A percentile heuristics and validated against a byte-level pixel-identity positive anchor and a large random inter-CPA negative anchor.
+The pipeline takes as input a corpus of PDF audit reports and produces five-way operational screening labels (§III-H.1) whose behaviour is characterised by pixel-identity positive-anchor capture checks and inter-CPA coincidence-rate calibration (§III-L).
 
 Throughout this paper we use the term *non-hand-signed* rather than "digitally replicated" to denote any signature produced by reproducing a previously stored image of the partner's signature---whether by administrative stamping workflows (dominant in the early years of the sample) or firm-level electronic signing systems (dominant in the later years).
 From the perspective of the output image the two workflows are equivalent: both can reproduce one or more stored signature images, producing same-CPA signatures that are identical or near-identical up to reproduction, scanning, compression, and template-variant noise.
 
 <!--
 [Figure 1: Pipeline Architecture - clean vector diagram]
-90,282 PDFs → VLM Pre-screening → 86,072 PDFs
+90,282 PDFs → VLM Pre-screening → 86,084 candidates → processing checks → 86,071 PDFs
 → YOLOv11 Detection → 182,328 signatures
 → ResNet-50 Features → 2048-dim embeddings
 → Dual-Descriptor Verification (Cosine + dHash)
-→ Firm A P7.5-anchored Classifier → Five-way classification
-→ Pixel-identity + Inter-CPA + Held-Out Firm A validation
+→ Anchor-Calibrated Five-Way Classifier → Five-way classification
+→ Pixel-identity Positive Anchor + Inter-CPA Coincidence-Rate Negative Anchor
 -->
 
 ## B. Data Collection
@@ -29,15 +29,16 @@ Each report is a multi-page PDF document containing, among other content, the au
 CPA names, affiliated accounting firms, and audit engagement tenure were obtained from a publicly available audit-firm tenure registry encompassing 758 unique CPAs across 15 document types, with the majority (86.4%) being standard audit reports.
 Table I summarizes the dataset composition.
 
-<!-- TABLE I: Dataset Summary
+**Table I.** Dataset Summary.
+
 | Attribute | Value |
 |-----------|-------|
 | Total PDF documents | 90,282 |
 | Date range | 2013–2023 |
-| Documents with signatures | 86,072 (95.4%) |
+| Signature-page candidates (VLM-positive) | 86,084 (95.3%) |
+| Processed for signature extraction | 86,071 (95.3%) |
 | Unique CPAs identified | 758 |
 | Accounting firms | >50 |
--->
 
 ## C. Signature Page Identification
 
@@ -47,8 +48,8 @@ The model was configured with temperature 0 for deterministic output.
 
 The scanning range was restricted to the first quartile of each document's page count, reflecting the regulatory structure of Taiwanese audit reports in which the auditor's report page is consistently located in the first quarter of the document.
 Scanning terminated upon the first positive detection.
-This process identified 86,072 documents with signature pages; the remaining 4,198 documents (4.6%) were classified as having no signatures and excluded.
-An additional 12 corrupted PDFs were excluded, yielding a final set of 86,071 documents.
+This process identified 86,084 documents with signature pages; the remaining 4,198 documents (4.6%) were classified as having no signatures and excluded.
+An additional 13 PDFs that could not be rendered (corruption or read errors) were excluded, yielding a final set of 86,071 documents.
 
 Cross-validation between the VLM and subsequent YOLO detection confirmed high agreement: YOLO successfully detected signature regions in 98.8% of VLM-positive documents.
 The 1.2% disagreement reflects the combined rate of (i) VLM false positives (pages incorrectly flagged as containing signatures) and (ii) YOLO false negatives (signature regions missed by the detector), and we do not attempt to attribute the residual to either source without further labeling.
@@ -61,20 +62,19 @@ A region was labeled as "signature" if it contained any Chinese handwritten cont
 
 The model was trained for 100 epochs on a 425/75 training/validation split with COCO pre-trained initialization, achieving strong detection performance (Table II).
 
-<!-- TABLE II: YOLO Detection Performance
+**Table II.** YOLO Detection Performance.
+
 | Metric | Value |
 |--------|-------|
 | Precision | 0.97–0.98 |
 | Recall | 0.95–0.98 |
 | mAP@0.50 | 0.98–0.99 |
 | mAP@0.50:0.95 | 0.85–0.90 |
--->
 
 Batch inference on all 86,071 documents extracted 182,328 signature images at a rate of 43.1 documents per second (8 workers).
 A red stamp removal step was applied to each cropped signature using HSV color-space filtering, replacing detected red regions with white pixels to isolate the handwritten content.
 
-Each signature was matched to its corresponding CPA using positional order (first or second signature on the page) against the official CPA registry, achieving a 92.6% match rate (168,755 of 182,328 signatures).
-The remaining 7.4% (13,573 signatures) could not be matched to a registered CPA name---typically because the auditor's report page format deviates from the standard two-signature layout, or because OCR of the printed CPA name on the page returns a name not present in the registry---and these signatures are excluded from all subsequent same-CPA pairwise analyses (a same-CPA best-match statistic is undefined when a signature has no assigned CPA). The 92.6% matched subset is the sample that flows into Sections IV-D through IV-H; the unmatched 7.4% are excluded for definitional reasons rather than discarded as noise.
+Each signature was matched to its corresponding CPA using positional order (first or second signature on the page) against the official CPA registry, achieving a 92.6% match rate (168,755 of 182,328 signatures). The matched records assume standard two-signature ordering; residual order-mismatch risk remains for nonstandard layouts. The remaining 7.4% (13,573 signatures) could not be matched to a registered CPA name---typically because the auditor's report page format deviates from the standard two-signature layout, or because OCR of the printed CPA name on the page returns a name not present in the registry---and these signatures are excluded from all subsequent same-CPA pairwise analyses (a same-CPA best-match statistic is undefined when a signature has no assigned CPA). The 92.6% matched subset forms the candidate pool for same-CPA analyses, before the Big-4 and descriptor-completeness restrictions described in §III-G.
 
 ## E. Feature Extraction
 
@@ -84,8 +84,8 @@ The final classification layer was removed, yielding the 2048-dimensional output
 Preprocessing consisted of resizing to 224×224 pixels with aspect-ratio preservation and white padding, followed by ImageNet channel normalization.
 All feature vectors were L2-normalized, ensuring that cosine similarity equals the dot product.
 
-The choice of ResNet-50 without fine-tuning was motivated by three considerations: (1) the task is similarity comparison rather than classification, making general-purpose discriminative features sufficient; (2) ImageNet features have been shown to transfer effectively to document analysis tasks [20], [21]; and (3) avoiding domain-specific fine-tuning reduces the risk of overfitting to dataset-specific artifacts, though we note that a fine-tuned model could potentially improve discriminative performance (see Section V-G).
-This design choice is validated by an ablation study (Section IV-I) comparing ResNet-50 against VGG-16 and EfficientNet-B0.
+The choice of ResNet-50 without fine-tuning was motivated by three considerations: (1) the task is similarity comparison rather than classification, making general-purpose discriminative features sufficient; (2) ImageNet features have been shown to transfer effectively to document analysis tasks [20], [21]; and (3) avoiding domain-specific fine-tuning reduces the risk of overfitting to dataset-specific artifacts, though we note that a fine-tuned model could potentially improve discriminative performance (see Section V-H, Engineering-level caveats).
+This design choice is supported by an ablation study (Section IV-L) comparing ResNet-50 against VGG-16 and EfficientNet-B0.
 
 ## F. Dual-Method Similarity Descriptors
 
@@ -105,207 +105,332 @@ Unlike DCT-based perceptual hashes, dHash is computationally lightweight and par
 
 These descriptors provide partially independent evidence.
 Cosine similarity is sensitive to the full feature distribution and reflects fine-grained execution variation; dHash captures only coarse perceptual structure and is robust to scanner-induced noise.
-Non-hand-signing yields extreme similarity under *both* descriptors, since the underlying image is identical up to reproduction noise.
-Hand-signing, by contrast, yields high dHash similarity (the overall layout of a signature is preserved across writing occasions) but measurably lower cosine similarity (fine execution varies).
+Non-hand-signing is expected to yield extreme similarity under *both* descriptors, since the underlying image is identical up to reproduction noise; scan-stage noise can in principle push a replicated pair off either extremum but rarely both.
+One working hypothesis is that some hand-signed repetitions may preserve coarse layout while varying in fine execution, producing relatively higher dHash similarity than cosine similarity within a same-CPA pair; the classifier does not require this hypothesis to hold for all CPAs, and the descriptor-level pattern is used only as input to the deployed rule, not as a within-CPA consistency claim.
 Convergence of the two descriptors is therefore a natural robustness check; when they disagree, the case is flagged as borderline.
 
-We did not use SSIM (Structural Similarity Index) [30] or pixel-level comparison as primary descriptors, and the reasons are specific to what each of those measures was designed to do rather than to how either happened to perform on our corpus.
+We do not use SSIM (Structural Similarity Index) [30] or pixel-level comparison as primary descriptors. SSIM was developed as a perceptual quality index for natural images and is by construction sensitive to the local-luminance and local-contrast perturbations routine in a print-scan cycle (JPEG block artefacts, scan-noise speckle, scanner-rule ghosts) — properties that penalise identically-reproduced signature crops at the very margins SSIM is designed to weight most heavily. Pixel-level distances ($L_1$, $L_2$, pixel-identity counting) are defined on geometrically aligned images at a common resolution and inflate under the sub-pixel offsets that scanner DPI, paper-handling alignment, and PDF-page rasterisation routinely introduce, so two scans of the same physical document cannot score near-identically. The supplementary materials contain the full design-level argument; pixel-identity counting is retained only as a threshold-free positive anchor (§III-K), because byte-identical pairs are necessarily produced by literal file reuse and so do not interact with the alignment-fragility argument.
 
-SSIM was developed by Wang et al. [30] as a perceptual quality index for *natural images*, and it factorises local-window image statistics into three components---luminance, contrast, and structural correlation---combined multiplicatively over a sliding window.
-Each of these components is computed at the pixel level on the original-resolution image and is *designed to be sensitive* to small fluctuations in local luminance and local contrast, because that is what makes SSIM track human perception of natural-image quality.
-Applied to a binarised auditor's signature crop, exactly those design choices become liabilities: the JPEG block artifacts, scan-noise speckle, and faint scanner-rule ghosts that are routine in a print-scan cycle perturb local luminance and local contrast in every window they touch, and SSIM amplifies those perturbations in the structural-correlation product.
-A signature reproduced twice from the same stored image---the very case that defines our positive class---is therefore one in which SSIM is structurally guaranteed to penalise the easily perturbed margins around the strokes, even though the strokes themselves are identical up to rendering noise.
-This is a property of how SSIM is constructed, not a finding about how it scored on our data; the empirical observation that the calibration firm exhibits a mean SSIM of only $0.70$ in our corpus is a confirmation of the design-level prediction rather than the basis for the rejection.
+Cosine similarity on L2-normalised deep embeddings and dHash both remain stable across the print-scan-rasterise cycle by design [14], [19], [21], [27]; together they constitute the dual descriptor used throughout the rest of this paper.
 
-Pixel-level comparison---whether $L_1$, $L_2$, or pixel-identity counting---fails on a stricter design ground.
-Pixel-level distances are defined on geometrically aligned images at a common resolution, and they treat any sub-pixel translation, rotation, or rescale as a large perturbation by construction (a one-pixel uniform translation flips a fraction of foreground pixels on a thin-stroke signature crop and inflates pixel L1 distance to the same magnitude as for a different signer's signature).
-Two scans of the same physical document, however, do not share a common pixel grid: scanner DPI, paper-handling alignment, and PDF-page rasterisation each contribute random sub-pixel offsets, and the print-scan cycle that intervenes between the stored stamp image and the audit-report PDF additionally introduces resolution mismatch and small geometric drift.
-A pixel-level descriptor cannot therefore satisfy the basic stability requirement for our task: two presentations of the same stored image must score nearly identically.
-We retain pixel-identity counting only as a *threshold-free anchor* (Section III-J), because byte-identical pairs in our corpus are necessarily produced by literal file reuse rather than by repeated scanning, and so they do not interact with the alignment-fragility argument; they are not used as a primary similarity descriptor.
+## G. Unit of Analysis and Scope
 
-Cosine similarity on deep embeddings and dHash, in contrast, both remain stable across the print-scan-rasterise cycle by design: cosine on L2-normalised pooled features is invariant to overall scale and bias and degrades gracefully under local-pixel noise that the convolutional backbone has been trained to absorb [14], [21], while dHash compresses the image to a $9 \times 8$ grayscale grid before computing horizontal-gradient signs, which removes the resolution and sub-pixel-alignment sensitivity that breaks pixel-level comparison [19], [27].
-Together they constitute the dual descriptor used throughout the rest of this paper.
+We analyse signatures at two **descriptor-summary** units of resolution. The **signature** — one signature image extracted from one report — is the operational unit of classification (§III-H.1) and of the signature-level analyses in §IV (notably §IV-J for the five-way per-signature category counts and the inter-CPA negative-anchor coincidence-rate analysis referenced in §IV-I). The **accountant** — one CPA aggregated over all of their signatures in the corpus — is the unit of mixture-model characterisation (§III-J), of per-CPA internal-consistency analysis (§III-K), and of the leave-one-firm-out reproducibility check (§III-K). At the accountant level we compute, for each CPA with $n_{\text{sig}} \geq 10$ signatures, the per-CPA mean of the per-signature best-match cosine ($\overline{\text{cos}}_a$) and the per-CPA mean of the independent-minimum dHash ($\overline{\text{dHash}}_a$). The minimum threshold of 10 signatures per CPA is required for the per-CPA mean to be a stable summary; CPAs below this threshold are excluded from the accountant-level analyses but remain in the per-signature analyses. §III-L additionally characterises the deployed rule's behaviour at three **operational reporting** units (per-comparison, per-signature, per-document), which are distinct from the descriptor-summary units defined here: the descriptor-summary units summarise input descriptors; the operational reporting units summarise rule outputs.
 
-## G. Unit of Analysis and Summary Statistics
+We make no within-year or across-year uniformity assumption about CPA signing mechanisms. Per-signature labels are signature-level quantities throughout this paper; we do not translate them to per-report or per-partner mechanism assignments, and we abstain from partner-level frequency inferences (such as "X% of CPAs hand-sign") that would require such a translation. A CPA's per-CPA mean is a *summary statistic* of their observed signatures, not a claim that all of their signatures share a single mechanism.
 
-Two unit-of-analysis choices are relevant for this study, ordered from finest to coarsest: (i) the *signature*---one signature image extracted from one report; and (ii) the *auditor-year*---all signatures by one CPA within one fiscal year.
-The signature is the operational unit of classification (Section III-K) and of all primary statistical analyses (Section IV-D, IV-F, IV-G).
-The auditor-year is used in the partner-level similarity ranking of Section IV-G.2 as a within-year aggregation unit: each auditor-year's mean is computed over its own fiscal-year signatures, although the per-signature best-match cosine that feeds the mean is computed against the full same-CPA cross-year pool (Section III-G's max-cosine / min-dHash definition).
-We do not use a coarser CPA-level cross-year unit, because pooling a CPA's signatures across the full 2013--2023 sample period would conflate distinct signing-mechanism regimes whenever a CPA's practice changes during the sample, and we make no claim about the within-CPA stability of signing mechanisms over time.
+We adopt one stipulation about same-CPA pair detectability:
 
-For per-signature classification we compute, for each signature, the maximum pairwise cosine similarity and the minimum dHash Hamming distance against every other signature attributed to the same CPA (over the full same-CPA set, not restricted to the same fiscal year).
-The max/min (rather than mean) formulation reflects the identification logic for non-hand-signing: if even one other signature of the same CPA is a pixel-level reproduction, that pair will dominate the extremes and reveal the non-hand-signed mechanism.
-Mean statistics would dilute this signal.
+> **(A1) Pair-detectability.** *If a CPA uses image replication anywhere in the corpus, then at least one same-CPA signature pair is near-identical (after reproduction noise) within the observed same-CPA candidate pool used by the max-cosine / min-dHash computation, pooled over the CPA's reports across years. A1 does not assume temporal stability of handwriting or scanning workflow within or across years.*
 
-For the dHash dimension we use the *independent minimum dHash*: the minimum Hamming distance from a signature to *any* other signature of the same CPA (over the full same-CPA set).
-The independent minimum is unconditional on the cosine-nearest pair and is therefore the conservative structural-similarity statistic; it is the dHash statistic used throughout the operational classifier (Section III-K) and all reported capture-rate analyses.
+A1 is plausible for high-volume stamping or firm-level electronic signing workflows but is not guaranteed when (i) the corpus contains only one observed replicated report for a CPA, (ii) multiple template variants are used in parallel, or (iii) scan-stage noise pushes a replicated pair outside the detection regime. A1 is the only assumption the per-signature detector requires to be sensitive to replication.
 
-We make one stipulation about same-CPA pair detectability.
+**Scope: the Big-4 sub-corpus.** The primary analyses (§III-I, §III-J, §III-K, §III-L, and the corresponding §IV-D through §IV-J and §IV-M tables) are restricted to the four largest accounting firms in Taiwan, pseudonymously labelled Firm A through Firm D throughout the manuscript. §IV-A through §IV-C and §IV-L report the corpus-wide pipeline performance and feature-backbone ablation that support the descriptor choice of §III-F; §IV-K reports a deliberately narrow full-dataset cross-check at $n = 686$ CPAs. The Big-4 sub-corpus comprises 437 CPAs (171 / 112 / 102 / 52 across Firms A through D) with $n_{\text{sig}} \geq 10$ — the threshold for accountant-level analyses — totalling 150,442 Big-4 signatures with both pre-computed descriptors available. Restricting the primary analyses to Big-4 is a methodological choice driven by four considerations:
 
-**(A1) Pair-detectability.** *If a CPA uses image replication anywhere in the corpus, then at least one same-CPA signature pair is near-identical (after reproduction noise) within the cross-year same-CPA pool used by the max-cosine / min-dHash computation above.*
-This is plausible for high-volume stamping or firm-level electronic-signing workflows---where a stored image is typically reused many times under similar scan and compression conditions---but it is *not* guaranteed when (i) the corpus contains only one observed replicated report for a CPA, (ii) multiple template variants are in use simultaneously, or (iii) scan-stage noise pushes a replicated pair outside the detection regime.
-A1 is a *cross-year pair-existence* property, not a within-year uniformity claim, and is the only assumption the per-signature detector requires to be sensitive to replication.
+1. **Restricted generalisability claim and Big-4 institutional comparability.** The primary claims are scoped to the Big-4 audit-report context, where the four firms share comparable institutional scale, document-production infrastructure, and CPA-volume regime; we do not assert that the same descriptive mixture structure or operational alert behaviour extends to mid/small firms. The 249 non-Big-4 CPAs enter only (a) as an external reference population in §III-H.2's reverse-anchor internal-consistency check, (b) as a robustness comparison in §IV-K, and (c) as a corroborating-population check on the dHash discrete-mass-point artefact in §III-I.4. Generalisation beyond Big-4 is left as future work.
 
-We make *no* within-year or across-year uniformity assumption about CPA signing mechanisms.
-Per-signature labels are signature-level quantities throughout this paper; we do not translate them to per-report or per-partner mechanism assignments, and we abstain from partner-level frequency inferences (such as "X% of CPAs hand-sign") that would require such a translation.
-A CPA's signing output within a single fiscal year may reflect a single replication template, multiple templates used in parallel (e.g., different stored images for different engagement positions or reporting pipelines), within-year mechanism mixing, or a combination; our signature-level analyses remain valid under all of these regimes, since they do not attempt mechanism attribution at the partner or report level.
+2. **Within-firm cross-CPA collision structure analysis.** §III-L.4 reports a Big-4 cross-firm hit-matrix analysis that quantifies the within-firm cross-CPA template-like collision pattern. The four-firm setting affords the cleanest signal for this analysis; replicating the same matrix structure on the heterogeneous mid/small-firm tail is left as future work.
 
-The intra-report consistency analysis in Section IV-G.3 is a firm-level homogeneity check---whether the *two co-signing CPAs on the same report* receive the same signature-level label under the operational classifier---rather than a test of within-partner or within-year uniformity.
+3. **Firm A as templated-end case study.** Firm A is empirically the firm whose CPAs are most concentrated in the high-cosine, low-dHash corner of the descriptor plane (§III-J K=3 component cross-tab; byte-level pair analysis referenced in §III-H.2). We retain Firm A within the Big-4 scope as a descriptive case study of the templated end rather than as the calibration anchor for thresholds.
 
-## H. Calibration Reference: Firm A as a Replication-Dominated Population
+4. **Leave-one-firm-out fold feasibility.** §III-K reports leave-one-firm-out (LOOO) cross-validation of the Big-4 K=3 fit. The Big-4 sub-corpus permits a four-fold LOOO at the firm level (one fold per Big-4 firm). No analogous firm-level fold is available outside Big-4 because mid/small firms have CPA counts of $O(1)$–$O(30)$ per firm.
 
-A distinctive aspect of our methodology is the use of Firm A---a major Big-4 accounting firm in Taiwan---as an empirical calibration reference.
-Rather than treating Firm A as a synthetic or laboratory positive control, we treat it as a naturally occurring *replication-dominated population*: a CPA population whose aggregate signing behavior is dominated by non-hand-signing but is not a pure positive class.
+**Sample-size reconciliation.** Two Big-4 signature counts appear in this section and §IV: $n = 150{,}442$ for analyses using the pre-computed per-signature descriptors $\text{cos}_s$ (`max_similarity_to_same_accountant`) and $\text{dHash}_s$ (`min_dhash_independent`), and $n = 150{,}453$ for analyses recomputing pair-level metrics directly from the stored feature and dHash byte vectors (Scripts 40b, 43, 44). The $11$-signature difference reflects descriptor-completion status: $11$ signatures have feature vectors and dHash byte vectors stored but lack the pre-computed extrema. The $11$ signatures are negligible at population scale and do not affect any reported coincidence rate within $0.01$ percentage point. The CPA counts $468$ (all Big-4 CPAs with both vectors stored) and $437$ (Big-4 CPAs with $n_{\text{sig}} \geq 10$ for accountant-level stability) likewise reflect a single uniform exclusion rule rather than analysis-specific subsetting.
 
-Practitioner knowledge motivated treating Firm A as a candidate calibration reference: the firm is understood within the audit profession to reproduce a stored signature image for the majority of certifying partners---originally via administrative stamping workflows and later via firm-level electronic signing systems---while not ruling out that a minority of partners may continue to hand-sign some or all of their reports.
-This practitioner background motivates Firm A's selection but is not used as evidence: the evidentiary basis in the analyses below---byte-identical same-CPA pairs, the Firm A per-signature similarity distribution, partner-ranking concentration, and intra-report consistency---is derived entirely from the audit-report images themselves and does not depend on any claim about firm-level signing practice.
+## H. Operational Classifier and Reference Populations
 
-We establish Firm A's replication-dominated status through two primary independent quantitative analyses plus a third strand comprising three complementary checks, each of which can be reproduced from the public audit-report corpus alone:
+### H.1. Deployed Operational Rule
 
-First, *automated byte-level pair analysis* (Section IV-F.1; reproduction artifact listed in Appendix B) identifies 145 Firm A signatures that are byte-identical to at least one other same-CPA signature from a different audit report, distributed across 50 distinct Firm A partners (of 180 registered); 35 of these byte-identical matches span different fiscal years.
-Byte-identity implies pixel-identity by construction, and independent hand-signing cannot produce pixel-identical images across distinct reports---these pairs therefore establish image reuse as a concrete, threshold-free phenomenon within Firm A and confirm that replication is widespread (50 of 180 registered partners) rather than confined to a handful of CPAs.
+Each Big-4 signature is assigned to one of five categories using the per-signature descriptor pair $(\text{cos}_s, \text{dHash}_s)$ where $\text{cos}_s$ is the maximum cosine similarity to another signature by the same CPA and $\text{dHash}_s$ is the minimum independent dHash to another signature by the same CPA. The five labels below name regions of the descriptor space and are operational rule outputs, not validated ground-truth classes; the label names reflect the screening hypothesis associated with each region and are subject to the unsupervised-setting caveats of §III-M:
 
-Second, *signature-level distributional evidence*: Firm A's per-signature best-match cosine distribution fails to reject unimodality (Hartigan dip test $p = 0.17$, $N = 60{,}448$ Firm A signatures; Section IV-D) and exhibits a long left tail, consistent with a dominant high-similarity regime plus residual within-firm heterogeneity rather than two cleanly separated mechanisms.
-92.5% of Firm A's per-signature best-match cosine similarities exceed 0.95 and the remaining 7.5% form the long left tail (we do not disaggregate partner-level mechanism here; see Section III-G for the scope of claims).
-The unimodal-long-tail shape, not the precise 92.5/7.5 split, is the structural evidence: it predicts that Firm A is replication-dominated rather than a clean two-class population, and a noise-only explanation of the left tail would predict a shrinking share as scan/PDF technology matured over 2013--2023, which is not what we observe (Section IV-G.1).
+1. **High-confidence non-hand-signed (HC):** Cosine $> 0.95$ AND $\text{dHash}_{\text{indep}} \leq 5$. Both descriptors converge on image-similarity evidence consistent with replication; mechanism attribution remains subject to §III-M.
+2. **Moderate-confidence non-hand-signed (MC):** Cosine $> 0.95$ AND $5 < \text{dHash}_{\text{indep}} \leq 15$. Feature-level similarity is strong; structural similarity is present but below the high-confidence cutoff.
+3. **High style consistency (HSC):** Cosine $> 0.95$ AND $\text{dHash}_{\text{indep}} > 15$. High feature-level similarity without structural corroboration; the descriptor signature is operationally distinguished from HC/MC, but the underlying mechanism (within-CPA signing style, lossy image reproduction with structural drift, or a hybrid) is not resolved by descriptor data alone.
+4. **Uncertain (UN):** Cosine between the all-pairs intra/inter KDE crossover ($0.837$) and $0.95$.
+5. **Likely hand-signed (LH):** Cosine $\leq 0.837$. The "Likely hand-signed" name reflects the screening hypothesis that low maximum same-CPA cosine similarity is more consistent with hand-signing variation than with image replication; the label is operational, not a verified hand-signed classification, since cross-year handwriting drift, scanner-workflow change, or template variant rotation within a CPA's reports can also yield a low max-cosine within a same-CPA pool.
 
-Third, we additionally validate the Firm A benchmark through three complementary analyses reported in Section IV-G. Only the partner-level ranking is fully threshold-free; the longitudinal-stability and intra-report analyses use the operational classifier and are interpreted as consistency checks on its firm-level output:
-  (a) *Longitudinal stability (Section IV-G.1).* The share of Firm A per-signature best-match cosine values below 0.95 is stable at 6-13% across 2013-2023, with the lowest share in 2023. The 0.95 cutoff is the whole-sample Firm A P7.5 heuristic (Section III-K; 92.5% of whole-sample Firm A signatures exceed this cutoff); the substantive finding here is the *temporal stability* of the rate, not the absolute rate at any single year.
-  (b) *Partner-level similarity ranking (Section IV-G.2).* When every auditor-year is ranked globally by its per-auditor-year mean best-match cosine (across all firms: Big-4 and Non-Big-4), Firm A auditor-years account for 95.9% of the top decile against a baseline share of 27.8% (a 3.5$\times$ concentration ratio), and this over-representation is stable across 2013-2023. This analysis uses only the ordinal ranking and is independent of any absolute cutoff.
-  (c) *Intra-report consistency (Section IV-G.3).* Because each Taiwanese statutory audit report is co-signed by two engagement partners, firm-wide stamping practice predicts that both signers on a given Firm A report should receive the same signature-level label under the classifier. Firm A exhibits 89.9% intra-report agreement against 62-67% at the other Big-4 firms. This test uses the operational classifier and is therefore a *consistency* check on the classifier's firm-level output rather than a threshold-free test; the cross-firm gap (not the absolute rate) is the substantive finding.
+Document-level labels are aggregated via the worst-case rule: each audit report inherits the most-replication-consistent category among its certifying-CPA signatures (rank order HC > MC > HSC > UN > LH). The thresholds ($\text{cos} = 0.95$ as the cosine operating point, $\text{cos} = 0.837$ as the all-pairs KDE crossover, $\text{dHash} = 5$ and $15$ as structural-similarity sub-band cutoffs) retain their prior calibration provenance (see supplementary materials). These thresholds define the deployed screening rule; the present analysis does not re-derive them as optimal cutoffs but characterises their behaviour under inter-CPA coincidence anchors (developed in §III-L).
 
-The 92.5% figure is a within-sample consistency check rather than an independent validation of Firm A's status; the validation role is played by the byte-level pixel-identity evidence, the unimodal-long-tail dip-test result, the three complementary analyses above, and the held-out Firm A fold (described in Section III-J; fold-level rate differences are disclosed in Section IV-F.2).
-Firm A's replication-dominated status itself was *not* derived from the thresholds we calibrate against it; it rests on the byte-level pair evidence and the dip-test-confirmed unimodal-long-tail shape, both of which are independent of any threshold choice.
-The "replication-dominated, not pure" framing is important both for internal consistency---it predicts and explains the long left tail observed in Firm A's cosine distribution (Section IV-D)---and for avoiding overclaim in downstream inference.
+The remainder of this section (§III-H.2) describes the reference populations used to calibrate and cross-check this rule. §III-I demonstrates that the descriptor distributions do not provide a within-population natural threshold; §III-J–§III-K develop the descriptive partition and internal-consistency cross-checks; §III-L develops the anchor-based threshold calibration; §III-M discloses the unsupervised-setting limits.
 
-## I. Signature-Level Threshold Characterisation
+### H.2. Reference Populations
 
-This section describes how we set the operational classifier's similarity threshold and how we characterise the per-signature similarity distribution that supports it.
-The two roles are kept separate by design.
+The supporting diagnostics use two reference populations: Firm A as a within-Big-4 templated-end case study, and the 249 non-Big-4 CPAs as an out-of-target reference for internal-consistency checking. Neither population is the calibration anchor for the deployed threshold; both are descriptive references that inform the cross-checks in §III-K.
 
-**Operational threshold (used by the classifier).** The cosine cut is anchored on the whole-sample Firm A P7.5 percentile (cos $> 0.95$; Section III-K).
+**Internal reference: Firm A as the templated-end case study.** Firm A is empirically the firm whose CPAs are most concentrated in the high-cosine, low-dHash corner of the Big-4 descriptor plane. In the Big-4 K=3 descriptive partition (§III-J; Scripts 35, 38), Firm A accounts for 0% of the C1 component (low-cos / high-dHash corner; cos $\approx 0.946$, dHash $\approx 9.17$, weight $\approx 0.143$), 17.5% of the C2 component (central region), and 82.5% of the C3 component (high-cos / low-dHash corner); the opposite pattern holds at Firm C (Script 35: 23.5% C1, 75.5% C2, 1.0% C3, hereafter referred to as "the Firm whose CPAs are most concentrated in C1"). Byte-level decomposition of these signatures (see supplementary materials) identifies 145 Firm A pixel-identical signatures, spanning 50 distinct Firm A partners of the 180 registered, with 35 byte-identical matches occurring across different fiscal years; the 145 are the Firm A portion of the 262 byte-identical Big-4 signatures.
 
-**Statistical characterisation (used to motivate the choice of anchor and to describe the distributional structure).** A Hartigan dip test, an EM-fitted Beta mixture (with logit-Gaussian robustness check), and a Burgstahler-Dichev / McCrary density-smoothness procedure---all applied at the per-signature level (Section IV-D).
+Firm A is *not* the calibration anchor for the operational threshold. Firm A enters the Big-4 mixture on equal footing with Firms B through D; the K=3 components are derived from the joint Big-4 distribution (§III-J), not from Firm A alone. Firm A's role in the methodology is descriptive: it is the Big-4 firm whose CPAs are most concentrated in the high-cosine, low-dHash corner of the descriptor plane, and the byte-level pair evidence above provides the firm-level signature-reuse evidence that anchors §III-K's pixel-identity positive-anchor miss rate.
 
-The reason for the split is empirical.
-The three statistical diagnostics jointly find that per-signature similarity forms a continuous quality spectrum (Section IV-D, summarised below): the dip test fails to reject unimodality for Firm A; BIC strongly prefers a 3-component over a 2-component Beta fit, so the 2-component crossing is a forced fit; and the BD/McCrary candidate transition lies inside the non-hand-signed mode rather than between modes (and is not bin-width-stable; Appendix A).
-Under these conditions the natural anchor for an operational cosine cut is a transparent percentile of a replication-dominated reference population (Firm A) rather than a mixture-fit crossing whose location depends on parametric assumptions the data do not support.
+**External reference: non-Big-4 as the reverse-anchor reference for internal-consistency checking.** The 249 non-Big-4 CPAs ($n_{\text{sig}} \geq 10$, drawn from $\sim$30 mid- and small-firms) constitute a population strictly outside the Big-4 target. Their per-CPA $(\overline{\text{cos}}_a, \overline{\text{dHash}}_a)$ distribution defines a 2D Gaussian reference (fit by Minimum Covariance Determinant with support fraction 0.85 for robustness; Script 38). This reference is used in §III-K's reverse-anchor internal-consistency check: each Big-4 CPA's location relative to the reference centre, measured as the marginal cosine cumulative-distribution-function value under the reference, is one of three feature-derived scores used as a cross-check on the per-signature classifier. The reverse-anchor reference is *not* a positive or negative anchor for threshold derivation — its role is to provide a strictly out-of-target benchmark against which the within-Big-4 mixture-derived ranking can be internally cross-checked.
 
-We describe the three diagnostics and the assumptions underlying each in the subsections below.
-The two threshold estimators rest on decreasing-in-strength assumptions: the KDE antimode/crossover requires only smoothness; the Beta mixture additionally requires a parametric specification, and the logit-Gaussian cross-check reports sensitivity to that form.
-The Burgstahler-Dichev / McCrary procedure is applied to the same distribution as a *density-smoothness diagnostic*: it would identify a sharp local density discontinuity if one existed at the boundary between two cleanly separated mechanisms.
-Because all three diagnostics are applied to the same sample rather than to independent experiments, agreement or disagreement among them is read as evidence about distributional structure rather than as a formal statistical guarantee.
+The reverse-anchor reference centre is at $\overline{\text{cos}} = 0.935$, $\overline{\text{dHash}} = 9.77$ (Script 38). The reference sits at a lower cosine and higher dHash than the Big-4 K=3 low-cos / high-dHash component (cos $= 0.946$, dHash $= 9.17$; §III-J); compared to the Big-4 high-cos / low-dHash component (cos $= 0.983$, dHash $= 2.41$; §III-J) the reference is markedly less replication-dominated. The reverse-anchor metric for a given Big-4 CPA is the percentile of $\overline{\text{cos}}_a$ within the reference marginal cosine distribution, sign-flipped so that lower percentile (further into the left tail of the reference) corresponds to a Big-4 CPA whose mean cosine sits further from the templated end of the descriptor plane. This is a "deviation in the less-replication-dominated descriptor-position direction" measure, not a "deviation toward the templated descriptor-position" measure; the reference is the less-replication-dominated population.
 
-### 1) Method 1: KDE Antimode / Crossover with Unimodality Test
+## I. Distributional Diagnostics: Why the Composition Path Does Not Yield a Natural Threshold
 
-We use two closely related KDE-based threshold estimators and apply each where it is appropriate.
-When two labeled populations are available (e.g., the all-pairs intra-class and inter-class similarity distributions of Section IV-C), the *KDE crossover* is the intersection point of the two kernel density estimates under Scott's rule for bandwidth selection [28]; under equal priors and symmetric misclassification costs it approximates the Bayes-optimal decision boundary between the two classes.
-When a single distribution is analysed (e.g., the per-signature best-match cosine distribution of Section IV-D) the *KDE antimode* is the local density minimum between two modes of the fitted density; it serves the same decision-theoretic role when the distribution is multimodal but is undefined when the distribution is unimodal.
-In either case we use the Hartigan & Hartigan dip test [37] as a formal test of unimodality.
-The dip test asks one question: *is the distribution single-peaked?*
-A non-significant $p$-value means we cannot reject the single-peak null (the data are consistent with one peak); a significant $p$-value means the distribution has *more than one peak* (it could be two, three, or more---the test does not specify how many).
-We use the test to decide whether a KDE antimode is well-defined (it is, only when there is more than one peak), not to assert any particular number of components.
-We additionally perform a sensitivity analysis varying the bandwidth over $\pm 50\%$ of the Scott's-rule value to verify threshold stability.
+This section characterises the joint distribution of accountant-level descriptor means $(\overline{\text{cos}}_a, \overline{\text{dHash}}_a)$ across the 437 Big-4 CPAs of §III-G and tests whether the distribution provides distributional support — in the form of within-population bimodality — for the deployed operational thresholds. We apply four diagnostic procedures in turn: a univariate unimodality test on each accountant-level marginal; a 2D Gaussian mixture fit (developed in §III-J); a density-smoothness diagnostic; and a composition decomposition that distinguishes within-population multimodality from between-firm location-shift artefacts. The four diagnostics jointly imply that the operational thresholds are *not* anchored by distributional bimodality: §III-L develops an anchor-based calibration framework that does not require this assumption.
 
-### 2) Method 2: Finite Mixture Model via EM
+**1. Hartigan dip test on each accountant-level marginal.** We apply the Hartigan & Hartigan dip test [37] to each of the two marginal distributions $\{\overline{\text{cos}}_a\}_{a=1}^{437}$ and $\{\overline{\text{dHash}}_a\}_{a=1}^{437}$, with bootstrap-based $p$-value estimation ($n_{\text{boot}} = 2000$). In both cases no bootstrap replicate exceeded the observed dip statistic, so the empirical $p$-value is bounded above by $5 \times 10^{-4}$; we report this in tables as $p < 5 \times 10^{-4}$ rather than $p = 0$ to reflect the bootstrap resolution (Script 34). For comparison, no rejection of unimodality holds in the comparison scopes tested in Script 32: Firm A pooled alone ($p_{\text{cos}} = 0.992$, $p_{\text{dHash}} = 0.924$, $n = 171$); Firms B + C + D pooled ($p_{\text{cos}} = 0.998$, $p_{\text{dHash}} = 0.906$, $n = 266$); all non-Firm-A CPAs pooled ($p_{\text{cos}} = 0.998$, $p_{\text{dHash}} = 0.907$, $n = 515$). Single-firm dip tests for Firms B, C, and D were not separately computed; the comparison scopes above sufficed to establish that no narrower-than-Big-4 *tested* scope at the accountant level rejected unimodality. The accountant-level Big-4 rejection is a descriptive observation; §III-I.4 below shows that the rejection is fully explained by between-firm location-shift effects rather than within-population bimodality.
 
-We fit a two-component Beta mixture to the cosine distribution via the EM algorithm [40] using method-of-moments M-step estimates (which are numerically stable for bounded proportion data).
-The first component represents non-hand-signed signatures (high mean, narrow spread) and the second represents hand-signed signatures (lower mean, wider spread).
-Under the fitted model the threshold is the crossing point of the two weighted component densities,
+**2. K=2 / K=3 Gaussian mixture fits (descriptive partition).** A 2-component 2D Gaussian Mixture Model (full covariance, $n_{\text{init}} = 15$, fixed seed 42; Script 34) recovers components at $(\overline{\text{cos}}, \overline{\text{dHash}}) = (0.954, 7.14)$, weight $0.689$, and $(0.983, 2.41)$, weight $0.311$. The marginal crossings of the K=2 fit are $\overline{\text{cos}}^* = 0.9755$ and $\overline{\text{dHash}}^* = 3.755$, with bootstrap 95% confidence intervals $[0.9742, 0.9772]$ and $[3.48, 3.97]$ over $n_{\text{boot}} = 500$ resamples. The 3-component fit (§III-J) is BIC-preferred — using the convention that lower BIC is preferred, $\text{BIC}(K{=}3) - \text{BIC}(K{=}2) = -3.48$ (Script 36). The $\Delta$BIC magnitude is small in absolute terms; we do not treat $\Delta\text{BIC} = 3.5$ alone as decisive evidence for K=3 as a population mixture. Following §III-I.4 we treat both K=2 and K=3 fits as *descriptive partitions* of the joint Big-4 distribution that reflect firm-composition structure (Firm A vs others; §III-J) rather than as inferential evidence for two or three latent population modes.
 
-$$\pi_1 \cdot \text{Beta}(x; \alpha_1, \beta_1) = (1 - \pi_1) \cdot \text{Beta}(x; \alpha_2, \beta_2),$$
+**3. Burgstahler-Dichev / McCrary density-smoothness diagnostic.** We apply the discontinuity test of [38, 39] as a *density-smoothness diagnostic* (rather than as a threshold estimator) on each accountant-level marginal axis (cosine in bins of $0.002$, dHash in integer bins). At the Big-4 scope, the diagnostic identifies no significant transition on either marginal at $\alpha = 0.05$ (Script 34). Outside Big-4, the diagnostic does flag dHash transitions in some subsets (Script 32: `big4_non_A` dHash transition at $10.8$; `all_non_A` dHash transition at $6.6$; pre-2018 and post-2020 time-stratified variants also exhibit one or more dHash transitions), but no cosine transition is identified in any subset. The Big-4-scope null on both axes is consistent with §III-I.4 below: under the composition decomposition the Big-4 marginals are unimodal once between-firm and integer-tie confounds are removed, so a local-discontinuity test correctly fails to flag a within-population transition.
 
-solved numerically via bracketed root-finding.
-As a robustness check against the Beta parametric form we fit a parallel two-component Gaussian mixture to the *logit-transformed* similarity, following standard practice for bounded proportion data.
-White's [41] quasi-MLE consistency result justifies interpreting the logit-Gaussian estimates as asymptotic approximations to the best Gaussian-family fit under misspecification; we use the cross-check between Beta and logit-Gaussian crossings as a diagnostic of parametric-form sensitivity rather than as a guarantee of distributional recovery.
+**4. Composition decomposition (Scripts 39b–39e).** §III-I.1 establishes that the accountant-level marginals reject unimodality at the Big-4 sub-corpus. The remaining question is whether the rejection reflects (a) genuine within-population bimodality at the signature or accountant level, (b) between-firm location-shift artefacts (firms with different mean descriptor positions pool to a multi-peaked distribution), or (c) integer mass-point artefacts on the integer-valued dHash axis (the dHash dip statistic is sensitive to spikes at integer values). We apply four diagnostics that decompose the rejection into these candidate sources:
 
-We fit 2- and 3-component variants of each mixture and report BIC for model selection.
-When BIC prefers the 3-component fit, the 2-component assumption itself is a forced fit; we report the resulting crossing only as a forced-fit descriptive reference and do not use it as an operational threshold.
+*Within-firm signature-level dip (Scripts 39b, 39c).* Repeating the dip test at the signature level inside each individual Big-4 firm (Script 39b) and inside each individual non-Big-4 firm with $\geq 500$ signatures (Script 39c) yields a consistent picture. The cosine marginal *fails* to reject unimodality in every single firm tested — all four Big-4 firms ($p_{\text{cos}} \in \{0.176, 0.991, 0.551, 0.976\}$ for Firms A through D; Script 39b) and ten non-Big-4 firms with $\geq 500$ signatures ($p_{\text{cos}} \in [0.59, 0.99]$; Script 39c). The raw dHash marginal *does* reject unimodality in every firm tested ($p < 5 \times 10^{-4}$ in all $14$ firms), but the raw dHash values are integer-valued in $\{0, 1, \ldots, 64\}$, leaving open the possibility of an integer-tie artefact.
 
-### 3) Density-Smoothness Diagnostic: Burgstahler-Dichev / McCrary
+*Integer-jitter robustness (Scripts 39d, 39e).* Adding independent uniform jitter $\sim \mathrm{U}[-0.5, +0.5]$ to break exact dHash ties and re-running the dip test on the perturbed signature cloud (5 seeds, $n_{\text{boot}} = 2000$; Script 39d) eliminates the dHash within-firm rejection in every Big-4 firm tested (Firm A jittered $p_{\text{median}} = 0.999$; B $0.996$; C $0.999$; D $0.9995$; $0$/$5$ seeds reject at $\alpha = 0.05$ in any firm). The pooled-Big-4 dHash dip *does* survive jitter alone ($p_{\text{median}} = 0$, $5$/$5$ seeds reject), but Firm A's mean dHash ($2.73$) is substantially below Firms B/C/D's ($6.46$, $7.39$, $7.21$) — a between-firm location shift. Script 39e applies a 2 \times 2 factorial correction (firm-mean centring $\times$ integer jitter) on the Big-4 pooled dHash:
 
-Complementing the two threshold estimators above, we apply the discontinuity test of Burgstahler and Dichev [38], made asymptotically rigorous by McCrary [39], as a *density-smoothness diagnostic* rather than as a third threshold estimator.
-We discretize each distribution (cosine into bins of width 0.005; $\text{dHash}_\text{indep}$ into integer bins) and compute, for each bin $i$ with count $n_i$, the standardized deviation from the smooth-null expectation of the average of its neighbours,
+| Condition | Firm-mean centred | Integer jitter | Median dip $p$ |  Reject at $\alpha = 0.05$ |
+|---|---|---|---|---|
+| 1 raw | — | — | $< 5 \times 10^{-4}$ | $5/5$ |
+| 2 centred only | $\checkmark$ | — | $< 5 \times 10^{-4}$ | $5/5$ |
+| 3 jittered only | — | $\checkmark$ | $< 5 \times 10^{-4}$ | $5/5$ |
+| 4 centred and jittered | $\checkmark$ | $\checkmark$ | $\mathbf{0.35}$ | $\mathbf{0/5}$ |
 
-$$Z_i = \frac{n_i - \tfrac{1}{2}(n_{i-1} + n_{i+1})}{\sqrt{N p_i (1-p_i) + \tfrac{1}{4} N (p_{i-1}+p_{i+1})(1 - p_{i-1} - p_{i+1})}},$$
+Removing *both* the between-firm location shift *and* the integer mass points eliminates the Big-4 dHash rejection. The Big-4 pooled dHash multimodality is therefore fully attributable to firm-composition contrast (primarily Firm A's mean $\text{dHash} = 2.73$ versus Firms B/C/D $\approx 6.5$–$7.4$) and integer-density artefacts, with no residual continuous within-firm bimodality.
 
-which is approximately $N(0,1)$ under the null of distributional smoothness.
-A candidate transition is identified at an adjacent bin pair where $Z_{i-1}$ is significantly negative and $Z_i$ is significantly positive (cosine) or the reverse (dHash).
-Appendix A reports a bin-width sensitivity sweep covering $\text{bin} \in \{0.003, 0.005, 0.010, 0.015\}$ for cosine and $\text{bin} \in \{1, 2, 3\}$ for dHash; the sweep shows that signature-level BD transitions are not bin-width-stable, consistent with histogram-resolution artifacts rather than a genuine cross-mode density discontinuity.
-We therefore do not treat the BD/McCrary procedure as a threshold estimator in our application but as diagnostic evidence about distributional smoothness.
+*Cosine analogue.* The cosine axis follows the same pattern by construction: the within-firm signature-level cosine dip tests above (Scripts 39b, 39c) fail to reject in every Big-4 firm and in every eligible non-Big-4 firm, so any pooled cosine multimodality must arise from between-firm composition rather than from within-population bimodality.
 
-### 4) Reading the Three Diagnostics Together
+*Integer-histogram valleys (Script 39d).* A genuine within-firm dHash antimode would appear as a strict local minimum in the count histogram with deep relative depth. Within each of the four Big-4 firms, the dHash histogram on bins $0$–$20$ exhibits no strict local minimum; the Big-4 pooled histogram exhibits one shallow valley at $\text{dHash} = 4$ with relative depth $0.021$ (a $2.1\%$ count drop). No valley near the deployed $\text{dHash} = 5$ operational boundary appears within any individual firm. The hypothesised dHash antimode near $\text{dHash} \approx 5$ is not empirically supported by the histogram analysis.
 
-The two threshold estimators rest on decreasing-in-strength assumptions: the KDE antimode/crossover requires only smoothness; the Beta mixture additionally requires a parametric specification (with logit-Gaussian as a robustness cross-check against that form).
-If the two estimated thresholds were to differ by less than a practically meaningful margin and the BD/McCrary procedure were to identify a sharp transition at the same level, that pattern would constitute convergent evidence for a clean two-mechanism boundary at that location.
+**5. Conclusion: no natural threshold from the descriptor distribution.** §III-I.4 jointly establishes that (a) the Big-4 accountant-level dip rejection is fully attributable to between-firm composition and integer mass-point artefacts; (b) within the Big-4 firms, the descriptor marginals at the signature level are unimodal once integer ties are broken (Scripts 39b, 39d); (c) eligible non-Big-4 checks provide corroborating raw-axis evidence on the cosine dimension (Script 39c) and corroborate the integer-mass-point reading of raw dHash, but are not used as calibration evidence for the deployed thresholds; and (d) no integer-histogram valley near the deployed $\text{dHash} = 5$ operational boundary exists within any Big-4 firm. The descriptor distributions therefore do not contain a within-population bimodal antimode that could anchor an operational threshold. The K=2 / K=3 mixture fits of §III-I.2 and §III-J are retained as *descriptive partitions* that reflect firm-composition contrast, not as inferential evidence for two or three population modes. §III-L develops the anchor-based threshold calibration framework, which derives operational rates from inter-CPA pair-level negative-anchor coincidences rather than from a distributional antimode.
 
-This is *not* the pattern we observe at the per-signature level.
-The two threshold estimators yield crossings spread across a wide range (Section IV-D); the BIC clearly prefers a 3-component over a 2-component Beta fit, indicating that the 2-component crossing is a forced fit reported only as a descriptive reference rather than as an operational threshold; and the BD/McCrary procedure locates its candidate transition *inside* the non-hand-signed mode rather than between modes (Appendix A).
-We interpret this jointly as evidence that per-signature similarity is a continuous quality spectrum rather than a clean two-mechanism mixture, and we accordingly anchor the operational classifier's cosine cut on whole-sample Firm A percentile heuristics (Section III-K) rather than on a mixture-fit crossing.
+## J. K=3 as a Descriptive Partition of Firm-Composition Contrast
 
-## J. Pixel-Identity, Inter-CPA, and Held-Out Firm A Validation (No Manual Annotation)
+This section develops the K=2 and K=3 Gaussian mixture fits to the Big-4 accountant-level distribution and clarifies their role. **Both fits are descriptive partitions of the joint Big-4 distribution; they reflect firm-composition contrast — primarily Firm A versus Firms B, C, D — rather than within-population mechanism modes.** §III-I.4 demonstrates that the apparent multimodality of the accountant-level marginals is fully explained by between-firm location shifts and integer mass-point artefacts, leaving no residual evidence for two or three latent within-population mechanism classes. Neither mixture is used to assign signature-level or document-level labels in the primary analysis. The operational classifier of §III-H.1 is calibrated in §III-L via inter-CPA negative-anchor coincidence rates, not via mixture-derived antimodes.
 
-Rather than construct a stratified manual-annotation validation set, we validate the classifier using four naturally occurring reference populations that require no human labeling:
+**K=2 fit.** Two components at $(\overline{\text{cos}}, \overline{\text{dHash}}) = (0.954, 7.14)$ (weight $0.689$) and $(0.983, 2.41)$ (weight $0.311$) (Script 34). $\text{BIC}(K{=}2) = -1108.45$. Marginal crossings: $\overline{\text{cos}}^* = 0.9755$, $\overline{\text{dHash}}^* = 3.755$. We refer to the components by index rather than by mechanism labels, since §III-I.4 establishes that the K=2 separation is firm-compositional rather than mechanistic.
 
-1. **Pixel-identical anchor (gold positive, conservative subset):** signatures whose nearest same-CPA match is byte-identical after crop and normalization.
-Handwriting physics makes byte-identity impossible under independent signing events, so a byte-identical same-CPA pair is pair-level proof of image reuse and---for the byte-identical subset---conservative ground truth for non-hand-signed signatures; the narrow exception, in which a genuinely hand-signed exemplar was subsequently reused as the stamping or e-signature template, is discussed as a Limitation in Section V-G.
-We further emphasize that this anchor is a *subset* of the true positive class---only those non-hand-signed signatures whose nearest match happens to be byte-identical---and perfect recall against this anchor therefore does not establish recall against the full non-hand-signed population (Section V-G discusses this further).
+**K=3 fit.** Three components, sorted by ascending cosine mean (Script 35; Script 38 reproduces):
 
-2. **Inter-CPA negative anchor (large gold negative):** $\sim$50,000 pairs of signatures randomly sampled from *different* CPAs.
-Inter-CPA pairs cannot arise from reuse of a single signer's stored signature image, so this population is a reliable negative class for threshold sweeps.
-This anchor is substantially larger than a simple low-similarity-same-CPA negative and yields tight Wilson 95% confidence intervals on FAR at each candidate threshold.
+| 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 |
 
-3. **Firm A anchor (replication-dominated prior positive):** Firm A signatures, treated as a majority-positive reference with within-firm heterogeneity in the left tail, as evidenced by the 7.5% of Firm A signatures whose per-signature best-match cosine falls at or below 0.95 (Section III-H, Section IV-D).
-Because Firm A is both used for empirical percentile calibration in Section III-H and as a validation anchor, we make the within-Firm-A sampling variance visible by splitting Firm A CPAs randomly (at the CPA level, not the signature level) into a 70% *calibration* fold and a 30% *heldout* fold.
-The calibration-fold percentiles used in thresholding---cosine median, P1, and P5 (lower-tail, since higher cosine indicates greater similarity), and dHash_indep median and P95 (upper-tail, since lower dHash indicates greater similarity)---are derived from the 70% calibration fold only.
-The heldout fold is used exclusively to report post-hoc capture rates with Wilson 95% confidence intervals.
+$\text{BIC}(K{=}3) = -1111.93$, lower than $K{=}2$ by $3.48$ (mild numerical preference for K=3 under standard BIC interpretation, but not by itself decisive). The "descriptive position" column refrains from any mechanism interpretation: §III-I.4 establishes that the cosine and dHash axes both lack within-population bimodality, so component centres are best interpreted as locations in a continuous descriptor space rather than as latent mechanism modes.
 
-4. **Low-similarity same-CPA anchor (supplementary negative):** signatures whose maximum same-CPA cosine similarity is below 0.70.
-This anchor is retained for continuity with prior work but is small in our dataset ($n = 35$) and is reported only as a supplementary reference; its confidence intervals are too wide for quantitative inference.
+**Per-firm component composition (Script 35 firm × cluster cross-tab).** The K=3 partition is dominated by firm membership:
 
-From these anchors we report FAR with Wilson 95% confidence intervals against the inter-CPA negative anchor.
-We do not report an Equal Error Rate or FRR column against the byte-identical positive anchor, because byte-identical pairs have cosine $\approx 1$ by construction and any FRR computed against that subset is trivially $0$ at every threshold below $1$; the conservative-subset role of the byte-identical anchor is instead discussed qualitatively in Section V-F.
-Precision and $F_1$ are not meaningful in this anchor-based evaluation because the positive and negative anchors are constructed from different sampling units (intra-CPA byte-identical pairs vs random inter-CPA pairs), so their relative prevalence in the combined set is an arbitrary construction rather than a population parameter; we therefore omit precision and $F_1$ from Table X.
-The 70/30 held-out Firm A fold of Section IV-F.2 additionally reports capture rates with Wilson 95% confidence intervals computed within the held-out fold, which is a valid population for rate inference.
+- Firm A: $0\%$ C1, $17.5\%$ C2, $82.5\%$ C3
+- Firm B: $8.9\%$ C1, $\sim 78\%$ C2, $\sim 13\%$ C3
+- Firm C: $23.5\%$ C1, $75.5\%$ C2, $1.0\%$ C3
+- Firm D: $11.5\%$ C1, $\sim 84\%$ C2, $\sim 4.5\%$ C3
 
-## K. Per-Document Classification
+Firm A accounts for $141$ of the $143$ C3-assigned CPAs; Firm C accounts for $24$ of the $40$ C1-assigned CPAs. The K=3 partition is therefore well-described as a firm-compositional decomposition: C3 is essentially "Firm A and any non-Firm-A CPA whose mean descriptors happen to land in the high-cos / low-dHash corner"; C1 is essentially "non-Firm-A CPAs whose mean descriptors land in the low-cos / high-dHash corner." The composition contrast that K=3 captures at the accountant level reappears at the deployment level in the cross-firm hit matrix of §III-L.4 (Script 44): under the deployed any-pair rule, within-firm collision concentration is $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms). The K=3 partition and the cross-firm hit matrix therefore describe the same underlying firm-compositional structure at two different units of analysis.
 
-The per-signature classifier operates at the signature level with operational thresholds anchored on whole-sample Firm A percentile heuristics: cos $> 0.95$ (Firm A P7.5) for the cosine dimension and dHash$_\text{indep} \leq 5$ / $> 15$ (Firm A median+P75 / style-consistency ceiling) for the structural dimension.
-This percentile-based anchor is the natural choice given the continuous-spectrum shape of the per-signature similarity distribution documented in Section IV-D; sensitivity to nearby alternatives is reported in Section IV-F.3.
-All dHash references in this section refer to the *independent-minimum* dHash defined in Section III-G---the smallest Hamming distance from a signature to any other same-CPA signature.
-We use a single dHash statistic throughout the operational classifier and the supporting capture-rate analyses (Tables IX, XI, XII, XVI), which keeps the classifier definition and its empirical evaluation arithmetically consistent.
+**Leave-one-firm-out stability (Scripts 36, 37).** Leave-one-firm-out cross-validation shows that K=2 is unstable across folds: holding Firm A out gives a fold rule cos $> 0.938$ AND dHash $\leq 8.79$, while holding any single non-Firm-A Big-4 firm out gives a fold rule near cos $> 0.975$ AND dHash $\leq 3.76$ (Script 36). The maximum absolute deviation of the four fold cosine crossings from their across-fold mean is $0.028$ (the corresponding pairwise across-fold range is $0.0376$, from $0.9380$ for the held-out-Firm-A fold to $0.9756$ for the held-out-Firm-D fold; Script 36 stability summary). The $0.028$ value is $5.6\times$ the report's $0.005$ across-fold stability tolerance. K=3 in contrast has a *reproducible component shape*: across the four folds the C1 cosine mean varies by at most $0.005$, the C1 dHash mean by at most $0.96$, and the C1 weight by at most $0.023$ (Script 37). K=3 hard-posterior membership for the held-out firm is composition-sensitive — for Firm C the held-out C1 rate is $36.3\%$ vs the full-Big-4 baseline of $23.5\%$, an absolute difference of $12.8$ pp; for Firm A the held-out C1 rate is $4.7\%$ vs baseline $0.0\%$; the report's own legend classifies this pattern as `P2_PARTIAL` ("the C1 cluster exists but membership is not well-predicted by the held-out fit"). We accordingly do not use K=3 hard-posterior membership as an operational label.
 
-We assign each signature to one of five signature-level categories using convergent evidence from both descriptors:
+We take the joint K=2 / K=3 LOOO evidence as supporting the following descriptive claims, all of which are used in §III-K and §V but none of which underwrites the operational classifier:
 
-1. **High-confidence non-hand-signed:** Cosine $> 0.95$ AND $\text{dHash}_\text{indep} \leq 5$.
-Both descriptors converge on strong replication evidence.
+- The Big-4 K=2 marginal crossing $(0.975, 3.76)$ is essentially a firm-mass separator between Firm A and Firms B + C + D, not a within-Big-4 mechanism boundary.
+- The Big-4 K=3 mixture exhibits a reproducible three-component component shape across LOOO folds at the descriptor-position level, with C1 reproducibly located at $\overline{\text{cos}} \approx 0.946$, $\overline{\text{dHash}} \approx 9.17$.
+- Hard-posterior K=3 membership is composition-sensitive across folds (max absolute deviation $12.8$ pp); K=3 is therefore not used to assign operational labels to CPAs.
 
-2. **Moderate-confidence non-hand-signed:** Cosine $> 0.95$ AND $5 < \text{dHash}_\text{indep} \leq 15$.
-Feature-level evidence is strong; structural similarity is present but below the high-confidence cutoff, potentially due to scan variations.
+The operational signature-level classifier of §III-H.1 is calibrated in §III-L against inter-CPA pair-level negative-anchor coincidence rates, not against mixture-derived antimodes. Cross-checks between the deployed five-way box rule and the K=3 partition appear in §III-K.
 
-3. **High style consistency:** Cosine $> 0.95$ AND $\text{dHash}_\text{indep} > 15$.
-High feature-level similarity without structural corroboration---consistent with a CPA who signs very consistently but not via image reproduction.
+## K. Convergent Internal-Consistency Checks
 
-4. **Uncertain:** Cosine between the all-pairs intra/inter KDE crossover (0.837) and 0.95 without sufficient convergent evidence for classification in either direction.
+The descriptive partition of §III-J is supported by three feature-derived per-CPA scores and a conservative hard-positive subset analysis. We caution at the outset that the three scores are **not statistically independent measurements** — all three are deterministic functions of the same per-CPA descriptor means $(\overline{\text{cos}}_a, \overline{\text{dHash}}_a)$ — so their high pairwise rank correlations are partly a mechanical consequence of shared inputs. Per §III-I.4, none of the three scores has a within-population bimodality interpretation; they are firm-compositional position scores at the accountant level. The checks below therefore document **internal consistency among feature-derived ranks**, not external validation against an independent hand-signed ground truth (which the corpus does not provide).
 
-5. **Likely hand-signed:** Cosine below the all-pairs KDE crossover threshold.
+**1. Three feature-derived per-CPA scores (Script 38).** For each Big-4 CPA we compute:
 
-We note three conventions about the thresholds.
-First, the cosine cutoff $0.95$ is the *operating point* chosen for the five-way classifier from a small grid of candidate cuts, on the basis of an explicit capture-vs-FAR tradeoff against the inter-CPA negative anchor of Section III-J---*not* a discovered natural boundary in the per-signature distribution.
-The candidate grid spans the calibration-fold P5 (0.9407), its rounded value (0.945), the operational anchor (0.95), and two reference points drawn from the signature-level threshold-estimator outputs of Section IV-D (the Firm A Beta-2 forced-fit crossing 0.977 and the BD/McCrary candidate transition 0.985); for each grid point Section IV-F.3 reports the Firm A capture rate, the non-Firm-A capture rate, and the inter-CPA FAR with Wilson 95% CI (Table XII-B).
-Three considerations motivate the operating point at 0.95.
-(i) *Inter-CPA specificity.* At cosine $> 0.95$ the inter-CPA FAR against the 50,000-pair anchor of Section IV-F.1 is $0.0005$ (Wilson 95% CI $[0.0003, 0.0007]$): one in two thousand random cross-CPA pairs exceeds the cut, an order-of-magnitude margin against the working assumption that random cross-CPA pairs do not arise from image reuse.
-(ii) *Capture stability under nearby alternatives.* Moving the cut to $0.945$ raises Firm A capture by 1.51 percentage points (operational dual rule cos $> t$ AND $\text{dHash}_\text{indep} \leq 15$; Section IV-F.3) and inter-CPA FAR by $0.00032$, while moving it to the calibration-fold P5 of $0.9407$ raises Firm A capture by 2.63 percentage points and inter-CPA FAR by $0.00076$; in either direction the qualitative finding---Firm A is replication-dominated, non-Firm-A capture is much lower at the same cut, and the inter-CPA noise floor is small---is preserved.
-(iii) *Interpretive transparency.* The complement $7.5\%$ corresponds to the whole-sample Firm A P7.5 of the per-signature best-match cosine distribution---that is, $92.5\%$ of whole-sample Firm A signatures exceed this cutoff and $7.5\%$ fall at or below it (Section III-H)---which gives the operational cut a transparent reading in the replication-dominated reference population without requiring a parametric mixture fit that the data of Section IV-D do not support.
-The cosine crossover $0.837$ is the all-pairs intra/inter KDE crossover; both $0.95$ and $0.837$ are derived from whole-sample distributions rather than from the 70% calibration fold, so the classifier inherits its operational cosine cuts from the whole-sample Firm A and all-pairs distributions.
-Section IV-F.2 reports both calibration-fold and held-out-fold capture rates for this classifier so that fold-level sampling variance is visible; Section IV-F.3 (Table XII-B) reports the full capture-vs-FAR tradeoff at the candidate grid above.
-Second, the dHash cutoffs $\leq 5$ and $> 15$ are chosen from the whole-sample Firm A $\text{dHash}_\text{indep}$ distribution: $\leq 5$ captures the upper tail of the high-similarity mode (whole-sample Firm A median $\text{dHash}_\text{indep} = 2$, P75 $\approx 4$, so $\leq 5$ is the band immediately above median), while $> 15$ marks the regime in which independent-minimum structural similarity is no longer indicative of image reproduction.
-Third, the signature-level threshold-estimator outputs of Section IV-D (KDE antimode, Beta-mixture and logit-Gaussian crossings, BD/McCrary diagnostic) are *not* the operational thresholds of this classifier: they are descriptive characterisation of the per-signature similarity distribution, and Section IV-D shows they do not converge to a clean two-mechanism boundary at the per-signature level---which is why the operational cosine cut is anchored on the whole-sample Firm A percentile rather than on any mixture-fit crossing.
+- **Score 1 (K=3 posterior on the low-cos / high-dHash component):** $P(\text{C1})$ from the K=3 fit of §III-J. Per §III-J this is a firm-compositional position score on the (cos, dHash) plane (not a probability of any latent "hand-signing mechanism") — a function of both descriptor means.
+- **Score 2 (reverse-anchor cosine percentile):** the marginal cosine CDF value of $\overline{\text{cos}}_a$ under the non-Big-4 reference Gaussian of §III-H.2, sign-flipped so that lower percentile (further into the reference's left tail) corresponds to a Big-4 CPA whose mean cosine sits further from the templated end. This is a function of $\overline{\text{cos}}_a$ alone.
+- **Score 3 (deployed binary high-confidence box rule rate):** the per-CPA fraction of signatures that do **not** satisfy the deployed binary high-confidence box rule (cos $> 0.95$ AND dHash $\leq 5$). This is a per-signature-aggregated function of the same descriptors.
 
-Because each audit report typically carries two certifying-CPA signatures (Section III-D), we aggregate signature-level outcomes to document-level labels using a worst-case rule: the document inherits the *most-replication-consistent* signature label (i.e., among the two signatures, the label rank ordered High-confidence $>$ Moderate-confidence $>$ Style-consistency $>$ Uncertain $>$ Likely-hand-signed determines the document's classification).
-This rule is consistent with the detection goal of flagging any potentially non-hand-signed report rather than requiring all signatures on the report to converge.
+Pairwise Spearman rank correlations among the three scores, $n = 437$ Big-4 CPAs (Script 38):
 
-## L. Data Source and Firm Anonymization
+| Pair | Spearman $\rho$ | $p$-value |
+|---|---|---|
+| Score 1 vs Score 3 | $+0.9627$ | $< 10^{-248}$ |
+| Score 2 vs Score 3 | $+0.8890$ | $< 10^{-149}$ |
+| Score 1 vs Score 2 | $+0.8794$ | $< 10^{-142}$ |
+
+We read this as the strongest internal-consistency signal in the analysis: three different summarisations of the same descriptor pair agree on the per-CPA descriptor-position ranking with $\rho > 0.87$. The three scores agree on placing Firm A as the most replication-dominated descriptor position and the three non-Firm-A Big-4 firms further from the templated end, but they do not all rank the non-Firm-A firms identically: the K=3 posterior P(C1) and the box-rule less-replication-dominated rate (Scores 1 and 3) place Firm C at the less-replication-dominated end of Big-4 (mean P(C1) $= 0.311$; mean box-rule less-replication-dominated rate $= 0.790$), while the reverse-anchor cosine percentile (Score 2) places Firm D fractionally higher than Firm C (mean reverse-anchor score $-0.7125$ vs Firm C $-0.7672$, with higher value indicating deeper into the reference left tail). The mean values for Firms B and D sit between Firms A and C on Scores 1 and 3 (Script 38 per-firm summary). We do not claim this constitutes external validation of any operational classifier; the deployed box rule is calibrated separately (§III-L), and the convergence above shows that a mixture-derived score and a reverse-anchor score concur with the box rule's per-CPA-aggregated outputs on the directional ordering, with a modest disagreement at the less-replication-dominated end between the three non-A Big-4 firms.
+
+**2. Per-signature consistency (Script 39).** Per-CPA aggregation could in principle reflect averaging across within-CPA heterogeneity rather than coherent within-CPA behaviour. We test this by repeating the K=3 fit at the signature level — fitting a fresh K=3 GMM to the 150,442 Big-4 signature-level $(\text{cos}, \text{dHash}_{\text{indep}})$ points (Script 39) — and comparing labels. The per-CPA and per-signature K=3 fits recover a broadly similar three-component ordering; per-CPA C1 is at $\overline{\text{cos}} = 0.946$, $\overline{\text{dHash}} = 9.17$ vs per-signature C1 at $\overline{\text{cos}} = 0.928$, $\overline{\text{dHash}} = 9.75$ (an absolute cosine drift of $0.018$). Cohen $\kappa$ on the binary collapse (replication-dominated vs less-replication-dominated):
+
+| Pair | Cohen $\kappa$ |
+|---|---|
+| Deployed binary high-confidence box rule vs per-CPA K=3 hard label | $0.662$ |
+| Deployed binary high-confidence box rule vs per-signature K=3 hard label | $0.559$ |
+| Per-CPA K=3 vs per-signature K=3 | $0.870$ |
+
+The $\kappa = 0.870$ between per-CPA-fit and per-signature-fit K=3 binary labels indicates that per-CPA aggregation does not collapse the broad three-component ordering. The lower $\kappa = 0.56\text{–}0.66$ between the binary box rule and either K=3 fit is consistent with two factors: different decision geometries (rectangular box vs Gaussian-mixture posterior boundary), and the fact that the binary box rule is a strict subset of the five-way rule. This comparison checks only the binary high-confidence rule (cos $> 0.95$ AND dHash $\leq 5$); §III-K does not directly check the five-way rule's `5 < \text{dHash} \leq 15` moderate-confidence band, whose calibration and capture-rate evidence is reported in the supplementary materials and not regenerated on the Big-4 subset.
+
+**3. Leave-one-firm-out reproducibility (Scripts 36, 37).** Discussed in §III-J above. We summarise the joint result for cross-reference:
+
+- *K=2 LOOO is unstable.* The maximum absolute deviation of the four fold cosine crossings from their across-fold mean is $0.028$, against the report's $0.005$ across-fold stability tolerance (Script 36; pairwise fold range $0.0376$, from $0.9380$ to $0.9756$). When Firm A is held out, the fold rule classifies $171/171$ of held-out Firm A CPAs as templated; when any non-Firm-A Big-4 firm is held out, the fold rule classifies $0$ of the held-out firm's CPAs as templated. This pattern indicates the K=2 boundary is essentially a Firm-A-vs-others separator rather than a within-Big-4 mechanism boundary.
+
+- *K=3 LOOO is partially stable.* The C1 (low-cos / high-dHash) component shape is reproducible across folds: max deviation from the full-Big-4 baseline is $0.005$ in cosine, $0.96$ in dHash, and $0.023$ in mixture weight (Script 37). Hard-posterior membership remains composition-sensitive — observed absolute differences are $1.8$–$12.8$ pp across the four folds, with the Firm C fold exceeding the report's $5$ pp viability bar; the report's own screening label is `P2_PARTIAL` ("K=3 is not predictively useful as an operational classifier"). We accordingly do not use K=3 hard-posterior membership as an operational label.
+
+**4. Positive-anchor miss rate on byte-identical signatures (Script 40).** The corpus provides one conservative hard-positive subset: 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 a conservative hard-positive subset for image replication. The Big-4 byte-identical subset comprises $n = 262$ signatures ($145 / 8 / 107 / 2$ across Firms A through D; Script 40).
+
+We report each candidate check's *positive-anchor miss rate* — the fraction of byte-identical signatures classified as belonging to the less-replication-dominated descriptor positions. This is a one-sided check against a conservative positive subset, **not a paired specificity metric in the usual two-class sense**; we do not report a paired negative-anchor metric here because no signature-level hand-signed ground truth exists. The corresponding signature-level inter-CPA negative-anchor ICCR evidence is developed in §III-L.1 (Big-4 sample) and the corpus-wide version cited at §IV-I:
+
+| Candidate check | Pixel-identity miss rate (Wilson 95% CI) |
+|---|---|
+| Deployed binary high-confidence box rule (cos $> 0.95$ AND dHash $\leq 5$) | $0\%$ $[0\%, 1.45\%]$ |
+| K=3 per-CPA hard label (C3 high-cos / low-dHash corner; descriptive only) | $0\%$ $[0\%, 1.45\%]$ |
+| Reverse-anchor with prevalence-calibrated cut | $0\%$ $[0\%, 1.45\%]$ |
+
+All three candidate scores correctly assign every byte-identical signature to the replicated class. We caution that for the box rule this result is close to tautological: byte-identical nearest-neighbour signatures have cosine $\approx 1$ and dHash $\approx 0$ by construction, so any threshold strictly below cos $= 1$ and strictly above dHash $= 0$ will capture them. The positive-anchor miss rate is therefore a necessary check (a classifier that *failed* this check would be disqualified), not a sufficient validation of the classifier's behaviour on the non-byte-identical replicated population. The reverse-anchor cut here is chosen by prevalence calibration against the box rule's overall replicated rate ($49.58\%$ of Big-4 signatures); this is a documented limitation since no signature-level hand-signed ground truth exists to permit direct ROC optimisation.
+
+## L. Anchor-Based Threshold Calibration
+
+The operational classifier defined in §III-H.1 is calibrated by characterising the deployed thresholds' inter-CPA pair-level negative-anchor coincidence behaviour and their pool-normalised per-signature and per-document alert behaviour, at multiple units of analysis. §III-I.4 establishes that the descriptor distributions do not contain a within-population bimodal antimode that could anchor an operational threshold; the K=3 mixture of §III-J is a descriptive firm-compositional partition, not a mechanism-cluster model. Throughout this section we report **inter-CPA coincidence rates** rather than "False Acceptance Rates"; we explain the terminological choice in §III-L.0.
+
+### L.0. Calibration methodology
+
+**Calibration role of the present analysis.** The deployed thresholds of §III-H.1 preserve continuity with the existing literature and the supplementary calibration evidence. §III-I.4 establishes that a recalibration cannot be anchored on distributional antimodes (no within-population bimodality exists); §III-L.1 below characterises the cosine threshold's specificity-proxy behaviour at the inter-CPA pair level and the structural-dimension threshold $\text{dHash} \leq 5$'s pair-level coincidence behaviour. The sub-band thresholds ($\text{dHash} = 15$, $\text{cos} = 0.837$) retain their supplementary calibration evidence; the present calibration does not provide independent rates for those sub-bands.
+
+**Three units of analysis.** We report inter-CPA negative-anchor coincidence behaviour at three units, each addressing a different operational question:
+
+- *Per comparison.* For a randomly drawn pair of signatures from different CPAs, what fraction satisfies the rule (cos $>$ cos\_threshold and / or dHash $\leq$ dHash\_threshold)? This is the conventional pairwise calibration unit in biometric verification. We report it for both the cosine and dHash dimensions, marginally and jointly (§III-L.1).
+- *Per signature pool.* For a Big-4 source signature $s$ with same-CPA pool of size $n_{\text{pool}}(s)$, what is the probability that the deployed rule fires *under the counterfactual* of replacing the source's same-CPA pool with $n_{\text{pool}}(s)$ random non-same-CPA candidates? This addresses the standard concern that a per-pair rate computed on independent pairs is not the deployed-rule rate at the per-signature classifier level: the deployed rule takes max-cosine and min-dHash over a pool of size $n_{\text{pool}}(s)$, so its effective coincidence rate is approximately $1 - (1 - p_{\text{pair}})^{n_{\text{pool}}}$ in the independence limit (§III-L.2).
+- *Per document.* For an audit report aggregated via the worst-case rule, what fraction of documents have at least one signature whose deployed pool-normalised rule fires under the same inter-CPA candidate-replacement counterfactual? This is the operational alarm-rate unit (§III-L.3).
+
+**Any-pair vs same-pair semantics.** The deployed rule uses independent extrema: a signature satisfies the HC rule if $\max_{\text{pool}} \text{cos} > 0.95$ AND $\min_{\text{pool}} \text{dHash} \leq 5$, *not* if a single candidate in the pool satisfies both. We refer to this as the **any-pair** rule. A stricter alternative — the **same-pair** rule — requires a single candidate to satisfy both inequalities; the deployed rule is any-pair, but we report same-pair as a stricter alternative classifier where useful (§III-L.2, §III-L.4).
+
+**Terminological note on "FAR".** The biometric-verification literature speaks of "False Acceptance Rate" (FAR) for a per-pair rate computed on independent inter-CPA pairs. We adopt **inter-CPA coincidence rate (ICCR)** as the metric name and *do not* use "FAR" in the manuscript prose, for two reasons: (a) FAR has a specific biometric-verification meaning that requires ground-truth negative labels (which the corpus does not provide at the signature level); (b) §III-L.4 shows that the inter-CPA negative-anchor assumption — that inter-CPA pairs are negative — is partially violated by within-firm cross-CPA template-like collision structures. Reading "inter-CPA coincidence rate" as a *specificity proxy* under an explicitly disclosed assumption is faithful to the evidence; reading it as a true biometric FAR would overstate the evidence.
+
+### L.1. Per-comparison inter-CPA coincidence rate (Script 40b)
+
+We sample $5 \times 10^5$ inter-CPA pairs uniformly at random from Big-4 signatures, computing for each pair the cosine similarity (feature dot product) and Hamming distance between the dHash byte vectors. Marginal and joint rates at threshold $k$ are reported with Wilson 95% confidence intervals (Script 40b).
+
+| Threshold | Per-comparison inter-CPA coincidence rate | 95% Wilson CI |
+|---|---|---|
+| Cosine $> 0.95$ | $0.00060$ | $[0.00053, 0.00067]$ |
+| Cosine $> 0.945$ (alternative operating point from supplementary calibration evidence) | $0.00081$ | $[0.00073, 0.00089]$ |
+| Cosine $> 0.97$ | $0.00024$ | $[0.00020, 0.00029]$ |
+| Cosine $> 0.98$ | $0.00009$ | $[0.00007, 0.00012]$ |
+| dHash $\leq 5$ | $0.00129$ | $[0.00120, 0.00140]$ |
+| dHash $\leq 4$ | $0.00050$ | $[0.00044, 0.00057]$ |
+| dHash $\leq 3$ | $0.00019$ | $[0.00015, 0.00023]$ |
+| dHash $\leq 2$ | $0.00006$ | $[0.00004, 0.00008]$ |
+| Joint: cos $> 0.95$ AND dHash $\leq 5$ (any-pair semantics) | $0.00014$ | $[0.00011, 0.00018]$ |
+| Joint: cos $> 0.95$ AND dHash $\leq 4$ (any-pair) | $0.00011$ | $[0.00008, 0.00014]$ |
+
+The cosine row at $\text{cos} > 0.95$ is consistent with the corpus-wide per-comparison rate of $0.0005$ reported in §IV-I on a similarly-sized inter-CPA sample; the present $5 \times 10^5$-pair sample yields $0.00060$, within that precision. The dHash row and joint row are reported here for the first time; the corpus-wide spike did not provide an inter-CPA pair-level coincidence rate for the structural dimension or the joint rule.
+
+The all-firms-scope sample yields slightly lower per-comparison coincidence rates (cos $> 0.95$: $0.00031$; dHash $\leq 5$: $0.00073$; joint: $0.00007$); the all-firms sample weights small CPAs more heavily under CPA-uniform pair sampling, so we treat the Big-4 sample as the primary calibration scope and report all-firms as a corroborating-scope robustness check.
+
+**Conditional inter-CPA coincidence rate.** A natural follow-up question is whether the dHash dimension provides marginal specificity beyond the cosine gate. For pairs with cos $> 0.95$, the conditional rate of dHash $\leq 5$ is $0.234$ (Wilson 95% CI $[0.190, 0.285]$; $70$ of $299$ pairs in the Big-4 sample). At cos $> 0.95$, dHash provides $\sim 4.3\times$ further per-comparison ICCR refinement (joint $0.00014$ vs cos-only $0.00060$).
+
+The per-comparison rate is a useful *specificity-proxy calibration* for the deployed rule's pair-level behaviour. It does *not* directly translate to the deployed-rule specificity at the per-signature classifier level, because the deployed classifier takes extrema over a same-CPA pool of size $n_{\text{pool}}$. The pool-normalised inter-CPA alert rate is reported in §III-L.2.
+
+### L.2. Pool-normalised inter-CPA alert rate (Script 43)
+
+The deployed rule uses $\max_{\text{pool}} \text{cos}$ and $\min_{\text{pool}} \text{dHash}$ over the same-CPA pool of size $n_{\text{pool}}(s)$ for each signature $s$. A per-comparison rate is therefore not the rate at which the deployed classifier fires per signature. To compute the per-signature inter-CPA-equivalent rate, for each Big-4 source signature $s$ we simulate one realisation of an inter-CPA candidate pool of the same size $n_{\text{pool}}(s)$, drawn uniformly from non-same-CPA signatures across all firms, compute the deployed extrema and rule indicator, and aggregate (Script 43; $n_{\text{sig}} = 150{,}453$ vector-complete in this analysis; CPA-block bootstrap 95% CIs reported below).
+
+**Headline rates (deployed any-pair rule, HC = cos $> 0.95$ AND dHash $\leq 5$).** Wilson 95% CIs on the point estimate, CPA-block bootstrap 95% CI on $n_{\text{boot}} = 1000$ replicates:
+
+| Rule semantics | Per-signature ICCR | Wilson 95% CI | CPA-bootstrap 95% CI |
+|---|---|---|---|
+| Any-pair (deployed) | $0.1102$ | $[0.1086, 0.1118]$ | $[0.0908, 0.1330]$ |
+| Same-pair (stricter alternative) | $0.0827$ | $[0.0813, 0.0841]$ | $[0.0668, 0.1021]$ |
+
+Per-firm any-pair rates (no bootstrap; descriptive):
+
+| Firm | $n_{\text{sig}}$ | Any-pair ICCR | Same-pair ICCR |
+|---|---|---|---|
+| Firm A | $60{,}450$ | $0.2594$ | $0.2018$ |
+| Firm B | $34{,}254$ | $0.0147$ | $0.0023$ |
+| Firm C | $38{,}616$ | $0.0053$ | $0.0019$ |
+| Firm D | $17{,}133$ | $0.0110$ | $0.0051$ |
+
+**Pool-size decile dependence.** The deployed rule's pool-normalised rate is monotonically (broadly) increasing in $n_{\text{pool}}$, consistent with the $1 - (1 - p_{\text{pair}})^{n_{\text{pool}}}$ form expected under inter-CPA independence (Script 43 decile table). This functional form is used as descriptive intuition for the broad monotone trend, not as an independence assumption used for estimation; the within-firm violation of inter-CPA independence (§III-L.4) bounds how literally the closed form can be read. Decile 1 (smallest pools, $n_{\text{pool}} \in [0, 201]$): any-pair ICCR $= 0.0249$. Decile 10 (largest, $n_{\text{pool}} \in [846, 1115]$): any-pair ICCR $= 0.1905$. The trend is broadly monotonic with two minor non-monotone reversals (decile 5 and decile 9 dip below their predecessors).
+
+**Threshold sensitivity at per-signature unit.** Tightening the HC rule from $\text{dHash} \leq 5$ to $\text{dHash} \leq 3$ (same-pair) reduces the per-signature ICCR from $0.0827$ to $0.0449$ (Big-4 pooled); tightening to $\text{dHash} \leq 4$ gives $0.0639$ (same-pair). A stricter operating point of dHash $\leq 3$ same-pair would correspond to a per-signature ICCR of $\approx 0.05$; the deployed HC any-pair rule with $\text{dHash} \leq 5$ corresponds to $\approx 0.11$. Stakeholders requiring a tighter specificity proxy could consider the dHash $\leq 3$ same-pair variant, with the unsupervised-setting caveats of §III-M.
+
+### L.3. Document-level inter-CPA proxy alert rate (Script 45)
+
+The deployed worst-case aggregation classifies each document by the most-replication-consistent category among its constituent signatures (§III-H.1). Three operationally meaningful document-level alarm definitions are reported, each as the fraction of documents whose worst-case signature category falls in the alarm set under the same inter-CPA candidate-pool counterfactual as §III-L.2 (Script 45; $n_{\text{docs}} = 75{,}233$ Big-4 documents):
+
+| Alarm definition | Alarm set | Document-level ICCR | Wilson 95% CI |
+|---|---|---|---|
+| D1 | HC only | $0.1797$ | $[0.1770, 0.1825]$ |
+| D2 | HC + MC ("any non-hand-signed screening label") | $0.3375$ | $[0.3342, 0.3409]$ |
+| D3 | HC + MC + HSC | $0.3384$ | $[0.3351, 0.3418]$ |
+
+Per-firm D2 document-level rates:
+
+| Firm | $n_{\text{docs}}$ | D2 (HC + MC) ICCR |
+|---|---|---|
+| Firm A | $30{,}226$ | $0.6201$ |
+| Firm B | $17{,}127$ | $0.1600$ |
+| Firm C | $19{,}501$ | $0.1635$ |
+| Firm D | $8{,}379$ | $0.0863$ |
+
+The document-level D2 rate of $33.75\%$ pooled over Big-4 is the most operationally relevant alarm-rate metric: it is the fraction of audit documents that would carry at least one signature flagged HC or MC under the counterfactual of inter-CPA candidate-pool replacement. The non-trivial per-document inter-CPA alarm rate (and its concentration in Firm A at $62\%$) motivates the positioning of the operational system as a **screening framework with human-in-the-loop review**, not as an autonomous forensic classifier (§III-M).
+
+### L.4. Firm heterogeneity (Script 44)
+
+§III-L.2 and §III-L.3 report large per-firm variation in the deployed rule's pool-normalised behaviour: Firm A's any-pair per-signature ICCR is $0.2594$, an order of magnitude larger than Firm B's $0.0147$, Firm C's $0.0053$, Firm D's $0.0110$. A natural alternative explanation is the pool-size confound: Firm A's median pool size ($\sim 285$) is larger than other firms', and pool size monotonically (broadly) increases the per-signature rate (§III-L.2 decile trend). We test the firm-vs-pool confound with a logistic regression of the per-signature hit indicator (any-pair HC) on firm dummies (Firm A = reference) and centred log pool size (Script 44):
+
+| Term | Odds ratio (vs Firm A) | Direction | Magnitude |
+|---|---|---|---|
+| Firm B | $0.053$ | $< 1$ | $\sim 19\times$ lower odds than Firm A |
+| Firm C | $0.010$ | $< 1$ | $\sim 100\times$ lower odds than Firm A |
+| Firm D | $0.027$ | $< 1$ | $\sim 37\times$ lower odds than Firm A |
+| log(pool size, centred) | $4.01$ | $> 1$ | $\sim 4\times$ higher odds per unit log pool size |
+
+The Firm B/C/D odds ratios are very small after controlling for pool size, indicating that firm membership accounts for a large multiplicative effect on the per-signature rate that is *not* explained by pool size alone. (We report odds ratios rather than $z$-scores because per-signature observations are clustered by CPA and firm, and naive standard errors would be unreliable under within-cluster correlation; a cluster-robust standard error analysis is left as a robustness check.)
+
+The per-decile per-firm breakdown (Script 44) confirms the pattern: within every pool-size decile, Firms B/C/D have rates of $0.0006$–$0.0358$, while Firm A's rate ranges $0.0541$–$0.5958$ across deciles. The firm gap is large within matched pool sizes, not driven by pool composition.
+
+**Cross-firm hit matrix.** Among Big-4 source signatures whose any-pair rule fires under the inter-CPA candidate-pool counterfactual, the candidate firm of the max-cosine partner is distributed as follows (Script 44):
+
+| Source firm | Firm A candidate | Firm B | Firm C | Firm D | non-Big-4 | 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 | $22$ | $2$ | $6$ | $106$ | $1$ | $137$ |
+
+For the same-pair joint event (a single candidate satisfying both $\text{cos} > 0.95$ and $\text{dHash} \leq 5$), the candidate firm is even more strongly concentrated within the source firm: Firm A source $\to$ Firm A candidate in $11{,}314$ of $11{,}319$ same-pair hits ($99.96\%$); Firm B source $\to$ Firm B candidate in $85$ of $87$ ($97.7\%$); Firm C source $\to$ Firm C candidate in $54$ of $55$ ($98.2\%$); Firm D source $\to$ Firm D candidate in $64$ of $66$ ($97.0\%$).
+
+**Interpretation.** Under the deployed any-pair rule, the within-firm collision concentration is $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D — Firm A's pattern is markedly more within-firm-concentrated than the other three firms', though every Big-4 firm still has more than three quarters of its any-pair collisions falling on candidates within the same firm. The stricter same-pair joint event — a single candidate satisfying both cos $> 0.95$ and dHash $\leq 5$ — saturates at $97.0$–$99.96\%$ within-firm across all four firms. This pattern is consistent with — but not by itself diagnostic of — firm-specific template, stamp, or document-production reuse: within-firm scanning workflows, common form templates, and shared report-generation infrastructure could produce visually similar signature crops across different CPAs within the same firm. Byte-level decomposition of Firm A's $145$ pixel-identical signatures across $\sim 50$ distinct certifying partners (supplementary materials; §III-H.2) provides direct evidence of image-level reuse among Firm A signatures; the distribution across many partners is consistent with a firm-level template or production workflow, and the broader inter-CPA collision pattern in §III-L.4 is consistent with similar, milder within-firm collision patterns at Firms B/C/D, whose mechanisms may include template-like reuse, digitisation-pipeline homogeneity, or signing-style homogeneity (§V-H). We report this as "inter-CPA collision concentration is within-firm" — a descriptive observation about deployed-rule behaviour — and refrain from inferring that the within-firm hits constitute deliberate or systematic template sharing.
+
+This connects back to §III-J: the K=3 firm-composition contrast at the accountant level (Firm A dominating C3; Firm C dominating C1) reappears at the deployment level in the cross-firm hit matrix, where the within-firm collision concentration is the dominant pattern at all four Big-4 firms — most strongly at Firm A ($98.8\%$ any-pair, $99.96\%$ same-pair) and at materially lower but still majority levels at Firms B/C/D ($76.7$–$83.7\%$ any-pair; $97.0$–$98.2\%$ same-pair).
+
+### L.5. Alert-rate sensitivity around deployed thresholds (Script 46)
+
+To test whether the deployed cosine threshold $0.95$ and dHash threshold $5$ coincide with a low-gradient (plateau-stable) region of the deployed-rule alert-rate surface — which would be weak distributional evidence that the deployed thresholds are stable operating points — we sweep each threshold across a range and report the per-signature alert rate on actual observed Big-4 same-CPA pools (not inter-CPA-replaced pools), comparing the local gradient at the deployed threshold to the median gradient across the sweep (Script 46).
+
+At the deployed HC operating point cos $> 0.95$ AND dHash $\leq 5$, the local gradient of the per-signature alert rate is substantially larger than the median gradient across the sweep (cosine: ratio $\approx 25\times$ at the $0.95$ point relative to median; dHash: ratio $\approx 3.8\times$ at the $5$ point relative to median; both Script 46). Reading these ratios descriptively, the deployed HC threshold is *locally sensitive* rather than plateau-stable: small threshold perturbations materially change the deployed alert rate (cosine sweep at dHash $\leq 5$ yields rates of $0.5091$ at cos $> 0.945$ vs $0.4789$ at cos $> 0.955$, a $3.0$ pp swing across a $0.01$ cosine perturbation; dHash sweep at cos $> 0.95$ yields rates of $0.4207$ at dHash $\leq 4$ vs $0.5639$ at dHash $\leq 6$, a $14.3$ pp swing across a single integer step). The local-gradient-to-median-gradient ratios are descriptive diagnostics, not formal plateau tests; the primary evidence for "no within-population bimodal antimode at these thresholds" comes from §III-I.4's composition decomposition, not from §III-L.5.
+
+The MC/HSC boundary at dHash $= 15$, by contrast, *is* in a low-gradient region (ratio $\approx 0.08$ to the median); the plateau-like behaviour around dHash $= 15$ is corroborating evidence that the high-end structural threshold lies in a regime where the rule's alert rate is approximately saturated, consistent with the high-dHash tail behaviour expected once near-identical pairs have been exhausted. The §III-L.5 non-plateau / local-sensitivity finding therefore applies specifically to the HC cutoff (cos $= 0.95$, dHash $= 5$); the MC/HSC sub-band boundary at dHash $= 15$ exhibits the opposite behaviour and is plateau-like.
+
+We interpret the deployed HC thresholds as **specificity-anchored operating points** chosen for the specificity-vs-alert-yield tradeoff (§III-L.1), *not* as distributional antimodes. Alternative operating points on the tradeoff curve can be characterised by inverting the per-comparison or pool-normalised ICCR curves (§III-L.1, §III-L.2) at the preferred specificity target.
+
+### L.6. Observed deployed alert rate on actual same-CPA pools
+
+The pool-normalised inter-CPA rates of §III-L.2 and §III-L.3 use the counterfactual of replacing the source signature's same-CPA pool with random non-same-CPA candidates. The **observed deployed alert rate** uses the source's actual same-CPA pool, i.e., the rate at which the deployed rule fires on the real corpus. For Big-4, the deployed HC any-pair rule fires on $49.58\%$ of signatures and $62.28\%$ of documents (Script 46; Script 42 reproduces the per-signature rate at $49.58\%$).
+
+The per-signature observed-deployed rate is $\sim 4.5\times$ the pool-normalised inter-CPA rate ($0.4958$ vs $0.1102$); the per-document observed-deployed rate is $\sim 3.5\times$ the pool-normalised inter-CPA D1 (HC) rate ($0.6228$ vs $0.1797$). We refer to this multiplicative gap as the **deployed-rate excess over the inter-CPA proxy**:
+
+- Per-signature: $0.4958 - 0.1102 = 0.3856$ ($38.6$ pp excess)
+- Per-document HC: $0.6228 - 0.1797 = 0.4431$ ($44.3$ pp excess)
+
+We *do not* interpret the deployed-rate excess as a presumed true-positive rate; the inferential limits of this interpretation are developed in §III-M. The deployed-rate excess is best read as an *observed same-CPA-pool excess* — a quantity that exceeds what random inter-CPA candidate replacement would produce — whose mechanism is not identifiable from descriptor-only data (§III-M); we do not attribute it to within-CPA handwriting repeatability or to image replication without further evidence.
+
+## M. Unsupervised Diagnostic Strategy and Limits
+
+The corpus lacks signature-level ground-truth replication labels: no signature is annotated as definitively hand-signed or definitively templated. The conservative positive anchor (pixel-identical same-CPA signatures; §III-K.4) is by construction near $\text{cos} = 1$ and $\text{dHash} = 0$, providing a tautological capture-check rather than a sensitivity estimate for the non-byte-identical replicated class. The corpus therefore does not admit standard supervised classifier validation: we cannot report False Rejection Rate, sensitivity, recall, Equal Error Rate, ROC-AUC, or precision against ground truth.
+
+Each diagnostic reported in this paper therefore addresses one specific failure mode of an unsupervised screening classifier; the full diagnostic-to-failure-mode-to-assumption map is given in Appendix A Table A.II. No single diagnostic provides ground-truth validation; together they define the limits of what can be supported in this corpus without signature-level ground truth.
+
+**Limits of the present analysis.** We do not claim a validated forensic detector or an autonomous classification system. We do not report False Rejection Rate, sensitivity, recall, EER, ROC-AUC, precision, or positive predictive value against ground truth, because no ground truth exists at the signature level. We do not interpret the deployed-rate excess of §III-L.6 as a presumed true-positive rate: that interpretation would require assuming that the within-firm same-CPA pool's collision rate equals the inter-CPA proxy rate in the absence of replication (i.e., that genuine same-CPA hand-signing would produce a collision rate no higher than random inter-CPA pairs). Two factors make the assumption unsafe: (a) a CPA who signs consistently can produce stylistically similar signatures across years that exceed inter-CPA similarity at the cosine axis; (b) within-firm template sharing (§III-L.4 cross-firm hit matrix; byte-level evidence of Firm A's pixel-identical signatures across partners, supplementary materials) places a substantial inter-CPA collision floor that itself reflects template-like reuse rather than independent inter-CPA random matching. We do not infer that the within-firm collision concentration of §III-L.4 constitutes deliberate template sharing; we describe it as "inter-CPA collision concentration is within-firm" and treat the mechanism as an open empirical question.
+
+**Scope of the present analysis.** The deployed signature-replication screening rule is characterised at three units of analysis (per-comparison, per-signature pool, per-document) against an inter-CPA negative-anchor coincidence-rate calibration. The per-comparison rates ($\leq 0.0006$ at cos $> 0.95$; $\leq 0.0013$ at dHash $\leq 5$; $\leq 0.00014$ jointly) are specificity-proxy-anchored operating points consistent with biometric-verification convention, with the proxy nature recorded in §III-L.0 and §III-M. The per-signature any-pair HC ICCR ($0.11$; Table XXII) and per-document HC+MC alarm-rate ICCR ($0.34$; Table XXIII) are operationally meaningful **alarm-yield** indicators rather than true error rates. Per-firm rates show substantial heterogeneity (Firm A's per-document HC + MC alarm at $0.62$ vs Firm B/C/D at $0.09$–$0.16$), driven by firm-level rather than pool-size effects, and concentrated in within-firm cross-CPA candidate matching. The framework is positioned as a **specificity-proxy-anchored screening tool with human-in-the-loop review**, not as a validated forensic classifier.
+
+**Specificity-alert-yield tradeoff.** Because sensitivity is unobservable, stakeholders cannot derive an operating point by optimising a ROC criterion. Instead, the specificity-proxy-anchored framework offers a *specificity-alert-yield tradeoff*: tighter operating points (e.g., cos $> 0.98$ AND dHash $\leq 3$) reduce both per-comparison ICCR (to $\approx 5 \times 10^{-5}$; §III-L.1 inversion) and per-signature alert yield (to $\approx 0.05$; §III-L.2), with an unknown effect on actual replication-detection recall. Tighter operating points are not necessarily preferable: any tightening reduces the alert rate but may also miss true replicated signatures whose noise has pushed them outside the tighter envelope. The deployment decision depends on the relative cost of manual review (per alarm) and missed-replication risk (per false negative) — neither directly observable from corpus data.
+
+## N. Data Source and Firm Anonymization
 
 **Audit-report corpus.** The 90,282 audit-report PDFs analyzed in this study were obtained from the Market Observation Post System (MOPS) operated by the Taiwan Stock Exchange Corporation.
 MOPS is the statutory public-disclosure platform for Taiwan-listed companies; every audit report filed on MOPS is already a publicly accessible regulatory document.

paper/paper_a_references_v3.md

@@ -1,6 +1,6 @@
 # References
 
-<!-- IEEE numbered style, sequential by first appearance in text. v3 adds statistical-method refs (37–41). -->
+<!-- IEEE numbered style, sequential by first appearance in text. -->
 
 [1] Taiwan Certified Public Accountant Act (會計師法), Art. 4; FSC Attestation Regulations (查核簽證核准準則), Art. 6. Available: https://law.moj.gov.tw/ENG/LawClass/LawAll.aspx?pcode=G0400067
 
@@ -84,4 +84,10 @@
 
 [41] H. White, "Maximum likelihood estimation of misspecified models," *Econometrica*, vol. 50, no. 1, pp. 1–25, 1982.
 
-<!-- Total: 41 references (v2: 36 + 5 new statistical methods refs) -->
+[42] M. Stone, "Cross-validatory choice and assessment of statistical predictions," *J. R. Statist. Soc. B*, vol. 36, no. 2, pp. 111–147, 1974.
+
+[43] S. Geisser, "The predictive sample reuse method with applications," *J. Amer. Statist. Assoc.*, vol. 70, no. 350, pp. 320–328, 1975.
+
+[44] A. Vehtari, A. Gelman, and J. Gabry, "Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC," *Stat. Comput.*, vol. 27, no. 5, pp. 1413–1432, 2017.
+
+<!-- Total: 44 references -->

paper/paper_a_related_work_v3.md

@@ -15,8 +15,8 @@ Hafemann et al. [16] further addressed the practical challenge of adapting to ne
 A common thread in this literature is the assumption that the primary threat is *identity fraud*: a forger attempting to produce a convincing imitation of another person's signature.
 Our work addresses a fundamentally different problem---detecting whether the *legitimate signer's* stored signature image has been reproduced across many documents---which requires analyzing the upper tail of the intra-signer similarity distribution rather than modeling inter-signer discriminability.
 
-Brimoh and Olisah [8] proposed a consensus-threshold approach that derives classification boundaries from known genuine reference pairs, the methodology most closely related to our calibration strategy.
-However, their method operates on standard verification benchmarks with laboratory-collected signatures, whereas our approach applies threshold calibration using a replication-dominated subpopulation identified through domain expertise in real-world regulatory documents.
+Brimoh and Olisah [8] are closest in spirit in using reference evidence to discipline threshold choice.
+Their setting, however, uses standard verification benchmarks with known genuine references, whereas our archival setting lacks signature-level labels and therefore characterises a fixed deployed screening rule through inter-CPA coincidence-rate anchors.
 
 ## B. Document Forensics and Copy Detection
 
@@ -51,13 +51,13 @@ Chamakh and Bounouh [22] confirmed that a simple ResNet backbone with cosine sim
 Babenko et al. [23] established that CNN-extracted neural codes with cosine similarity provide an effective framework for image retrieval and matching, a finding that underpins our feature-comparison approach.
 These findings collectively suggest that pre-trained CNN features, when L2-normalized and compared via cosine similarity, provide a robust and computationally efficient representation for signature comparison---particularly suitable for large-scale applications where the computational overhead of Siamese training or metric learning is impractical.
 
-## E. Statistical Methods for Threshold Determination
+## E. Statistical Methods for Threshold Characterisation and Calibration
 
-Our threshold-determination framework combines three families of methods developed in statistics and accounting-econometrics.
+Our threshold-characterisation and calibration framework combines three families of methods developed in statistics and accounting-econometrics.
 
 *Non-parametric density estimation.*
 Kernel density estimation [28] provides a smooth estimate of a similarity distribution without parametric assumptions.
-Where the distribution is bimodal, the local density minimum (antimode) between the two modes is the Bayes-optimal decision boundary under equal priors.
+In idealized two-class mixture settings with equal priors and equal misclassification costs, the local density minimum (antimode) between the two modes coincides with the Bayes-optimal decision boundary.
 The statistical validity of the unimodality-vs-multimodality dichotomy can be tested via the Hartigan & Hartigan dip test [37], which tests the null of unimodality; we use rejection of this null as evidence consistent with (though not a direct test for) bimodality.
 
 *Discontinuity tests on empirical distributions.*
@@ -71,9 +71,12 @@ When the empirical distribution is viewed as a weighted sum of two (or more) lat
 For observations bounded on $[0,1]$---such as cosine similarity and normalized Hamming-based dHash similarity---the Beta distribution is the natural parametric choice, with applications spanning bioinformatics and Bayesian estimation.
 Under mild regularity conditions, White's quasi-MLE result [41] supports interpreting maximum-likelihood estimates under a mis-specified parametric family as consistent estimators of the pseudo-true parameter that minimizes the Kullback-Leibler divergence to the data-generating distribution within that family; we use this result to justify the Beta-mixture fit as a principled approximation rather than as a guarantee that the true distribution is Beta.
 
-The present study combines all three families, using each to produce an independent threshold estimate and treating cross-method convergence---or principled divergence---as evidence of where in the analysis hierarchy the mixture structure is statistically supported.
+The present study uses these tools diagnostically: first to test whether the descriptor distribution supports a natural operating boundary, and then, when that support fails under composition decomposition, to motivate anchor-based ICCR calibration of a fixed deployed rule.
+
+*Cross-validation in a small-cluster scope.*
+Cross-validation methodology in the leave-one-out tradition has been developed extensively in statistics since Stone [42] and Geisser [43], and modern surveys including Vehtari et al. [44] discuss its application to mixture models. In document-forensics calibration the technique has been used selectively, typically with the individual document or signature as the hold-out unit. Our application in §III-K differs in two respects from the standard usage: (i) the hold-out unit is the *firm* (not the individual CPA or signature), so the analysis directly probes cross-firm reproducibility of the fitted mixture rather than within-firm sampling variance; and (ii) the held-out predictions are interpreted as a *composition-sensitivity band* on the candidate mixture boundary, not as a sufficiency claim for the deployed five-way operational classifier (§III-H.1; calibrated separately in §III-L). We treat LOOO drift as descriptive information about how the mixture characterisation moves when training composition changes, not as a pass/fail test for the operational classifier.
 <!--
-REFERENCES for Related Work (see paper_a_references_v3.md for full list):
+REFERENCES for Related Work (full list in the References section):
 [3]  Bromley et al. 1993 — Siamese TDNN (NeurIPS)
 [4]  Dey et al. 2017 — SigNet
 [5]  Kao & Wen 2020 — Single-sample SV with forgery detection
@@ -101,4 +104,7 @@ REFERENCES for Related Work (see paper_a_references_v3.md for full list):
 [39] McCrary 2008 — density discontinuity test
 [40] Dempster, Laird & Rubin 1977 — EM algorithm
 [41] White 1982 — quasi-MLE consistency
+[42] Stone 1974 — cross-validatory choice
+[43] Geisser 1975 — predictive sample reuse
+[44] Vehtari, Gelman & Gabry 2017 — practical Bayesian LOO/WAIC
 -->

paper/paper_a_results_v3.md

@@ -1,5 +1,7 @@
 # IV. Experiments and Results
 
+Section IV reports the empirical results that calibrate and characterise the operational classifier of §III-H.1 (calibration developed in §III-L). The primary analyses (§IV-D through §IV-J, and the anchor-based ICCR calibration consolidated in §IV-M) 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. §IV-K reports a full-dataset (686 CPAs) robustness check on the K=3 mixture and per-CPA score-rank convergence; §IV-A through §IV-C and §IV-L report the corpus-wide pipeline performance and feature-backbone ablation that support the descriptor choice of §III-F.
+
 ## A. Experimental Setup
 
 Experiments used mixed hardware: YOLOv11n training and inference for signature detection, and ResNet-50 forward inference for feature extraction over all 182,328 detected signatures, were performed on an Nvidia RTX 4090 (CUDA); the downstream statistical analyses (KDE antimode, Hartigan dip test, Beta-mixture EM with logit-Gaussian robustness check, Burgstahler-Dichev/McCrary density-smoothness diagnostic, and pairwise cosine/dHash computations) were performed on an Apple Silicon workstation with Metal Performance Shaders (MPS) acceleration.
@@ -11,10 +13,11 @@ Because all steps rely on deterministic forward inference over fixed pre-trained
 
 The YOLOv11n model achieved high detection performance on the validation set (Table II), with all loss components converging by epoch 60 and no significant overfitting despite the relatively small training set (425 images).
 We note that Table II reports validation-set metrics, as no separate hold-out test set was reserved given the small annotation budget (500 images total).
-However, the subsequent production deployment provides practical validation: batch inference on 86,071 documents yielded 182,328 extracted signatures (Table III), with an average of 2.14 signatures per document, consistent with the standard practice of two certifying CPAs per audit report.
+However, the subsequent production deployment provides a practical consistency check: batch inference on 86,071 documents yielded 182,328 extracted signatures (Table III), with an average of 2.14 signatures per document, consistent with the standard practice of two certifying CPAs per audit report.
 The high VLM--YOLO agreement rate (98.8%) further corroborates detection reliability at scale.
 
-<!-- TABLE III: Extraction Results
+**Table III.** Extraction Results.
+
 | Metric | Value |
 |--------|-------|
 | Documents processed | 86,071 |
@@ -23,15 +26,17 @@ The high VLM--YOLO agreement rate (98.8%) further corroborates detection reliabi
 | Avg. signatures per document | 2.14 |
 | CPA-matched signatures | 168,755 (92.6%) |
 | Processing rate | 43.1 docs/sec |
--->
+
+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 the primary analyses of §IV-D through §IV-J.
 
 ## C. All-Pairs Intra-vs-Inter Class Distribution Analysis
 
 Fig. 2 presents the cosine similarity distributions computed over the full set of *pairwise comparisons* under two groupings: intra-class (all signature pairs belonging to the same CPA) and inter-class (signature pairs from different CPAs).
-This all-pairs analysis is a different unit from the per-signature best-match statistics used in Sections IV-D onward; we report it first because it supplies the reference point for the KDE crossover used in per-document classification (Section III-K).
+This all-pairs analysis is a different unit from the per-signature best-match statistics used in Sections IV-D onward; we report it first because it supplies the reference point for the KDE crossover used in per-signature classification (Section III-H.1).
 Table IV summarizes the distributional statistics.
 
-<!-- TABLE IV: Cosine Similarity Distribution Statistics
+**Table IV.** Cosine Similarity Distribution Statistics.
+
 | Statistic | Intra-class | Inter-class |
 |-----------|-------------|-------------|
 | N (pairs) | 41,352,824 | 500,000 |
@@ -40,431 +45,261 @@ Table IV summarizes the distributional statistics.
 | Median | 0.836 | 0.774 |
 | Skewness | −0.711 | −0.851 |
 | Kurtosis | 0.550 | 1.027 |
--->
 
 Both distributions are left-skewed and leptokurtic.
 Shapiro-Wilk and Kolmogorov-Smirnov tests rejected normality for both ($p < 0.001$), confirming that parametric thresholds based on normality assumptions would be inappropriate.
-Distribution fitting identified the lognormal distribution as the best parametric fit (lowest AIC) for both classes, though we use this result only descriptively; all subsequent threshold-estimator outputs reported in Section IV-D are derived via the methods of Section III-I to avoid single-family distributional assumptions.
+Distribution fitting identified the lognormal distribution as the best parametric fit (lowest AIC) for both classes, though we use this result only descriptively; the subsequent distributional diagnostics in Section IV-D are produced via the methods of Section III-I to avoid single-family distributional assumptions.
 
-The KDE crossover---where the two density functions intersect---was located at 0.837 (Table V).
-Under equal prior probabilities and equal misclassification costs, this crossover approximates the Bayes-optimal boundary between the two classes.
+The KDE crossover---where the two density functions intersect---was located at 0.837.
+Under equal prior probabilities and equal misclassification costs, this crossover is a candidate decision boundary between the two classes; we adopt it only as the operational LH/UN boundary in §III-H.1, not as a natural distributional threshold.
 Statistical tests confirmed significant separation between the two distributions (Cohen's $d = 0.669$, Mann-Whitney [36] $p < 0.001$, K-S 2-sample $p < 0.001$).
 
 We emphasize that pairwise observations are not independent---the same signature participates in multiple pairs---which inflates the effective sample size and renders $p$-values unreliable as measures of evidence strength.
 We therefore rely primarily on Cohen's $d$ as an effect-size measure that is less sensitive to sample size.
 A Cohen's $d$ of 0.669 indicates a medium effect size [29], confirming that the distributional difference is practically meaningful, not merely an artifact of the large sample count.
 
-## D. Signature-Level Distributional Characterisation
-
-This section applies the threshold-estimator and density-smoothness diagnostic of Section III-I to the per-signature similarity distribution.
-The joint reading is that per-signature similarity is a continuous quality spectrum rather than a clean two-mechanism mixture, which is why the operational classifier (Section III-K) anchors its cosine cut on the whole-sample Firm A P7.5 percentile rather than on any mixture-fit crossing.
-
-### 1) Hartigan Dip Test: Unimodality at the Signature Level
-
-Applying the Hartigan & Hartigan dip test [37] to the per-signature best-match distributions reveals a critical structural finding (Table V).
-The $N = 168{,}740$ count used in Table V and in the downstream same-CPA per-signature best-match analyses (Tables V and XII, and the Firm-A per-signature rows of Tables XIII and XVIII) is $15$ signatures smaller than the $168{,}755$ CPA-matched count reported in Table III: these $15$ signatures belong to CPAs with exactly one signature in the entire corpus, for whom no same-CPA pairwise best-match statistic can be computed, and are therefore excluded from all same-CPA similarity analyses.
-
-<!-- TABLE V: Hartigan Dip Test Results
-| Distribution | N | dip | p-value | Verdict (α=0.05) |
-|--------------|---|-----|---------|------------------|
-| Firm A cosine (max-sim) | 60,448 | 0.0019 | 0.169 | Unimodal |
-| Firm A min dHash (independent) | 60,448 | 0.1051 | <0.001 | Multimodal |
-| All-CPA cosine (max-sim) | 168,740 | 0.0035 | <0.001 | Multimodal |
-| All-CPA min dHash (independent) | 168,740 | 0.0468 | <0.001 | Multimodal |
--->
-
-Firm A's per-signature cosine distribution *fails to reject unimodality* ($p = 0.17$), a pattern consistent with a dominant high-similarity regime plus a long left tail attributable to within-firm heterogeneity in signing outputs (Section III-G discusses the scope of partner-level claims).
-The all-CPA cosine distribution, which mixes many firms with heterogeneous signing practices, is *multimodal* ($p < 0.001$).
-The Firm A unimodal-long-tail finding is, in conjunction with the byte-identity, partner-ranking, and intra-report evidence reported below, consistent with the replication-dominated framing (Section III-H): a dominant high-similarity regime plus residual within-firm heterogeneity, rather than two cleanly separated mechanisms.
-
-### 2) Burgstahler-Dichev / McCrary Density-Smoothness Diagnostic
-
-Applying the BD/McCrary procedure (Section III-I.3) to the per-signature cosine distribution yields a nominally significant $Z^- \rightarrow Z^+$ transition at cosine 0.985 for Firm A and 0.985 for the full sample; the min-dHash distributions exhibit a transition at Hamming distance 2 for both Firm A and the full sample under the bin width ($0.005$ / $1$) used here.
-Two cautions, however, prevent us from treating these signature-level transitions as thresholds.
-First, the cosine transition at 0.985 lies *inside* the non-hand-signed mode rather than at the separation between two mechanisms, consistent with the dip-test finding that per-signature cosine is not cleanly bimodal.
-Second, Appendix A documents that the signature-level transition locations are not bin-width-stable (Firm A cosine drifts across 0.987, 0.985, 0.980, 0.975 as the bin width is widened from 0.003 to 0.015, and full-sample dHash transitions drift across 2, 10, 9 as bin width grows from 1 to 3), which is characteristic of a histogram-resolution artifact rather than of a genuine density discontinuity between two mechanisms.
-We therefore use BD/McCrary as a density-smoothness diagnostic rather than as an independent threshold estimator.
-
-### 3) Beta Mixture at Signature Level: A Forced Fit
-
-Fitting 2- and 3-component Beta mixtures to Firm A's per-signature cosine via EM yields a clear BIC preference for the 3-component fit ($\Delta\text{BIC} = 381$), with a parallel preference under the logit-GMM robustness check.
-For the full-sample cosine the 3-component fit is likewise strongly preferred ($\Delta\text{BIC} = 10{,}175$).
-Under the forced 2-component fit the Firm A Beta crossing lies at 0.977 and the logit-GMM crossing at 0.999---values sharply inconsistent with each other, indicating that the 2-component parametric structure is not supported by the data.
-Under the full-sample 2-component forced fit no Beta crossing is identified; the logit-GMM crossing is at 0.980.
-
-### 4) Joint Reading of the Three Diagnostics
-
-The three diagnostics agree that per-signature similarity does not form a clean two-mechanism mixture:
-(i) the Hartigan dip test fails to reject unimodality for Firm A and rejects it for the heterogeneous-firm pooled sample;
-(ii) BIC strongly prefers a 3-component over a 2-component Beta fit, so the 2-component crossing is a *forced fit* and the Beta-vs-logit-Gaussian disagreement (0.977 vs 0.999 for Firm A) reflects parametric-form sensitivity rather than a stable two-mechanism boundary;
-(iii) the BD/McCrary procedure locates its candidate transition *inside* the non-hand-signed mode rather than between modes, and the transition is not bin-width-stable.
-
-Table VI summarises the signature-level threshold-estimator outputs for cross-method comparison.
-
-<!-- TABLE VI: Signature-Level Threshold-Estimator Summary
-| Population | Method | Cosine threshold | dHash threshold | Status |
-|------------|--------|------------------|-----------------|--------|
-| **Threshold estimators (signature-level distributional fits)** | | | | |
-| Firm A signature-level    | KDE antimode + Hartigan dip (Section III-I.1)  | undefined           | —    | unimodal at $\alpha=0.05$ ($p=0.169$); antimode not defined for unimodal data |
-| Firm A signature-level    | Beta-2 EM crossing (Section III-I.2)           | 0.977               | —    | forced fit; BIC strongly prefers $K{=}3$ ($\Delta\text{BIC} = 381$) |
-| Firm A signature-level    | logit-Gaussian-2 crossing (robustness check)   | 0.999               | —    | forced fit; sharply inconsistent with Beta-2 crossing—reflects parametric-form sensitivity |
-| Full-sample signature-l.  | KDE antimode + Hartigan dip                    | (multiple modes)    | —    | multimodal ($p<0.001$); KDE crossover at full-sample is dominated by between-firm heterogeneity |
-| Full-sample signature-l.  | Beta-2 EM crossing                             | no crossing         | —    | forced fit; component densities do not cross over $[0,1]$ under recovered parameters |
-| Full-sample signature-l.  | logit-Gaussian-2 crossing                      | 0.980               | —    | forced fit; BIC strongly prefers $K{=}3$ ($\Delta\text{BIC} = 10{,}175$) |
-| **Density-smoothness diagnostics (not threshold estimators)** | | | | |
-| Firm A signature-level    | BD/McCrary candidate transition (Section III-I.3) | 0.985 (bin 0.005)| 2.0 (bin 1) | bin-unstable across $\{0.003, 0.005, 0.010, 0.015\}$ (Appendix A); transition lies *inside* the non-hand-signed mode |
-| Full-sample signature-l.  | BD/McCrary candidate transition                | 0.985 (bin 0.005)   | 2.0 (bin 1) | bin-unstable across $\{0.003, 0.005, 0.010, 0.015\}$ (Appendix A) |
-| **Reference: between-class KDE (different unit of analysis)** | | | | |
-| All-pairs intra/inter (pair-level; Section IV-C) | KDE crossover | 0.837 | — | reference point for the Uncertain/Likely-hand-signed boundary in the operational classifier |
-| **Operational classifier anchors and percentile cross-references** | | | | |
-| Firm A whole-sample       | P7.5 (operational anchor; Section III-K)       | 0.95                | —    | operational cosine cut for the five-way classifier |
-| Firm A whole-sample       | dHash$_\text{indep}$ P75                       | —                   | 4    | informs the $\leq 5$ high-confidence band edge in the classifier |
-| Firm A whole-sample       | dHash$_\text{indep}$ style-consistency ceiling | —                   | 15   | operational $> 15$ style-consistency boundary |
-| Firm A calibration fold (70%) | cosine P5 (Section IV-F.2)                  | 0.9407              | —    | calibration-fold cross-reference; held-out fold reports rates at this cut |
-| Firm A calibration fold (70%) | dHash$_\text{indep}$ P95                    | —                   | 9    | calibration-fold cross-reference (Tables IX and XI report rates at the rounded $\leq 8$ cut for continuity) |
-
-Read this table by *population × method*: each row reports one method applied to one population.
-The first three blocks (threshold estimators; density-smoothness diagnostics; between-class KDE) are *characterisation* outputs; the bottom block is the operational anchor set used by the classifier of Section III-K.
-The disagreement between Firm A Beta-2 (0.977) and Firm A logit-Gaussian-2 (0.999) is the parametric-form sensitivity referenced in the prose of Section IV-D.3; it cannot be resolved from the data because BIC rejects the underlying $K{=}2$ assumption itself.
--->
-
-Non-hand-signed replication quality is therefore best read as a continuous spectrum produced by firm-specific reproduction technologies (administrative stamping in early years, firm-level e-signing later) acting on a common stored exemplar.
-This finding has a direct methodological pay-off: it is *why* the operational cosine cut is anchored on the whole-sample Firm A P7.5 percentile (Section III-K), and it is *why* the byte-level pixel-identity anchor (Section IV-F.1) is the natural threshold-free positive reference for downstream validation.
-
-## E. Calibration Validation with Firm A
-
-Fig. 3 presents the per-signature cosine and dHash distributions of Firm A compared to the overall population.
-Table IX reports the proportion of Firm A signatures crossing each candidate threshold; these rates play the role of calibration-validation metrics (what fraction of a known replication-dominated population does each threshold capture?).
-
-<!-- TABLE IX: Firm A Whole-Sample Capture Rates (consistency check, NOT external validation)
-| Rule | Firm A rate | k / N |
-|------|-------------|-------|
-| **Cosine-only marginal rates** | | |
-| cosine > 0.837 (all-pairs KDE crossover)                                  | 99.93% | 60,408 / 60,448 |
-| cosine > 0.9407 (calibration-fold P5)                                     | 95.15% | 57,518 / 60,448 |
-| cosine > 0.945 (calibration-fold P5 rounded)                              | 94.02% | 56,836 / 60,448 |
-| cosine > 0.95 (operational; whole-sample Firm A P7.5)                     | 92.51% | 55,922 / 60,448 |
-| **dHash-only marginal rates** | | |
-| dHash_indep ≤ 5 (operational high-confidence cap)                         | 84.20% | 50,897 / 60,448 |
-| dHash_indep ≤ 8 (calibration-fold P95 rounded)                            | 95.17% | 57,527 / 60,448 |
-| dHash_indep ≤ 15 (operational style-consistency boundary)                 | 99.83% | 60,348 / 60,448 |
-| **Operational classifier dual rules (Section III-K)** | | |
-| cosine > 0.95 AND dHash_indep ≤ 5 (high-confidence non-hand-signed)       | 81.70% | 49,389 / 60,448 |
-| cosine > 0.95 AND 5 < dHash_indep ≤ 15 (moderate-confidence)              | 10.76% | 6,503 / 60,448  |
-| cosine > 0.95 AND dHash_indep ≤ 15 (combined non-hand-signed)             | 92.46% | 55,892 / 60,448 |
-| **Calibration-fold-adjacent cross-reference (not the operational classifier rule)** | | |
-| cosine > 0.95 AND dHash_indep ≤ 8                                         | 89.95% | 54,370 / 60,448 |
-
-All rates computed exactly from the full Firm A sample (N = 60,448 signatures); per-rule counts and codes are available in the supplementary materials.
-The two operational dHash cuts ($\leq 5$ for the high-confidence cap and $\leq 15$ for the style-consistency boundary) come from the classifier definition in Section III-K and are the rules used by the five-way classifier of Tables XII and XVII; the dHash $\leq 8$ row is *not* an operational classifier rule but a calibration-fold-adjacent reference (Section IV-F.2 calibration-fold dHash P95 = 9; we report the $\leq 8$ rate as the integer-valued threshold immediately below P95, included here so that Firm A capture in the calibration-fold-P95 neighbourhood can be read off the same table).
--->
-
-Table IX is a whole-sample consistency check rather than an external validation: the cosine cut $0.95$ and the operational dHash band edges ($\leq 5$ high-confidence cap and $\leq 15$ style-consistency boundary) are themselves anchored to the whole-sample Firm A distribution described in Section III-K (the 70/30 calibration-fold thresholds of Table XI are separate and slightly different, e.g., calibration-fold cosine P5 = 0.9407 rather than the whole-sample heuristic 0.95).
-The operational dual rule used by the five-way classifier of Section III-K---cosine $> 0.95$ AND $\text{dHash}_\text{indep} \leq 15$ (the union of the high-confidence and moderate-confidence non-hand-signed buckets)---captures 92.46% of Firm A; the high-confidence component alone (cosine $> 0.95$ AND $\text{dHash}_\text{indep} \leq 5$) captures 81.70%.
-For continuity with prior calibration-fold reporting (Section IV-F.2 reports the calibration-fold rate at the calibration-fold-P95-adjacent cut $\text{dHash}_\text{indep} \leq 8$), Table IX also lists the cosine $> 0.95$ AND $\text{dHash}_\text{indep} \leq 8$ rate of 89.95%; this is *not* the operational classifier rule but a cross-reference value.
-Both operational rates are consistent with the dip-test-confirmed unimodal-long-tail shape of Firm A's per-signature cosine distribution (Section IV-D.1) and the 92.5% / 7.5% signature-level split (Section III-H).
-Section IV-F.2 reports the corresponding rates on the 30% Firm A hold-out fold, which provides the external check these whole-sample rates cannot.
-
-## F. Pixel-Identity, Inter-CPA, and Held-Out Firm A Validation
-
-We report three validation analyses corresponding to the anchors of Section III-J.
-
-### 1) Pixel-Identity Positive Anchor with Inter-CPA Negative Anchor
-
-Of the 182,328 extracted signatures, 310 have a same-CPA nearest match that is byte-identical after crop and normalization (pixel-identical-to-closest = 1); these form the byte-identity positive anchor---a pair-level proof of image reuse that serves as conservative ground truth for non-hand-signed signatures, subject to the source-template edge case discussed in Section V-G.
-Within Firm A specifically, 145 of these byte-identical signatures are distributed across 50 distinct partners (of 180 registered Firm A partners), with 35 of the byte-identical pairs spanning different fiscal years; reproduction artifact for this Firm A decomposition is listed in Appendix B.
-As the gold-negative anchor we sample 50,000 i.i.d. random cross-CPA signature pairs from the full 168,755-signature matched corpus (inter-CPA cosine: mean $= 0.763$, $P_{95} = 0.886$, $P_{99} = 0.915$, max $= 0.992$).
-Because the positive and negative anchor populations are constructed from different sampling units (byte-identical same-CPA pairs vs random inter-CPA pairs), their relative prevalence in the combined anchor set is arbitrary, and precision / $F_1$ / recall therefore have no meaningful population interpretation.
-We accordingly report FAR with Wilson 95% confidence intervals against the large inter-CPA negative anchor in Table X.
-The primary quantity reported by Table X is FAR: the probability that a random pair of signatures from *different* CPAs exceeds the candidate threshold.
-We do not report an Equal Error Rate: EER is meaningful only when the positive and negative error-rate curves cross in a nontrivial interior region, but byte-identical positives all sit at cosine $\approx 1$ by construction, so FRR against that subset is trivially $0$ at every threshold below $1$. An EER calculation against this anchor would be arithmetic tautology rather than biometric performance, and we therefore omit it.
-
-<!-- TABLE X: Cosine Threshold Sweep — FAR Against 50,000 Inter-CPA Negative Pairs
-| Threshold | FAR | FAR 95% Wilson CI |
-|-----------|-----|-------------------|
-| 0.837 (all-pairs KDE crossover) | 0.2101 | [0.2066, 0.2137] |
-| 0.900                            | 0.0250 | [0.0237, 0.0264] |
-| 0.945 (calibration-fold P5 rounded) | 0.0008 | [0.0006, 0.0011] |
-| 0.950 (whole-sample Firm A P7.5; operational cut)  | 0.0005 | [0.0003, 0.0007] |
-| 0.977 (Firm A Beta-2 forced-fit crossing; Section IV-D)  | 0.00014 | [0.00007, 0.00029] |
-| 0.985 (BD/McCrary candidate transition; Appendix A) | 0.00004 | [0.00001, 0.00015] |
-
-Table note: We do not include FRR against the byte-identical positive anchor as a column here: the byte-identical subset has cosine $\approx 1$ by construction, so FRR against that subset is trivially $0$ at every threshold below $1$ and carries no biometric information beyond verifying that the threshold does not exceed $1$. The conservative-subset FRR role of the byte-identical anchor is instead discussed qualitatively in Section V-F.
--->
-
-Two caveats apply.
-First, the byte-identical positive anchor referenced above is a *conservative subset* of the true non-hand-signed population: it captures only those non-hand-signed signatures whose nearest match happens to be byte-identical, not those that are near-identical but not bytewise identical.
-A would-be FRR computed against this subset is definitionally $0$ at every threshold below $1$ (since byte-identical pairs have cosine $\approx 1$), so such an FRR is a mathematical boundary check rather than an empirical miss-rate estimate; we discuss the generalization limits of this conservative-subset framing in Section V-F.
-Second, the 0.945 / 0.95 thresholds are derived from the Firm A whole-sample and calibration-fold percentiles rather than from this anchor set, so the FAR values in Table X are post-hoc-fit-free evaluations of thresholds that were not chosen to optimize Table X.
-The very low FAR at the operational cut is therefore informative about specificity against a realistic inter-CPA negative population.
-
-### 2) Held-Out Firm A Validation (within-Firm-A sampling variance disclosure)
-
-We split Firm A CPAs randomly 70 / 30 at the CPA level into a calibration fold (124 CPAs, 45,116 signatures) and a held-out fold (54 CPAs, 15,332 signatures).
-The total of 178 Firm A CPAs differs from the 180 in the Firm A registry by two registered Firm A partners whose signatures in the corpus are singletons (only one signature each, so the per-signature best-match cosine is undefined and they do not appear in the same-CPA matched-signature table that script `24_validation_recalibration.py` reads); they are therefore not represented in either fold by construction rather than by an explicit exclusion rule.
-Thresholds are re-derived from calibration-fold percentiles only.
-Table XI reports both calibration-fold and held-out-fold capture rates with Wilson 95% CIs and a two-proportion $z$-test.
-
-<!-- TABLE XI: Held-Out vs Calibration Firm A Capture Rates and Generalization Test
-| Rule | Calibration 70% fold (CI) | Held-out 30% fold (CI) | 2-prop z | p | k/n calib | k/n held |
-|------|---------------------------|-------------------------|----------|---|-----------|----------|
-| cosine > 0.837                       | 99.94% [99.91%, 99.96%] | 99.93% [99.87%, 99.96%] | +0.31  | 0.756 n.s. | 45,087/45,116 | 15,321/15,332 |
-| cosine > 0.9407 (calib-fold P5)      | 94.99% [94.79%, 95.19%] | 95.63% [95.29%, 95.94%] | -3.19  | 0.001      | 42,856/45,116 | 14,662/15,332 |
-| cosine > 0.945 (calib-fold P5 rounded) | 93.77% [93.54%, 93.99%] | 94.78% [94.41%, 95.12%] | -4.54  | <0.001     | 42,305/45,116 | 14,531/15,332 |
-| cosine > 0.950 (whole-sample P7.5; operational cut) | 92.14% [91.89%, 92.38%] | 93.61% [93.21%, 93.98%] | -5.97  | <0.001     | 41,570/45,116 | 14,352/15,332 |
-| dHash_indep ≤ 5                      | 82.96% [82.61%, 83.31%] | 87.84% [87.31%, 88.34%] | -14.29 | <0.001     | 37,430/45,116 | 13,467/15,332 |
-| dHash_indep ≤ 8                      | 94.84% [94.63%, 95.04%] | 96.13% [95.82%, 96.43%] | -6.45  | <0.001     | 42,788/45,116 | 14,739/15,332 |
-| dHash_indep ≤ 9 (calib-fold P95)     | 96.65% [96.48%, 96.81%] | 97.48% [97.22%, 97.71%] | -5.07  | <0.001     | 43,604/45,116 | 14,945/15,332 |
-| dHash_indep ≤ 15                     | 99.83% [99.79%, 99.87%] | 99.84% [99.77%, 99.89%] | -0.31  | 0.754 n.s. | 45,040/45,116 | 15,308/15,332 |
-| cosine > 0.95 AND dHash_indep ≤ 8 (calibration-fold P95-adjacent reference; P95 = 9) | 89.40% [89.12%, 89.68%] | 91.54% [91.09%, 91.97%] | -7.60  | <0.001     | 40,335/45,116 | 14,035/15,332 |
-| cosine > 0.95 AND dHash_indep ≤ 15 (operational classifier rule, Section III-K) | 92.09% [91.84%, 92.34%] | 93.56% [93.16%, 93.93%] | -5.93  | <0.001     | 41,548/45,116 | 14,344/15,332 |
-
-Calibration-fold thresholds: Firm A cosine median = 0.9862, P1 = 0.9067, P5 = 0.9407; dHash_indep median = 2, P95 = 9. Counts and z/p values are reproducible from the supplementary materials (fixed random seed).
--->
-
-We report fold-versus-fold comparisons rather than fold-versus-whole-sample comparisons, because the whole-sample rate is a weighted average of the two folds and therefore cannot, in general, fall inside the Wilson CI of either fold when the folds differ in rate; the correct generalization reference is the calibration fold, which produced the thresholds.
-
-Under this proper test the two extreme rules agree across folds (cosine $> 0.837$ and $\text{dHash}_\text{indep} \leq 15$; both $p > 0.7$).
-The operationally relevant rules in the 85–95% capture band differ between folds by 1–5 percentage points ($p < 0.001$ given the $n \approx 45\text{k}/15\text{k}$ fold sizes).
-Both folds nevertheless sit in the same replication-dominated regime: every calibration-fold rate in the 85–99% range has a held-out counterpart in the 87–99% range, and the calibration-fold-adjacent reference rule cosine $> 0.95$ AND $\text{dHash}_\text{indep} \leq 8$ (the integer cut immediately below the calibration-fold dHash P95 of 9) captures 89.40% of the calibration fold and 91.54% of the held-out fold; the operational classifier rule cosine $> 0.95$ AND $\text{dHash}_\text{indep} \leq 15$ used by the five-way classifier of Section III-K captures still higher rates in both folds (calibration 92.09%, 41,548 / 45,116; held-out 93.56%, 14,344 / 15,332).
-The modest fold gap is consistent with within-Firm-A heterogeneity in replication intensity: the random 30% CPA sample evidently contained proportionally more high-replication CPAs.
-We therefore interpret the held-out fold as confirming the qualitative finding (Firm A is strongly replication-dominated across both folds) while cautioning that exact rates carry fold-level sampling noise that a single 30% split cannot eliminate; the threshold-independent partner-ranking analysis (Section IV-G.2) is the cross-check that is robust to this fold variance.
-
-### 3) Operational-Threshold Sensitivity: cos $> 0.95$ vs cos $> 0.945$
-
-The per-signature classifier (Section III-K) uses cos $> 0.95$ as its operational cosine cut, anchored on the whole-sample Firm A P7.5 heuristic (i.e., 7.5% of whole-sample Firm A signatures lie at or below 0.95; see Section III-H).
-We report a sensitivity check in which this round-number cut is replaced by the slightly stricter calibration-fold P5 rounded value cos $> 0.945$ (calibration-fold P5 = 0.9407, see Table XI).
-Table XII reports the five-way classifier output under each cut.
-
-<!-- TABLE XII: Classifier Sensitivity to the Operational Cosine Cut (All-Sample Five-Way Output, N = 168,740 signatures)
-| Cosine cut | High-confidence | Moderate-confidence | High style consistency | Uncertain | Likely hand-signed |
-|------------|-----------------|---------------------|------------------------|-----------|--------------------|
-| cos > 0.940                | 81,069 (48.04%) | 55,308 (32.78%) | 801 (0.47%) |  31,026 (18.39%) | 536 (0.32%) |
-| cos > 0.945                | 79,278 (46.98%) | 50,001 (29.63%) | 665 (0.39%) |  38,260 (22.67%) | 536 (0.32%) |
-| cos > 0.950 (operational)  | 76,984 (45.62%) | 43,906 (26.02%) | 546 (0.32%) |  46,768 (27.72%) | 536 (0.32%) |
-| cos > 0.960                | 70,250 (41.63%) | 29,450 (17.45%) | 288 (0.17%) |  68,216 (40.43%) | 536 (0.32%) |
-| cos > 0.970                | 60,247 (35.70%) | 14,865 ( 8.81%) | 117 (0.07%) |  92,975 (55.10%) | 536 (0.32%) |
-| cos > 0.985                | 37,368 (22.15%) |  2,231 ( 1.32%) |  10 (0.01%) | 128,595 (76.21%) | 536 (0.32%) |
-
-The dHash band edges ($\leq 5$ for high-confidence, $5 < \text{dHash}_\text{indep} \leq 15$ for moderate-confidence, $> 15$ for style) are held fixed across the grid; only the cosine cut varies. The Likely-hand-signed count is invariant across the grid because it depends only on the all-pairs KDE crossover cosine $= 0.837$.
--->
-
-At the aggregate firm-level, the calibration-fold-adjacent reference dual rule cos $> 0.95$ AND $\text{dHash}_\text{indep} \leq 8$ captures 89.95% of whole Firm A under the 0.95 cut and 91.14% under the 0.945 cut---a shift of 1.19 percentage points.
-The operational classifier rule cos $> 0.95$ AND $\text{dHash}_\text{indep} \leq 15$ used by the five-way classifier of Section III-K captures 92.46% under the 0.95 cut and 93.97% under the 0.945 cut---a shift of 1.51 percentage points.
-
-Reading the wider grid in Table XII: the High-confidence and Moderate-confidence shares shift by less than 5 percentage points across the 0.940-0.950 neighbourhood, while pushing the cosine cut to 0.970 or 0.985 produces qualitatively different classifier behaviour (Moderate-confidence collapses from 26.02% at $0.95$ to 8.81% at $0.97$ and 1.32% at $0.985$, with the displaced mass landing in Uncertain rather than reclassifying out of the corpus).
-The classifier output is therefore robust to small (~0.005-cosine) perturbations of the operational cut but not to wholesale reanchoring at the threshold-estimator outputs of Section IV-D, which is consistent with our reading that those outputs are not classifier thresholds.
-At the per-signature categorization level, replacing 0.95 by 0.945 reclassifies 8,508 signatures (5.04% of the corpus) out of the Uncertain band; 6,095 of them migrate to Moderate-confidence non-hand-signed, 2,294 to High-confidence non-hand-signed, and 119 to High style consistency.
-The Likely-hand-signed category is unaffected because it depends only on the fixed all-pairs KDE crossover cosine $= 0.837$.
-The High-confidence non-hand-signed share grows from 45.62% to 46.98%.
-
-We interpret this sensitivity pattern as indicating that the classifier's aggregate and high-confidence output is robust to the choice of operational cut within a 0.005-cosine neighbourhood of the Firm A P7.5 anchor, and that the movement is concentrated at the Uncertain/Moderate-confidence boundary.
-
-To make the operating-point selection (Section III-K) auditable rather than presented as a single fixed value, Table XII-B reports the capture-vs-FAR tradeoff over the candidate threshold grid spanning the calibration-fold P5 (0.9407), its rounded value (0.945), the operational anchor (0.95), the Firm A Beta-2 forced-fit crossing from Section IV-D.3 (0.977), and the BD/McCrary candidate transition from Section IV-D.2 (0.985).
-For each grid point we report Firm A capture (under both the cosine-only marginal and the operational dual rule cos $> t$ AND $\text{dHash}_\text{indep} \leq 15$ used by the five-way classifier of Section III-K), non-Firm-A capture (the cosine-only marginal in the 108,292 non-Firm-A matched signatures), and inter-CPA FAR with Wilson 95% CI against the 50,000-pair anchor of Section IV-F.1.
-
-<!-- TABLE XII-B: Cosine-Threshold Tradeoff: Capture vs Inter-CPA FAR
-| Cosine cut t | Firm A capture (cos > t) | Firm A capture (cos > t AND dHash_indep ≤ 15) | Non-Firm-A capture (cos > t) | Inter-CPA FAR | Inter-CPA FAR Wilson 95% CI |
-|--------------|--------------------------|------------------------------------------------|------------------------------|---------------|------------------------------|
-| 0.9407 (calibration-fold P5)               | 95.15% (57,518/60,448) | 95.09% (57,482/60,448) | 72.68% (78,710/108,292) | 0.00126 | [0.00099, 0.00161] |
-| 0.945 (calibration-fold P5 rounded)        | 94.02% (56,836/60,448) | 93.97% (56,804/60,448) | 67.51% (73,108/108,292) | 0.00082 | [0.00061, 0.00111] |
-| 0.95 (whole-sample Firm A P7.5; **operational cut**) | **92.51%** (55,922/60,448) | **92.46%** (55,892/60,448) | 60.50% (65,514/108,292) | **0.00050** | [0.00034, 0.00074] |
-| 0.977 (Firm A Beta-2 forced-fit crossing)  | 74.53% (45,050/60,448) | 74.51% (45,038/60,448) | 13.14% (14,233/108,292) | 0.00014 | [0.00007, 0.00029] |
-| 0.985 (BD/McCrary candidate transition)    | 55.27% (33,409/60,448) | 55.26% (33,406/60,448) |  5.73%  (6,200/108,292) | 0.00004 | [0.00001, 0.00015] |
-
-Inter-CPA FAR computed against 50,000 i.i.d. inter-CPA pairs (random seed 42, reproducing the anchor of Section IV-F.1 / Table X). Capture and FAR percentages are exact ratios of the displayed integer counts; gap arithmetic in the surrounding prose is computed from those exact counts and rounded to two decimal places. The dual-rule column is the operational classifier rule of Section III-K; for cuts above the dHash-15 saturation point (Firm A dHash$_\text{indep}$ $> 15$ rate is only 0.17%, Table IX), the dual-rule and cosine-only columns coincide to within the dHash$_\text{indep}$ $> 15$ residual.
--->
-
-Reading Table XII-B, three patterns motivate the choice of $0.95$ as the operating point.
-First, *Firm A capture* on the operational dual rule decays smoothly from 95.09% at $t = 0.9407$ to 55.26% at $t = 0.985$.
-Relaxing the cut from $0.95$ to $0.945$ buys 1.51 percentage points of additional Firm A capture, and to $0.9407$ buys 2.63 percentage points; tightening from $0.95$ to $0.977$ costs 17.96 percentage points and to $0.985$ costs 37.20 percentage points.
-The selected cut at $0.95$ is the strictest cut on this grid at which Firm A capture remains above $90\%$ on the operational dual rule.
-Second, *inter-CPA FAR* is small in absolute terms across the entire candidate grid ($0.00126$ at $0.9407$, falling to $0.00004$ at $0.985$): under any of these operating points the classifier's specificity against random cross-CPA pairs is in the per-mille range or better, so FAR alone does not determine the choice.
-The marginal FAR cost of relaxing from $0.95$ to $0.945$ is $+0.00032$ ($25 \to 41$ false positives per 50,000 pairs) and to $0.9407$ is $+0.00076$ ($25 \to 63$); the marginal FAR savings from tightening to $0.977$ and $0.985$ are $-0.00036$ and $-0.00046$ respectively.
-The FAR savings from going stricter are small in absolute terms compared with the corresponding Firm A capture loss, which makes $0.95$ a balanced operating point on this grid rather than a uniquely optimal one.
-Third, *non-Firm-A capture* (the cosine-only marginal in the 108,292 non-Firm-A signatures) decays from 67.51% at $0.945$ to 60.50% at $0.95$, 13.14% at $0.977$, and 5.73% at $0.985$.
-The Firm-A-minus-non-Firm-A gap widens with strictness through $0.977$ and then contracts (22.41 percentage points at $0.9407$; 26.46 at $0.945$; 31.97 at $0.95$; 61.36 at $0.977$; 49.54 at $0.985$): on the $0.95 \to 0.977$ segment non-Firm-A capture falls faster than Firm A capture in absolute terms ($-47.35$ vs $-17.96$ percentage points), so the widening is dominated by non-Firm-A removal rather than by an intrinsic property of Firm A; on the $0.977 \to 0.985$ segment Firm A capture falls faster than non-Firm-A's already-low residual, so the gap contracts.
-We do *not* read the gap pattern as evidence for a particular cut; it is reported here as cross-firm replication heterogeneity rather than as a selection criterion.
-The operating point at $0.95$ is therefore a defensible---not unique---selection in this neighbourhood, motivated by (i) keeping Firm A capture above $90\%$ on the operational dual rule, (ii) achieving an FAR of $0.0005$ at which marginal further savings from tightening are small relative to the corresponding capture loss, and (iii) preserving the interpretive transparency of the whole-sample Firm A P7.5 reading.
-It is *not* derived from the threshold-estimator outputs of Section IV-D, which the data do not support as classifier thresholds.
-
-The paper therefore retains cos $> 0.95$ as the primary operational cut and reports the 0.945 result of Table XII as a sensitivity check rather than as a deployed alternative; downstream document-level rates (Table XVII) and intra-report agreement (Table XVI) are robust to moderate cutoff shifts within the 0.945--0.95 neighbourhood as long as the same cutoff is applied uniformly across firms.
-
-## G. Additional Firm A Benchmark Validation
-
-Before presenting the three threshold-robust analyses, Fig. 4 summarises the per-firm yearly per-signature best-match cosine distribution that motivates them.
-The left panel reports the mean per-signature best-match cosine within each firm bucket and fiscal year (a threshold-free statistic); the right panel reports the share of each firm-bucket-year with per-signature best-match cosine $\geq 0.95$ (the operational cut of Section III-K).
-Both panels show Firm A above the other Big-4 firms in every year of the 2013-2023 sample, with non-Big-4 firms below all four Big-4 firms throughout, and the cross-firm ordering is stable across the sample period.
-The mean-cosine separation between Firm A and the other Big-4 firms is on the order of 0.02-0.04 throughout the sample (e.g., 2013: Firm A $0.9733$ vs Firm B $0.9498$, Firm C $0.9464$, Firm D $0.9395$, Non-Big-4 $0.9227$; 2023: $0.9860$ vs $0.9668$, $0.9662$, $0.9525$, $0.9346$); the share-above-0.95 separation is wider (2013: Firm A $87.2\%$ vs $61.8\%$, $56.2\%$, $38.5\%$, $27.5\%$).
-This visual is the most direct cross-firm evidence in the paper that Firm A's high-similarity behaviour is firm-specific rather than corpus-wide; the three subsections below decompose this gap along three threshold-free or threshold-robust dimensions.
-
-<!-- FIGURE 4: Per-firm yearly per-signature best-match cosine
-File: reports/figures/fig_yearly_big4_comparison.png (and .pdf)
-Generated by: signature_analysis/30_yearly_big4_comparison.py
-Caption: Per-firm yearly per-signature best-match cosine, 2013-2023.
-(a) Mean per-signature best-match cosine by firm bucket and fiscal year
-(threshold-free). (b) Share of per-signature best-match cosine $\geq 0.95$
-(operational cut of Section III-K). Five lines: Firm A, B, C, D, Non-Big-4.
-Firm A is above the other Big-4 firms in every year; Non-Big-4 is below all
-four Big-4 firms in every year. Per-firm signature counts and exact values
-are in `reports/firm_yearly_comparison/firm_yearly_comparison.{json,md}`.
--->
-
-The capture rates of Section IV-E are an *internal* consistency check: they ask "how much of Firm A does our threshold capture?", but the threshold was itself derived from Firm A's percentiles, so a high capture rate is not surprising.
-To go beyond this circular check, we report three further analyses, each chosen so that the *informative quantity* does not depend on the threshold's absolute value:
-
-- **§IV-G.1 (year-by-year stability).** Holds the cosine cutoff fixed at 0.95 and asks whether the share of Firm A below the cutoff is *stable across years*. The information is in the temporal trend, not in the absolute rate; under a noise-only explanation of the left tail, the share should shrink as scan/PDF technology matured.
-- **§IV-G.2 (partner-level similarity ranking).** Uses *no threshold at all*: every auditor-year is ranked by mean similarity, and we measure Firm A's share of the top decile against its baseline share. The information is in the concentration ratio, which is invariant to the choice of cutoff.
-- **§IV-G.3 (intra-report agreement).** Applies the calibrated classifier and measures whether the *two co-signing CPAs on the same Firm A report* receive the same classifier label, then compares Firm A's intra-report agreement rate to the other firms'. The information is in the *cross-firm gap*; the absolute agreement rate at any one firm depends on the cutoff, but the gap is robust to moderate cutoff shifts as long as the same cutoff is applied uniformly across firms.
-
-Together these three analyses provide threshold-free or threshold-robust evidence that complements the within-sample capture rates of Section IV-E.
-
-### 1) Year-by-Year Stability of the Firm A Left Tail
-
-Table XIII reports the proportion of Firm A signatures with per-signature best-match cosine below 0.95, disaggregated by fiscal year.
-Under the replication-dominated interpretation (Section III-H), this signature-level left-tail rate reflects within-firm heterogeneity in signing outputs at Firm A.
-Consistent with the scope-of-claims framing in Section III-G, we report the rate as a signature-level quantity without disaggregating the underlying mechanism (which may span a minority of hand-signing partners, multi-template replication workflows within the firm, or a combination); partner-level mechanism attribution is not attempted.
-Under the alternative hypothesis that the left tail is an artifact of scan or compression noise, the share should shrink as scanning and PDF-compression technology improved over 2013-2023.
-
-<!-- TABLE XIII: Firm A Per-Year Cosine Distribution
-| Year | N sigs | mean best-match cosine | % below 0.95 |
-|------|--------|-------------|--------------|
-| 2013 | 2,167 | 0.9733 | 12.78% |
-| 2014 | 5,256 | 0.9781 | 8.69% |
-| 2015 | 5,484 | 0.9793 | 7.46% |
-| 2016 | 5,739 | 0.9811 | 6.92% |
-| 2017 | 5,796 | 0.9814 | 6.69% |
-| 2018 | 5,986 | 0.9808 | 6.58% |
-| 2019 | 6,122 | 0.9780 | 8.71% |
-| 2020 | 6,122 | 0.9770 | 9.46% |
-| 2021 | 5,996 | 0.9792 | 8.37% |
-| 2022 | 5,918 | 0.9819 | 6.25% |
-| 2023 | 5,862 | 0.9860 | 3.75% |
--->
-
-The left tail is stable at 6-13% throughout the sample period and shows no pre/post-2020 level shift: the 2013-2019 mean left-tail share is 8.26% and the 2020-2023 mean is 6.96%.
-The lowest observed share is in 2023 (3.75%), consistent with firm-level electronic signing systems producing more uniform output than earlier manual scanning-and-stamping, not less.
-This stability supports the replication-dominated framing: a persistent within-firm heterogeneity component is consistent with a Beta left tail that is stable across production technologies, whereas a noise-only explanation would predict a shrinking share as technology improved.
-
-### 2) Partner-Level Similarity Ranking
-
-If Firm A applies firm-wide stamping while the other Big-4 firms use stamping only for a subset of partners, Firm A auditor-years should disproportionately occupy the top of the similarity distribution among all auditor-years (across all firms).
-We test this prediction directly.
-
-For each auditor-year (CPA $\times$ fiscal year) with at least 5 signatures we compute the mean best-match cosine similarity across the year's signatures, yielding 4,629 auditor-years across 2013-2023.
-Firm A accounts for 1,287 of these (27.8% baseline share).
-Table XIV reports per-firm occupancy of the top $K\%$ of the ranked distribution.
-The per-signature best-match cosine underlying each auditor-year mean is taken over the full same-CPA pool (Section III-G), consistent with the unit-of-analysis framing in Section III-G.
-
-<!-- TABLE XIV: Top-K Similarity Rank Occupancy by Firm (pooled 2013-2023)
-| Top-K | k in bucket | Firm A | Firm B | Firm C | Firm D | Non-Big-4 | Firm A share |
-|-------|-------------|--------|--------|--------|--------|-----------|--------------|
-| 10%   | 462         | 443    | 2      | 3      | 0      | 14        | 95.9% |
-| 20%   | 925         | 877    | 9      | 14     | 2      | 23        | 94.8% |
-| 25%   | 1,157       | 1,043  | 32     | 23     | 9      | 50        | 90.1% |
-| 30%   | 1,388       | 1,129  | 105    | 52     | 25     | 77        | 81.3% |
-| 50%   | 2,314       | 1,220  | 473    | 273    | 102    | 246       | 52.7% |
--->
-
-Firm A occupies 95.9% of the top 10%, 94.8% of the top 20%, 90.1% of the top 25%, and 81.3% of the top 30% of auditor-years by similarity, against its baseline share of 27.8%---a concentration ratio of $3.5\times$ at the top decile, $3.4\times$ at the top quintile, and $2.9\times$ at the top tercile.
-Firm A's share decays monotonically as the bracket widens (95.9% $\to$ 94.8% $\to$ 90.1% $\to$ 81.3% $\to$ 52.7% across top-10/20/25/30/50%), and only at the top 50% does its share approach its baseline; the over-representation is therefore concentrated in the very top of the distribution rather than spread uniformly through the upper half.
-Year-by-year (Table XV), the top-10% Firm A share ranges from 88.4% (2020) to 100% (2013, 2014, 2017, 2018, 2019), showing that the concentration is stable across the sample period.
-
-<!-- TABLE XV: Firm A Share of Top-K Similarity by Year (K = 10%, 20%, 30%)
-| Year | N auditor-years | Top-10% share | Top-20% share | Top-30% share | Firm A baseline |
-|------|-----------------|---------------|---------------|---------------|-----------------|
-| 2013 | 324 | 100.0% (32/32) | 98.4% (63/64) | 89.7% (87/97) | 32.4% |
-| 2014 | 399 | 100.0% (39/39) | 98.7% (78/79) | 82.4% (98/119) | 27.8% |
-| 2015 | 394 | 97.4% (38/39) | 96.2% (75/78) | 84.7% (100/118) | 27.7% |
-| 2016 | 413 | 95.1% (39/41) | 96.3% (79/82) | 81.3% (100/123) | 26.2% |
-| 2017 | 415 | 100.0% (41/41) | 97.6% (81/83) | 83.9% (104/124) | 27.2% |
-| 2018 | 434 | 100.0% (43/43) | 97.7% (84/86) | 80.0% (104/130) | 26.5% |
-| 2019 | 429 | 100.0% (42/42) | 97.6% (83/85) | 78.9% (101/128) | 27.0% |
-| 2020 | 430 | 88.4% (38/43)  | 91.9% (79/86) | 76.0% (98/129)  | 27.7% |
-| 2021 | 450 | 97.8% (44/45)  | 96.7% (87/90) | 81.5% (110/135) | 28.7% |
-| 2022 | 467 | 93.5% (43/46)  | 95.7% (89/93) | 84.3% (118/140) | 28.3% |
-| 2023 | 474 | 97.9% (46/47)  | 94.7% (89/94) | 83.8% (119/142) | 27.4% |
-
-Per-cell entries are "share (k_FirmA / k_total)". Top-25% and top-50% pooled values are reported in Table XIV; per-year top-25/50 columns are omitted from this table to reduce visual width but are reproducible from the supplementary materials.
--->
-
-This over-representation is consistent with firm-wide non-hand-signing practice at Firm A and is not derived from any threshold we subsequently calibrate.
-It therefore constitutes genuine cross-firm evidence for Firm A's benchmark status.
-
-### 3) Intra-Report Consistency
-
-Taiwanese statutory audit reports are co-signed by two engagement partners (a primary and a secondary signer).
-Under firm-wide stamping practice at a given firm, both signers on the same report should receive the same signature-level classification.
-Disagreement between the two signers on a report is informative about whether the stamping practice is firm-wide or partner-specific.
-
-For each report with exactly two signatures and complete per-signature data (84,354 reports total: 83,970 single-firm reports, in which both signers are at the same firm, and 384 mixed-firm reports, in which the two signers are at different firms), we classify each signature using the dual-descriptor rules of Section III-K and record whether the two classifications agree.
-Table XVI reports per-firm intra-report agreement for the 83,970 single-firm reports only (firm-assignment defined by the common firm identity of both signers); the 384 mixed-firm reports (0.46% of the 2-signature corpus) are excluded from the intra-report analysis because firm-level agreement is not well defined when the two signers are at different firms.
-
-<!-- TABLE XVI: Intra-Report Classification Agreement by Firm
-| Firm | Total 2-signer reports | Both non-hand-signed | Both uncertain | Both style | Both hand-signed | Mixed | Agreement rate |
-|------|-----------------------|----------------------|----------------|------------|------------------|-------|----------------|
-| Firm A  | 30,222 | 26,435 | 734 | 0 | 4 | 3,049 | **89.91%** |
-| Firm B  | 17,121 | 9,260  | 2,159| 5 | 6 | 5,691 | 66.76% |
-| Firm C  | 19,112 | 8,983  | 3,035| 3 | 5 | 7,086 | 62.92% |
-| Firm D  | 8,375  | 3,028  | 2,376| 0 | 3 | 2,968 | 64.56% |
-| Non-Big-4 | 9,140  | 1,671  | 3,945| 18| 27| 3,479 | 61.94% |
-
-A report is "in agreement" if both signature labels fall in the same coarse bucket
-(non-hand-signed = high+moderate; uncertain; style consistency; or likely hand-signed).
--->
-
-Firm A achieves 89.9% intra-report agreement, with 87.5% of Firm A reports having *both* signers classified as non-hand-signed and only 4 reports (0.01%) having both classified as likely hand-signed.
-The other Big-4 firms (B, C, D) and non-Big-4 firms cluster at 62-67% agreement, a 23-28 percentage-point gap.
-This 23-28 percentage-point gap in intra-report agreement between Firm A and the other firms is consistent with firm-wide (rather than partner-specific) non-hand-signing practice; we do not claim a sharp discontinuity in the formal sense, since classifier calibration, firm-specific document-production pipelines, and signer-mix differences could each contribute to gap magnitude.
-
-We note that this test uses the calibrated classifier of Section III-K rather than a threshold-free statistic; the substantive evidence lies in the *cross-firm gap* between Firm A and the other firms rather than in the absolute agreement rate at any single firm, and that gap is robust to moderate shifts in the absolute cutoff so long as the cutoff is applied uniformly across firms.
-
-## H. Classification Results
-
-Table XVII presents the final classification results under the dual-descriptor framework with Firm A-calibrated thresholds for 84,386 documents (656 documents excluded from the 85,042-document YOLO-detection cohort because no signature on the document could be matched to a registered CPA; see Table XVII note).
-We emphasize that the document-level proportions below reflect the *worst-case aggregation rule* of Section III-K: a report carrying one stamped signature and one hand-signed signature is labeled with the most-replication-consistent of the two signature-level verdicts.
-Document-level rates therefore represent the share of reports in which *at least one* signature is non-hand-signed rather than the share in which *both* are; the intra-report agreement analysis of Section IV-G.3 (Table XVI) reports how frequently the two co-signers share the same signature-level label within each firm, so that readers can judge what fraction of the non-hand-signed document-level share corresponds to fully non-hand-signed reports versus mixed reports.
-
-<!-- TABLE XVII: Document-Level Classification (Dual-Descriptor: Cosine + dHash)
-| Verdict | N (PDFs) | % | Firm A | Firm A % |
-|---------|----------|---|--------|----------|
-| High-confidence non-hand-signed | 29,529 | 35.0% | 22,970 | 76.0% |
-| Moderate-confidence non-hand-signed | 36,994 | 43.8% | 6,311 | 20.9% |
-| High style consistency | 5,133 | 6.1% | 183 | 0.6% |
-| Uncertain | 12,683 | 15.0% | 758 | 2.5% |
-| Likely hand-signed | 47 | 0.1% | 4 | 0.0% |
-
-Per the worst-case aggregation rule of Section III-K, a document with two signatures inherits the most-replication-consistent of the two signature-level labels.
-The 84,386-document cohort excludes 656 documents (relative to the 85,042 YOLO-detected cohort of Table III) for which no signature could be matched to a registered CPA: the per-document classifier requires at least one CPA-matched signature so that a same-CPA best-match similarity is defined. The exclusion is definitional rather than discretionary; typical causes are auditor's-report-page formats deviating from the standard two-signature layout, or OCR returning a printed CPA name not present in the registry.
--->
-
-Within the 71,656 documents exceeding cosine $0.95$, the dHash dimension stratifies them into three distinct populations:
-29,529 (41.2%) show converging structural evidence of non-hand-signing (dHash $\leq 5$);
-36,994 (51.7%) show partial structural similarity (dHash in $[6, 15]$) consistent with replication degraded by scan variations;
-and 5,133 (7.2%) show no structural corroboration (dHash $> 15$), suggesting high signing consistency rather than image reproduction.
-A cosine-only classifier would treat all 71,656 identically; the dual-descriptor framework separates them into populations with fundamentally different interpretations.
-
-### 1) Firm A Capture Profile (Consistency Check)
-
-96.9% of Firm A's documents fall into the high- or moderate-confidence non-hand-signed categories, 0.6% into high-style-consistency, and 2.5% into uncertain.
-This pattern is consistent with the replication-dominated framing: the large majority is captured by non-hand-signed rules, while the small residual is consistent with the within-firm heterogeneity implied by the dip-test-confirmed unimodal-long-tail shape of Firm A's per-signature cosine distribution (Section IV-D.1) and the 7.5% signature-level left tail (Section III-H).
-The near-zero "likely hand-signed" rate (4 of 30,226 Firm A documents, 0.013%; the 30,226 denominator is documents with at least one Firm A signer under the 84,386-document classification cohort, which differs from the 30,222 single-firm two-signer subset of Table XVI by 4 mixed-firm reports excluded from the firm-level intra-report comparison) indicates that the within-firm heterogeneity implied by the 7.5% signature-level left tail (Section IV-D) does not project into the lowest-cosine document-level category under the dual-descriptor rules; it is absorbed instead into the uncertain or high-style-consistency categories at this threshold set.
-We note that because the non-hand-signed thresholds are themselves calibrated to Firm A's empirical percentiles (Section III-H), these rates are an internal consistency check rather than an external validation; the held-out Firm A validation of Section IV-F.2 is the corresponding external check.
-
-### 2) Cross-Firm Comparison of Dual-Descriptor Convergence
-
-Among the 65,514 non-Firm-A signatures with per-signature best-match cosine $> 0.95$, 42.12% have $\text{dHash}_\text{indep} \leq 5$, compared to 88.32% of the 55,922 Firm A signatures meeting the same cosine condition---a $\sim 2.1\times$ difference that the structural-verification layer makes visible.
-The Firm A denominator (55,922) matches Table IX exactly: both Table IX and the cross-firm decomposition define Firm A membership via the CPA registry (`accountants.firm`), and the cross-firm analysis additionally requires a non-null independent-min dHash record, which all 55,922 Firm A cosine-eligible signatures have in the current database.
-This cross-firm gap is consistent with firm-wide non-hand-signing practice at Firm A versus partner-specific or per-engagement replication at other firms; it complements the partner-level ranking (Section IV-G.2) and intra-report consistency (Section IV-G.3) findings.
-Reproduction artifact for these counts is listed in Appendix B.
-
-## I. Ablation Study: Feature Backbone Comparison
-
-To validate the choice of ResNet-50 as the feature extraction backbone, we conducted an ablation study comparing three pre-trained architectures: ResNet-50 (2048-dim), VGG-16 (4096-dim), and EfficientNet-B0 (1280-dim).
+## 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. The accountant-level dip-test rejection reported in Table V is, per §III-I.4, fully attributable to between-firm location shifts and integer mass-point artefacts rather than to within-population bimodality; the 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 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 deployed box-rule less-replication-dominated rate | $+0.9627$ | $< 10^{-248}$ |
+| Reverse-anchor cosine percentile vs deployed 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 deployed 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 $\kappa$ |
+|---|---|
+| deployed binary high-confidence box rule (cos $> 0.95$ AND dHash $\leq 5$) vs per-CPA K=3 hard label | 0.662 |
+| deployed 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.) 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 ($5 < \text{dHash} \leq 15$) retains its prior calibration and capture-rate evidence (supplementary materials; cross-referenced in §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; screening 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 conservative hard-positive 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 a conservative hard-positive subset for image replication. 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 |
+|---|---|---|---|
+| deployed 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 deployed 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). The reverse-anchor cut is chosen by *prevalence calibration* against the box rule's overall replicated rate of $49.58\%$ across Big-4 signatures; this is a documented limitation since no signature-level hand-signed ground truth exists to permit direct ROC optimisation.
+
+## I. Inter-CPA Pair-Level Coincidence Rate
+
+The metric reported here is the inter-CPA pair-level coincidence rate (ICCR). It is the per-pair rate at which two signatures from different CPAs satisfy the deployed rule. We do not label it as a False Acceptance Rate because (a) FAR has a biometric-verification meaning that requires ground-truth negative labels, and (b) the inter-CPA negative-anchor assumption is partially violated by within-firm cross-CPA template-like collision structures (§III-L.4 cross-firm hit matrix).
+
+A corpus-wide spike on $\sim 50{,}000$ inter-CPA pairs gives a per-comparison rate of $0.0005$ (Wilson 95% CI $[0.0003, 0.0007]$) at the cosine cut $0.95$. The Big-4-scope spike at higher sample size ($5 \times 10^5$ inter-CPA pairs) replicates this number, adds the structural dimension (dHash), and adds joint-rule rates; the §III-L.1 numbers are referenced rather than duplicated here, and the consolidated ICCR calibration appears in §IV-M Tables XXI–XXVI.
+
+## J. Five-Way Per-Signature + Document-Level Classification Output
+
+This section reports the five-way per-signature + document-level worst-case classifier output on the Big-4 sub-corpus. See §III-H.1 for the five-way category definitions and the cosine and dHash cuts; calibration is in §III-L.
+
+**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 XVII: 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 XVII). 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 worst-case rule (HC > MC > HSC > UN > LH; §III-H.1), applied to the Big-4 sub-corpus.
+
+**Table XVI.** 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 $5 < \text{dHash} \leq 15$) retains its prior calibration (supplementary materials); it is **not separately re-characterised by Scripts 38–40**, which checked only the binary high-confidence rule (cos $> 0.95$ AND dHash $\leq 5$). The moderate-band cuts are not re-derived 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 capture-rate calibration evidence for the moderate band is reported in the supplementary materials 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 XVII.** 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 XVI above.**
+
+## K. Full-Dataset Robustness (light scope)
+
+This section reports the 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 + deployed operational-rule per-CPA less-replication-dominated rate analysis, sufficient to demonstrate that the K=3 + deployed-rule convergence reproduces at the wider scope. The §III-H.1 five-way classifier and the §IV-G LOOO analyses are not re-run at the full scope. The five-way moderate-confidence band retains its prior calibration (supplementary materials; §IV-J).
+
+**Table XVIII.** 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 XIX.** Spearman rank correlation between K=3 P(C1) and deployed operational less-replication-dominated rate, Big-4 sub-corpus vs full dataset.
+
+| Scope | $n$ CPAs | Spearman $\rho$ (P(C1) vs deployed 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 deployed 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 + deployed-rule 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 primary methodology is restricted to Big-4 by design (§III-G item 4).
+
+## L. Ablation Study: Feature Backbone Comparison
+
+To support the choice of ResNet-50 as the feature extraction backbone, we conducted an ablation study comparing three pre-trained architectures: ResNet-50 (2048-dim), VGG-16 (4096-dim), and EfficientNet-B0 (1280-dim).
 All models used ImageNet pre-trained weights without fine-tuning, with identical preprocessing and L2 normalization.
-Table XVIII presents the comparison.
+The comparison summary is reported in the supplementary materials (backbone-ablation table; not the same table as Table XIX in this section, which reports Big-4 vs full-dataset Spearman drift in §IV-K).
 
-<!-- TABLE XVIII: Backbone Comparison
+<!-- BACKBONE ABLATION TABLE (rendered in supplementary materials):
 | Metric | ResNet-50 | VGG-16 | EfficientNet-B0 |
 |--------|-----------|--------|-----------------|
 | Feature dim | 2048 | 4096 | 1280 |
@@ -484,11 +319,114 @@ single closest match from the same CPA.
 -->
 
 EfficientNet-B0 achieves the highest Cohen's $d$ (0.707), indicating the greatest statistical separation between intra-class and inter-class distributions.
-However, it also exhibits the widest distributional spread (intra std $= 0.123$ vs. ResNet-50's $0.098$), resulting in lower per-sample classification confidence.
+However, it also exhibits the widest distributional spread (intra std $= 0.123$ vs. ResNet-50's $0.098$), i.e., a wider descriptor dispersion per signature.
 VGG-16 performs worst on all key metrics despite having the highest feature dimensionality (4096), suggesting that additional dimensions do not contribute discriminative information for this task.
 
 ResNet-50 provides the best overall balance:
 (1) Cohen's $d$ of 0.669 is competitive with EfficientNet-B0's 0.707;
-(2) its tighter distributions yield more reliable individual classifications;
-(3) the highest Firm A all-pairs 1st percentile (0.543) indicates that known-replication signatures are least likely to produce low-similarity outlier pairs under this backbone; and
+(2) its tighter distributions yield more stable descriptor behaviour at the per-signature level;
+(3) the highest Firm A all-pairs 1st percentile (0.543) indicates that Firm A replication-dominated signatures are least likely to produce low-similarity outlier pairs under this backbone; and
 (4) its 2048-dimensional features offer a practical compromise between discriminative capacity and computational/storage efficiency for processing 182K+ signatures.
+
+## M. Anchor-Based ICCR Calibration Results
+
+This section consolidates the empirical results that support the §III-L anchor-based threshold calibration framework.
+
+### 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 deployed 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).
+
+| Threshold | Per-comparison ICCR | 95% Wilson CI |
+|---|---|---|
+| cos $> 0.945$ (alternative operating point from supplementary calibration evidence) | $0.00081$ | $[0.00073, 0.00089]$ |
+| cos $> 0.95$ (deployed 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 $\leq 5$ (deployed 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) | $0.00014$ | $[0.00011, 0.00018]$ |
+| Joint: cos $> 0.95$ AND dHash $\leq 4$ (any-pair) | $0.00011$ | $[0.00008, 0.00014]$ |
+
+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 is consistent with the corpus-wide spike of §IV-I (per-comparison rate $0.0005$). The dHash row and joint row are reported here for the first time on this corpus.
+
+### 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 XVI'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 documents that an alphabetically-ordered tie-break assigns to Firm C; full implementation detail in the supplementary materials). The four per-firm denominators here therefore sum to the full $75{,}233$, whereas Table XVI'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 deployed HC threshold (Script 46)
+
+**Table XXVI.** Local-gradient / median-gradient ratio at deployed 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 $0.3856$ ($38.6$ pp) per-signature and $0.4431$ ($44.3$ pp) per-document; this excess is reported as an observed same-CPA-pool excess under §III-M caveats, not as a presumed true-positive rate and not attributed to within-CPA handwriting repeatability.

paper/v13_build/CANONICAL_NUMBERS_rev9.1.md

@@ -0,0 +1,95 @@
+# 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.

paper/v13_build/paper_v13_filled.md

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+# Anchor-Calibrated, Label-Free Screening of Non-Hand-Signed Signatures in Large-Scale Audit Reports
+
+*(Authors removed for double-blind review)*
+
+## Abstract
+
+An audit report must carry each certifying accountant's signature as the mark of an individual act of endorsement, yet in digital workflows a saved image can instead be pasted onto many reports — by stamping or by an automated signing system — producing what we term non-hand-signed signatures. The signer is genuine; the question is whether signing occurred for each report, and at archive scale this carries no labels. We present a label-free screening system and apply it to 86,071 Taiwanese statutory audit reports (2013–2023), within which the four largest firms contribute 150,442 analyzable signatures. The system finds the signature page, detects signatures, extracts deep features, and computes two similarities to the same accountant's other signatures — a style cosine and a perceptual-hash (dHash) structural distance — on the logic that a consistent hand keeps style high while structure varies, whereas a reused image keeps both extreme. With no labels and no natural gap in the data (a unimodality test gives median p = 0.35 once firm effects and the hash's integer steps are removed), no cutoff can be learned; instead we calibrate a five-way rule by how often it fires by chance between unrelated accountants in a clean reference group (non-Firm-A firms, 2013–2019), where the strict rule fires on about 1.2% of reports and a looser advisory band on about 17.5%. Held out as a known-positive benchmark — one firm independently described by interviews as a stamping firm (a confirmatory check, not a blind test) — it fires the strict rule on 82% of its own signatures against 24–35% elsewhere, with its cross-firm rate at the clean floor, so the signal is entirely within-firm; the contrast survives pool-size stratification and accountant-level resampling, and 262 byte-identical signatures are direct evidence of reuse. The screen thus locates where reuse concentrates and confines human review to exceptions. We report a between-accountant specificity proxy — not a within-accountant error rate, nor a bound on one — and we neither separate signing practice from a firm's imaging pipeline nor label any single signature. Calibrated on a large Chinese-signature corpus with script-agnostic descriptors, it serves as an operator-set reference point for comparable pipelines.
+
+**Keywords:** signature analysis, document forensics, perceptual hashing, deep features, unsupervised calibration, audit reports, anchor-based screening.
+
+## I. Introduction
+
+An audit report is one of the main ways a company is held accountable to investors, and the certifying accountant's signature is the visible sign that a named professional takes responsibility for it. In Taiwan, the Certified Public Accountant Act and the attestation rules of the Financial Supervisory Commission require certifying CPAs to put their signature or seal on each audit report [1]. The law accepts either a handwritten signature or a seal, but the point of the requirement is the same in both cases: the mark on each report should stand for a deliberate, individual act of endorsement for that particular engagement [2].
+
+Going digital makes that harder to guarantee. Because reports are now created, sent, and stored as electronic files, it is easy to copy an accountant's saved signature image onto many reports instead of signing each one. This can happen in two ways: a staff member can overlay a scanned signature onto the finished report (a stamping workflow), or a firm-wide electronic-signing system can do the same step automatically. We call signatures produced either way non-hand-signed. To fix the term operationally before any method is introduced: a signature is *non-hand-signed* when the mark on the report is a reproduction of a stored signature image rather than a fresh signing act for that engagement. This spans manual overlay (stamping), automated firm-wide e-signing that pastes a saved image, and proxy application of a stored image by another person. It excludes a freshly handwritten signature or a hand-applied seal made for that specific report (the in-scope "hand-signed" case), and it is distinct from a cryptographic digital signature, which binds a document mathematically rather than reproducing an image. The criterion is therefore the visible outcome — image reuse, the same stored image recurring across reports — not the intent, the actor, or the legal status, and it is this outcome that our two measures and five categories track. The worry is not about legality; it is about meaning. A single image pasted onto hundreds of reports may not carry the individual endorsement the rule assumes — a concern the literature on signatures connects to behavior, and, in auditing specifically, to rules that name and identify the engagement partner [31], [32], [33]. This is also why the problem is not forgery: a non-hand-signed signature reuses the real signer's own image, and at scale no reader can see the difference.
+
+That difference matters for the method too. Almost all work on offline signature analysis is about forgery — deciding whether a questioned signature was really written by the person it claims to be [3]–[8]. In our setting the identity is not in doubt; the accountant is genuine. What we want to know is whether the person actually signed each report, or whether one signing was copied as an image. This removes the need to model clever forgers, but it adds a new difficulty: we must separate a person who signs consistently from a reused image. Someone who signs in a very steady hand will produce signatures that look alike year after year; a process that reuses one stored file will produce signatures that are structurally identical. The method has to tell these two cases apart.
+
+Two facts make the obvious approach — pick a similarity cutoff and call everything above it a copy — unworkable, and they shape our design. First, archives like ours have no labels at the level of individual signatures: no signature is marked as "definitely hand-signed" or "definitely reused." Without such labels, any cutoff we choose has unknown error rates; we cannot measure how often it would wrongly flag a genuine signature or miss a reused one. Second, even setting labels aside, the data themselves do not contain a natural cutoff. As we show in Section V, the raw numbers look at first as if they split into two groups, but that appearance comes from differences between firms and from the fact that the hash takes only whole-number values; once we remove those two effects, the distribution is a single smooth spread, not two clusters. You cannot read a dividing line off a distribution that has no gap, and you cannot test a line against labels that do not exist. So the method must get its cutoff some other way.
+
+Our two similarity measures are chosen precisely to expose the distinction the problem turns on. For each signature we compute two numbers against the same accountant's other signatures: a cosine similarity on deep ResNet-50 features, and an independent perceptual hash (dHash) distance. They carry different information. Cosine similarity measures overall style, and it is high both when an image is reused and when a person signs consistently. The dHash distance measures structure almost pixel by pixel, and a very small distance is the sign most specific to a reused image. But neither measure is enough on its own. Cosine alone over-flags a steady hand, because consistent signing also keeps it high. dHash alone has the opposite weakness: it is brittle to how an image is captured — a reused signature that has been re-scaled, re-cropped, or re-compressed can show a larger dHash distance and slip past a structure-only test — and a small dHash distance carries no meaning between two signatures whose styles do not match in the first place. The two are complementary precisely because they fail in different directions: cosine first establishes that the styles match, which catches reuse even when the image has been mildly altered, and dHash then asks whether the match is also near-identical in structure, which is what separates a reused image from a merely steady hand. A single similarity number blurs these two cases; two measures keep them apart. The implication between them runs one way only: a near-identical structure (a tiny dHash) forces a high cosine, but a high cosine in no way implies a near-identical structure — which is why the two-measure plane cannot be collapsed onto either single axis. This complementarity also shapes the rule (Section III-D): because a small dHash distance is only meaningful once cosine is already high, the structural cut subdivides the high-cosine cases rather than the low-cosine ones. This is the heart of the design.
+
+On this basis we build and study a complete screening system. The pipeline takes raw PDF reports through four steps — find the signature page, detect each signature, turn it into features, and compute the two similarities — and sorts each signature into one of five categories. Because there is no natural cutoff to read off the data and no labels to learn one from, we instead measure how often the rule fires by chance between unrelated accountants in a clean reference group. That chance rate is a *between-accountant* coincidence rate, which we treat as a proxy for the rule's specificity: it gives us a principled way to choose an operating point, and — just as important — it tells us what each category's flag is worth among unrelated accountants. It is not the within-accountant false-positive rate (how often a genuine consistent hand-signer would fire the rule), which the reuse question would ideally use but which no labels let us estimate (Section III-E).
+
+What is the screen for? Two things. Run over a large archive, it discovers where reuse concentrates — which firms, which periods — without being told where to look. And it keeps human review at the scale of exceptions. In a reuse-dominated population (a stamping firm, a firm with an electronic-signing system), the high-confidence tier routes most signatures directly to a high-specificity candidate list, and the small residual goes through a defined review protocol (specified in Section IV-B) — side-by-side overlay inspection, secondary image-artifact checks, and bounded per-accountant sampling — that also accumulates labels for later calibration. In a mixed population, where hand-signing and informal stamping coexist, the ambiguous middle is larger, and the same disposition machinery delivers the same promise one level up, at the accountant: the low-specificity advisory band is demoted rather than worked, accountant-level scores concentrate attention on the few high-ranked or mixed cases, and byte-identity hits supply proof where proof exists, confirming that an accountant's stored image is in circulation. What the screen does not deliver there — and we say so plainly when we report the category proportions (Section IV-B) — is a per-signature verdict for the ambiguous middle. In every case the output is bounded triage, not a verdict on any single signature.
+
+The Taiwan setting suits this study well. The Market Observation Post System offers a large, standardized, public collection of statutory audit reports, each with the same two-signature format, which makes large-scale extraction practical. In addition, anonymized interviews with certifying partners and signing-system staff at all four firms give us institutional facts about how each firm signs and about when each firm adopted a formal electronic-signing system — adoptions that were staggered from 2020 onward. This gives the study a natural before-and-after structure in time, and outside information against which to read the firm-level results (Section III-A).
+
+We make four contributions:
+
+1. An end-to-end screening pipeline that turns raw audit-report PDFs into operational risk strata for hundreds of thousands of signatures.
+2. A dual descriptor that separates style consistency from image reproduction — a distinction a single similarity measure blurs.
+3. The methodological core: a label-free way to *construct and characterize* a screening operating point when no signature-level labels exist — the question this paper is really organized around. With neither a natural cutoff in the data nor labels to learn one from, we set a tunable rule by measuring how often it fires by chance in a clean reference group, and we say plainly what that measure can and cannot support. This is the part we expect to transfer beyond the present setting; we demonstrate and stress-test it at scale on audit signatures rather than claiming it as a finished, fully general framework. Concretely the result is not only a calibration method but a concrete operating point — the high-confidence rule and its measured specificity proxy — that practitioners working with comparable Chinese-signature image pipelines can use as a starting reference (not transplant unchanged, since the proxy is conditional on a similar preprocessing and reference-group setup), together with a defined disposition path for the ambiguous middle (calibrated demotion of the low-specificity band, aggregation to the accountant level, byte-identity escalation, and a bounded manual protocol) that keeps human review at the scale of exceptions.
+4. A demonstration on Chinese signatures, a structurally complex and comparatively under-served script for signature analysis. Because our descriptors work on the image rather than on script-specific strokes, the approach does not depend on Latin-script assumptions and is a candidate for other scripts.
+
+The paper is organized to move from the problem to the evidence. Section II reviews related work and states the gap. Section III describes the study design — the data split, the pipeline, the five-way rule, and the calibration logic — and explains why each piece is built the way it is. Section IV reports the results: the calibration baseline, which category needs human review, and the held-out benchmark on Firm A. Section V collects supporting analyses, including the diagnostic showing that no natural cutoff exists. Section VI concludes.
+
+## II. Related Work and Research Gap
+
+Why reproduction matters: signatures carry symbolic weight. A signature is valuable mainly as a symbol — it stands for the signer's identity and intent. Recent experiments show that this symbolism does not survive a change in how one signs. In studies that take the reader's point of view, Chou [41] finds that electronic signatures give a weaker sense of the signer's presence than handwritten ones, and that readers therefore judge an e-signed document as less valid and expect more non-compliance; across five kinds of e-signature (a checked box, a PIN, an avatar, a typed name, and a software-generated signature), the software-generated kind felt the most "present" of the electronic options but still less than a handwritten signature. In studies that take the signer's point of view, Chou [42] finds that electronic signatures give a weaker sense of self-presence — the signer's felt attachment to the mark — and that this, in turn, makes people more willing to cheat; the work singles out signing by proxy (an autopen) as cutting the tie between the document and the signer. These results matter for us because the practice we detect — a stored signature image laid onto a report by staff or by software — is, in this scheme, one of the lowest-presence modes: it looks like a software-generated signature and is executed like a proxy signature, because the accountant performs no signing act for the report. These effects are robust rather than one-off: in a pre-registered, multi-study replication with meta-analysis, Tzelios and Williams [43] reproduce Chou's reader-side result — an avatar e-signature lowers the sense of the signer's presence and raises the expectation that the contract will be breached. In their general discussion the same authors point to accounting as a next setting — noting the spread of online tax filing and asking how digital signatures affect an evaluator's assessment of the legitimacy of claims, while cautioning that accounting documents may prove less sensitive to signature form than legal ones. We read that call precisely: their "auditors" are the readers of digitally signed filings — those who evaluate the claims — not the certifying accountants who sign. The signer-side question in auditing — what it means when the certifying professional's own signature is reproduced rather than performed — is not addressed in that literature. Both questions, reader-side and signer-side, presuppose the same missing capability: a way to measure non-hand-signing at scale. The lesson we draw is not that non-hand-signing harms audit quality — that is a separate question we leave to a companion study (Section VI) — but that whether it matters is a real question, and one nobody can study without first being able to measure non-hand-signing at scale.
+
+Signature analysis to date is about forgery, not reuse. The obvious toolkit for that measurement is signature analysis, but its main concern is the wrong one for us. Bromley et al. [3] introduced the Siamese network that still anchors the field; SigNet [4] extended it to compare writers it had never seen; Kao and Wen [5] worked from a single genuine sample; TransOSV [6] brought in a Vision Transformer; and meta-learning has been used to cut the effort of enrolling new signers [16]. All of this targets imitation by another hand, so it learns to tell different people apart. Our task is the opposite: spotting reuse of the genuine signer's own image, which lives in the most-similar tail of one person's signatures. The closest idea uses reference examples to set a sensible cutoff [8], but on benchmark data with known genuine references — whereas our archive has no signature-level labels at all. This body of work is also overwhelmingly built on Western, Latin-script signatures; non-Latin scripts such as Chinese are comparatively under-served, and reported accuracies for them are lower [44]. Chinese signatures are structurally distinctive — many strokes, with wide variation between writers — and the forensic literature on them is thin; the closest precedent, Chen [45], analyzes Chinese signatures with a maximum-similarity-to-same-class statistic that directly parallels our use of the maximum cosine to the same accountant. Our descriptors, however, work on the image rather than on script-specific strokes, so the method itself does not depend on the script.
+
+Image-duplication and document forensics: useful parts, different setting. A second line of work looks directly at duplicated images. Copy-move detection finds regions copied within an image [11], and Abramova and Böhme [10] adapted it to scanned documents, noting that ordinary repeated characters confuse the standard methods. Self-supervised copy detection on everyday photos [13] shows that pretrained CNN features with cosine similarity make a strong baseline for spotting near-duplicates. Closest in pipeline terms, Woodruff et al. [9] pull signatures from corporate filings for anti-money-laundering work — but to group signatures by who signed them, not to detect one signer's image being reused across documents. The building blocks exist; the specific setting — one signer's image reused across many scanned financial reports — does not seem to have been addressed.
+
+Deep features and perceptual hashing as ready-made parts. Features from a pretrained CNN transfer well to document images without any retraining [20], [21], and perceptual hashes are built to survive the print–scan–rasterize cycle [27]. Jakhar and Borah [12] show that combining a perceptual hash with deep features beats either one alone for near-duplicate detection — a direct precedent for our two-measure design, though they work on natural images rather than signatures.
+
+The recurring obstacle is the missing label. None of these lines solves the problem we face, because real archives carry no signature-level ground truth, and a similarity screen without it falls back on a hand-chosen cutoff whose error behavior is unknown. (The statistical tools we use to test for a natural cutoff and to describe the rule once we find none are introduced where they are used, in Section III and Section V, since they are part of our method rather than prior work on this problem.)
+
+The gap, and our contribution. Two gaps follow. First, large-scale screening for non-hand-signed auditor signatures has not been done, even though there is good reason (above) to think it matters. Second, and more broadly, similarity-based screening has no principled way to set and describe an operating point when labels are missing. Our contribution sits exactly here: a label-free calibration that replaces both the arbitrary cutoff and the unavailable labeled validation with a chance-rate measured in a clean reference group, together with the pipeline and dual descriptor that make the screening possible (contributions listed in Section I).
+
+It is worth being explicit about a design choice this implies, because it is easily mistaken for a missing component. A natural reflex would be to learn the discriminator — to fine-tune a Siamese or contrastive network to separate reused from hand-signed signatures. We deliberately do not, and the reason is not expedience but the defining constraint of the setting: supervised metric learning requires labeled pairs (genuine-vs-reused), which is exactly the ground truth the archive does not contain. Training such a network would require either fabricating labels or importing them from a different distribution (e.g., forgery datasets), reintroducing the unverifiable assumptions our calibration is designed to avoid; the resulting boundary would again have unknown error behavior on the real archive. Label-free operation is therefore not a weaker version of a supervised method but the only honest option when no labels exist, and the contribution is correspondingly methodological — a way to set and *characterize* an operating point by measured chance behavior — rather than a new network architecture. Off-the-shelf pretrained features are used precisely because they introduce no task-specific supervision; supervised fine-tuning is the right tool once a labeled sample exists, which is why we frame the review protocol's first run (Section IV-B, Section V) as the route to that sample and to any future supervised validation.
+
+## III. Research Background and Study Design
+
+This section explains how the study is built and why. We report no computed numbers here; all results appear in Section IV.
+
+### A. Institutional Background
+
+To pin down the signing practices that we need in order to interpret the results, we held semi-structured interviews with certifying partners and signing-system staff at all four firms in the study.¹ Three points do real work later. First, all four firms allow handwritten signing but none require it. Second, formal firm-wide electronic signing or sealing systems were adopted on staggered dates from 2020 onward. Third, one firm — which we call Firm A throughout — has used scanned-image overlay stamping as its usual practice since at least 2013. We use these facts only as background, not as labels for individual signatures: they guide how we split the data below and how we read the firm-level results in Section IV-C, but they do not tell us the status of any single signature. A further caution applies to how the interviews are used as corroboration. They are self-reported, anonymized, and not independently reproducible, so when the screen's firm-level output agrees with them (Section IV-C) that agreement is evidence of consistency with domain knowledge, not a measurement of the screen's accuracy or recall — quantifying those would require signature-level labels, which the archive does not provide. To be unambiguous about their role: the interviews are used only to *contextualize* the firm-level findings and are not treated as validation. They are corroborative, not confirmatory, and not independently reproducible; the empirical claims of this paper rest on the calibration and the byte-identical evidence, which stand without them. Their one load-bearing use is to motivate why Firm A is read as a known-positive benchmark rather than a blinded test (Section IV-C) — a framing that, if anything, lowers the evidentiary status we claim for that firm. The practical implication is that the years before the formal systems (before 2020) are the right "normal" period to use for calibration.
+
+> ¹ Footnote — institutional detail. The interviews were conducted under institutional research-ethics approval and are reported in anonymized, aggregated form; firms are labeled A–D and no individual can be identified. The formal systems were reported to have been adopted at roughly one firm in early 2020, one in 2021, and one in late 2022 (exact firm-level dates are withheld for anonymity; see supplementary materials). Interviewees attributed this timing partly to the COVID-19 pandemic, which forced remote review and signing, and to firm-wide paperless and environmental (ESG) initiatives — both of which accelerated the move to formal electronic signing at Firms B/C/D. For Firm A, the reported workflow is that the certifying accountant approves the finished report electronically, after which the print room overlays the accountant's stored seal or signature image onto the PDF and prints it; the stored image is rarely changed, and although handwritten signing is allowed it is reported to be very rare, and rarer over time. Before the formal systems, the other firms' practice varied: some used informal scan- or photocopy-based stamping alongside handwritten signing, and at least one reported mostly handwritten signing before its system. The property the calibration relies on (Section III-E) is that, in the pre-2020 baseline firms, different accountants did not share a common template — not that every signature was handwritten.
+
+### B. Data and Analysis Design
+
+The corpus is all retrievable Taiwan statutory audit reports for fiscal years 2013–2023; the four largest firms (A–D) form the primary analysis sample, and non-Big-4 firms enter only in the crossover-scope robustness check (Section V-C), never in the calibration or the headline rates. Signatures are extracted as described in Section III-C. To be precise about the headline denominator, since it recurs throughout: "150,442 analyzable signatures" means exactly those Big-4 signatures that are valid and have both similarity measures computed, assigned by each accountant's registered firm (Firm A 60,448, Firm B 34,248, Firm C 38,613, Firm D 17,133). We then split the Big-4 corpus by firm and by period, giving each part a distinct job (Fig. 1):
+
+- Calibration (the clean reference group): Firms B/C/D, 2013–2019.
+- Held-out benchmark 1: Firm A, 2013–2023 (a known positive, not a blinded test).
+- Held-out test 2 (secondary): Firms B/C/D, 2020–2023.
+
+We explain the reason for each part in Section III-E. The key idea is simple: we calibrate only on the clean cell — the non-Firm-A firms in the years before formal systems — and test everything else against it. No numbers appear here; the calibration results start in Section IV-A.
+
+![](figures/fig1.png)
+
+*Figure 1. The data split. Rows are Firms A–D; columns are 2013–2019 and 2020–2023. The B/C/D × 2013–2019 cells are the clean calibration group; Firm A (both periods) is held-out benchmark 1 (a known positive); B/C/D × 2020–2023 is the secondary held-out test. We calibrate only on the clean cell and test everything else against it.*
+
+### C. Pipeline
+
+The pipeline turns a raw PDF report into labeled signatures in five steps (Fig. 2).
+
+Finding the signature page. A vision-language model [24], [35] scans only the first quarter of each document — where the auditor's report page reliably sits — and stops as soon as it finds the page.
+
+Detecting signatures. A YOLOv11n detector [25], [34], trained on 500 hand-labeled signature pages (425 for training, 75 for validation; 100 epochs; started from COCO weights), draws a box around each signature. A region counts as a signature if it holds handwritten content that belongs to a personal signature, even where it overlaps an official stamp. A red-stamp removal step (filtering in HSV color space) then strips away overlapping red seals, leaving the handwritten part.
+
+Turning signatures into features. Each detected signature is passed through an ImageNet-pretrained ResNet-50 [26] used as a fixed feature extractor — we take the 2,048-number output of its global-average-pooling layer and drop the classification head. We resize each image to 224×224 while keeping its aspect ratio (padding with white), apply the standard ImageNet normalization, and scale the feature vector to unit length, so that cosine similarity is just the dot product. We use these off-the-shelf features rather than fine-tuning the network, for three reasons: the task is comparing similarity, not classifying; ImageNet features are known to transfer well to document images [20], [21]; and not fine-tuning avoids the risk of learning quirks of our particular dataset. The backbone choice is checked in Section V-C.
+
+Assigning each signature to an accountant. Each signature is matched to a registered accountant by its position on the page (first or second) against the official registry. Signatures we cannot match are left out of the same-accountant comparisons, because the "most similar signature by the same accountant" measure has no meaning without an assigned accountant.
+
+(Detection accuracy, signature counts, match rates, and the resulting analysis sample are reported in Section IV-A.)
+
+![](figures/fig2.png)
+
+*Figure 2. The screening pipeline. A raw PDF passes through page-finding (a vision-language model), signature detection (YOLOv11) with red-stamp removal, feature extraction (ResNet-50), the two per-signature similarities (cosine for style; the smallest dHash to the same accountant for structure), and a five-way label.*
+
+### D. The Two Similarity Measures and the Five-Way Rule
+
+For each signature we compute two numbers, both against the same accountant's other signatures: cos, its highest cosine similarity to another of that accountant's signatures, and dHash, its smallest perceptual-hash distance to another of them. As explained in Section I, the point of using two measures is to separate two things that one measure blurs. A high cos means the signatures look alike in style, which happens both when an image is reused and when a person signs consistently. A small dHash means the signatures are alike almost pixel for pixel, which is the sign most specific to a reused image. Together they are far more telling than either alone: a steady hand gives a high cos but a dHash that still varies, while a reused image gives a high cos and a tiny dHash.
+
+The rule places each signature in one of five categories, with cosine acting as the primary gate and the structural (dHash) distance refining only the cases where cosine is already high. Each name states the screening hypothesis its region suggests — a candidate reading, not a confirmed determination:
+
+- HC — high-confidence reuse candidate: cosine above the high cut and structure at or below the near-identical cut. Both measures point to a reused image.
+- MC — moderate-confidence, advisory: cosine above the high cut and structure between the two structural cuts. Style is very similar, but structure is below the strict bar.
+- HSC — high style-consistency: cosine above the high cut and structure above the upper structural cut. Style is similar with no structural support.
+- UN — uncertain: cosine between the low cut (the same-vs-different-accountant crossover) and the high cut.
+- LH — low reuse-similarity: cosine at or below the low cut.
+
+A report takes the strongest label among its signatures (HC > MC > HSC > UN > LH). Table I-a summarizes the five categories, the thresholds that define them, and the notation used throughout; the high cut is cosine 0.95, the low cut is 0.8547 (the same-vs-different-accountant crossover; Section IV-A), and the two structural cuts are dHash 5 and 15.
+
+**Table I-a — Category definitions, thresholds, and notation.**
+
+| Label | Name | Condition (cosine *c*, structure dHash *d*) | Role |
+|---|---|---|---|
+| HC | high-confidence reuse candidate | *c* > 0.95 and *d* ≤ 5 | self-certifying flag |
+| MC | moderate-confidence | *c* > 0.95 and 5 < *d* ≤ 15 | advisory |
+| HSC | high style-consistency | *c* > 0.95 and *d* > 15 | no structural support; no weight |
+| UN | uncertain | 0.8547 < *c* ≤ 0.95 | ambiguous middle |
+| LH | low reuse-similarity | *c* ≤ 0.8547 | likely hand-signed |
+
+Other abbreviations used throughout: ICCR — inter-CPA coincidence rate, the between-accountant chance-firing rate that calibrates the rule (Section III-E); *c* — cosine similarity to the same accountant's other signatures (style); *d* — smallest dHash distance (structure).
+
+Why the partition has this shape (five categories, not nine). As explained in Section I, a near-identical structure is decision-relevant only once the styles already match, so the two cosine cuts come first — splitting signatures into three style bands (low, uncertain, high) — and the two structural cuts subdivide only the high band. Three facts pin this shape down. First, structure carries little standalone decision weight in the two lower bands: between signatures whose styles do not clearly match, a moderate structural distance is hash noise, not evidence of reproduction — and even the near-identical structural matches that do appear below the style cut (quantified next) are not assigned HC; their structural information re-enters only through accountant-level aggregation and byte-identity review (Section IV-B), not through a separate cell. Second, the cells of the full 3×3 grid that pair a lower style band with a near-identical structure are sparsely populated rather than ignored — and the empirical reading is more precise than a simple "they are empty." An explicit count makes this exact: of the 150,442 Big-4 signatures, 7,681 (5.1%) combine a near-identical structural match (dHash ≤ 5) with a sub-0.95 cosine, so the one-way implication of Section I (a tiny dHash forces a high cosine) holds approximately, not strictly. But the residents' mass sits immediately below the high-cosine cut — 7,311 of them (95.2%) fall in cosine 0.90–0.95, and only 370 signatures (0.25% of the corpus) reach the genuinely low-cosine bands, of which just 38 lie below the LH/UN crossover (cosine ≤ 0.8547). These residents are not degenerate crops: their image size (mean 33k px) and detection confidence (0.875) match the rest of the corpus (28k px, 0.877). Under the coherent same-pair definition — style and structure satisfied on the same partner signature — the count falls further to 874 (0.58%). The point is therefore not that these cells are empty but that subdividing the lower style bands by structure changes no disposition: because cosine is the primary gate, a near-identical structural match beneath the style cut is already handled as UN, and the residual structural information re-enters through the accountant-level aggregation and byte-identity escalation of Section IV-B rather than through a separate cell. Third, a partition should cut only where the resulting actions differ: subdividing the two lower bands by structure would create cells whose dispositions (Section IV-B) are identical — all demoted or aggregated the same way — adding calibration burden without operational consequence, whereas the three structural cells inside the high band exist precisely because their dispositions differ. (Count from the deployed-rule descriptor columns; any-pair definition, full Big-4 corpus.)
+
+The cuts are operator-tunable operating points, not learned boundaries: there is no natural gap to read off the data (Section V-A) and no signature-level labels to learn one from, so the cuts are chosen and their specificity is measured, not learned. The four cut values, and where each one comes from — two are read directly from this study's data — are given in Section IV-A, alongside the chance-rate calibration that characterizes them and the figure of the two-measure plane (Fig. 3).
+
+Any-pair versus same-pair: how the two extrema combine. One construction detail deserves to be explicit, because a careful reader will ask. The two per-signature values are independent extrema over the same accountant's other signatures — the highest cosine and the smallest dHash, each taken on its own — so the two values may come from different partner signatures. We call this the any-pair rule, and the choice is deliberate, for three reasons. First, the two descriptors have different invariances: cosine survives re-scaling and re-compression; dHash does not. For a genuinely reused image that crossed different scan or compression pipelines, the style-nearest copy and the pixel-nearest copy can therefore legitimately be different reports — forcing both extrema onto one pair would miss exactly that most realistic positive case. Second, dHash takes whole-number values and ties are massive in duplicate-heavy pools: which tied copy wins the minimum is essentially arbitrary, so whether the two extrema land on the same file is largely tie-breaking noise — both point into the same duplicate cluster. Third, the chance-rate calibration of Section IV-A applies the same any-pair rule to the clean reference group, so the high-specificity claim rests on the absolute clean-group rate (the HC rule fires by chance on only ~1.2% of clean-group reports), not on any firm-versus-floor ratio; the same rule is applied to every firm and to the reference group alike. The stricter same-pair variant, in which a single partner signature must satisfy both inequalities at once, is reported as a robustness check (Section V-C) and leaves every conclusion unchanged — the within-firm concentration of cross-accountant matches is in fact *higher* under same-pair (97.0–99.96% across the four firms) than under the deployed any-pair rule (76.7–98.8%) — because in the high-confidence region the two rules nearly coincide: a partner within the near-identical structural cut is pixel-near-identical and therefore clears the high style cut by itself.
+
+Limitations, stated up front. Three follow directly from the design. (i) Because the cutoffs are chosen rather than learned, the system has a tunable operating point, not an optimal one. The dial moves in only one direction: a reviewer who wants more conservatism can tighten it for higher specificity, but cannot trade in the other direction toward higher recall, because recall is unobservable without labels — there is no precision–recall frontier to navigate (Section V). (ii) The chance rate we report is a *between-accountant* coincidence rate — a proxy for specificity, not a true false-acceptance rate — because we have no labeled negatives; it speaks to how often unrelated accountants collide, not to how often a genuinely consistent hand-signer would fire the rule, which is the quantity the reuse question needs and which we cannot estimate (Section III-E). (iii) For any single signature, the two measures cannot tell us why it is so similar to another: reuse of an image, a shared scanning pipeline, and a very uniform house style all push the numbers the same way, and we do not try to choose between them at the level of one signature. These limits apply to every claim that follows.
+
+### E. The Clean Reference Group and the Chance Rate
+
+With no labeled negatives to learn from, the calibration uses a stand-in: a group in which the rule should fire only by chance — unrelated accountants whose signatures happen to look alike now and then. Choosing this group well is the central design decision, and two requirements force the choice.
+
+Why not all four firms. As Section IV-C will show, almost all of one firm's between-accountant matches fall on other accountants of the same firm, and we have byte-level proof of image reuse across about fifty of that firm's partners. If we put Firm A into the reference group, we would be filling the "by chance" rate with exactly the within-firm matches the rule is supposed to catch — a circular calibration. So we use Firms B/C/D as the clean reference group and keep Firm A as a test case; we report the all-four-firm number only to show how much Firm A contaminates it.
+
+Why 2013–2019. We further limit the reference group to the years before formal firm-wide electronic-signing systems (adopted from 2020 onward; Section III-A). What this buys us is the absence of a shared template across accountants — not a guarantee that every signature was handwritten. The interviews say some baseline firms used informal individual stamping before 2020, but each accountant's stored image was their own, so different accountants' signatures still match only by chance; the chance rate is about matches between accountants, which individual stamping does not inflate. One further channel deserves to be named, because it is not the template and we cannot fully exclude it: accountants at the same firm pass through a shared imaging pipeline — common scanners, PDF-assembly software, and the red-stamp-removal step (Section III-C, Section V-B) — and a shared pipeline can imprint correlated artifacts on otherwise-unrelated signatures, which would lift the inter-CPA rate above true chance. The pipeline audit of Section V-B confirms that such shared production paths exist and change over time. This is a reason to read the ICCR as a *specificity proxy* rather than a literal coincidence rate; its bias, like reference contamination, runs toward a higher floor, which makes the Firm-A contrast more conservative rather than less. After 2020, formal systems standardize how reports are assembled, so that period is not a clean reference — and indeed the chance rate rises after 2020 (Section V-B). We therefore calibrate on the Firms-B/C/D 2013–2019 cell and score every held-out cell against it.
+
+We report the rule's chance rate at three levels, because the rule takes the best match over a pool and so the per-signature rate is not the same as the per-pair rate: per comparison (sampled pairs of different accountants), per signature, and per report, each with a confidence interval. We call this the inter-CPA coincidence rate (ICCR) rather than a "false-acceptance rate," which we reserve for settings that have labeled negatives. The ICCR is a *between-accountant* coincidence rate: how often the rule fires on the signatures of two *different* accountants. It is therefore at best a proxy for specificity, and only under the stated assumption (no shared template across accountants). It is important to be exact about what it is not. The quantity the reuse question actually needs is the *within-accountant* false-positive rate — how often the rule would fire on a genuinely consistent hand-signer's own signatures — and that rate is not estimable here, because no accountant in the corpus is labeled as a known hand-signer. We considered benchmarking it against an external corpus of genuine repeated signatures (a public signature dataset supplies many authentic samples per writer), but such corpora are a different population and script acquired under a different pipeline, so the resulting rate would not transfer to this setting; importing it would reintroduce exactly the kind of unverifiable cross-distribution assumption our label-free calibration is built to avoid. We therefore report the limitation rather than a misleading proxy. The ICCR is not even a bound on it: a uniform individual hand keeps cosine high by design, so a true hand-signer's within-accountant fire rate can sit far *above* the between-accountant coincidence rate. Any statement that divides a firm's within-accountant fire rate by this between-accountant floor (an "X× the floor" comparison) therefore overstates the gap — the bias runs in the anti-conservative direction — and we do not report such ratios as effect sizes. Read as a between-accountant specificity proxy under the stated assumption, the ICCR is faithful to the evidence; read as a true error rate for the reuse question, it would claim more than we can show.
+
+One further assumption deserves to be stated rather than buried, because it concerns how the clean group was chosen. The floor is *conditional on the reference group actually being clean* — it is a coincidence rate among accountants we take to be independent hand-signers, and the group (non-Firm-A firms, pre-2020) was selected partly because its rates are low and its practices, by the interviews, are not stamping-dominated. That selection is mild but not innocent: if some baseline accountants in fact reuse images undetected, the reference is contaminated. The direction of that error, however, is reassuring for the Firm-A contrast. Undetected reuse inside the baseline would only *raise* the between-accountant coincidence floor, which makes Firm A's gap above it *smaller*, not larger — so contamination of the clean group biases the headline contrast conservatively, against our conclusion rather than toward it. Two pieces of evidence bound the concern empirically. First, the three baseline firms are mutually consistent and uniformly low (Firms B and C within about 3.5× of each other, none close to Firm A; Section IV-A), so the floor does not hinge on any single firm and a leave-one-baseline-firm-out reading does not move it materially. Second, the one data-derived threshold, the low cosine cut, is stable when the group composition is changed — 0.8547 on the calibration cell, 0.8302 with the non-Big-4 firms folded in, a shift of at most 0.025 (Section V-C) — so widening or narrowing the reference at its boundary does not move the operating point. We therefore treat the clean-group assumption as a stated limitation with a known-safe error direction, not as a hidden premise.
+
+### F. What HC Means and Does Not Mean
+
+One sentence prevents the most common misreading of everything that follows. *HC is not a reuse label.* HC denotes an extreme *within-accountant repetition pattern* — a signature whose closest match among the same accountant's own signatures is both stylistically near-identical (cosine > 0.95) and structurally near-identical (dHash ≤ 5) — that is *statistically rare between unrelated accountants*, by the ICCR calibration of Section III-E. That is the whole of what the rule, on its own, establishes. Reuse of a stored image is one interpretation of an HC pattern, and the most economical one, but the rule does not imply it: a very steady hand, a fixed scanning-and-assembly pipeline, or a uniform house style can each raise within-accountant repetition (Section V-A, Section V-B). Where we go further than "extreme repetition" — as we do for Firm A — the additional weight comes from outside the rule: byte-identical signatures, which independent hand-signing cannot produce, and the institutional context, neither of which is implied by HC alone. For Firms B/C/D we make no reuse claim at all; their HC signatures are reported as a within-accountant repetition rate, not as detected reuse. Read this way, HC is a calibrated, reproducible screening category, and "reuse" is a conclusion that has to be earned separately — firm by firm, or signature by signature — rather than read off the label.
+
+## IV. Findings
+
+This section reports the numbers. It starts with the calibration baseline (Firms B/C/D, 2013–2019), then says which category needs human review, then presents the held-out benchmark on Firm A.
+
+### A. Detection Sample (Whole Corpus) and the Calibration Baseline (Firms B/C/D, 2013–2019)
+
+Detection and the analysis sample (whole corpus). Two scopes appear in this section and must not be confused: detection and the analysis sample here are computed on the whole corpus, whereas both data-derived calibration quantities — the chance-rate ICCR and the low cosine cut (Section IV-C) — are computed only on the clean Firms-B/C/D 2013–2019 cell. Of the 90,282 reports, the page-finder flagged 86,084 as having a signature page (the other 4,198, or 4.6%, had none); 13 of those 86,084 could not be rendered, leaving 86,071 documents processed. On the validation set, the YOLOv11n detector reached precision 0.97–0.98, recall 0.95–0.98, mAP@0.50 0.98–0.99, and mAP@0.50:0.95 0.85–0.90. Across the corpus it extracted 182,328 signatures — 2.14 per document with detections, where two certifying accountants per report implies 2.00. The ≈6.7% excess is explained by extra detections rather than missed accountants: of the 13,573 detections (7.4%) that could not be matched to a registered accountant and were excluded, 8,901 (66%) are third-or-later detections on a page — boxes beyond the two certifying signatures — and the unmatched set as a whole carries lower detection confidence than the matched set (mean 0.826 vs 0.874), consistent with these being extra boxes and low-confidence noise; the remaining 4,672 are first/second-position detections that failed registry matching. Throughput was 43.1 documents per second, and the detector agreed with the vision-language model on 98.8% of documents. Matching by position assigned 92.6% of signatures (168,755 of 182,328) to a registered accountant; of these, 168,740 have both similarity measures computed (the 15-signature difference is accountants with a single signature in the corpus, for whom no same-accountant comparison exists, so the full-corpus distributional statistics in the Appendix are reported on 168,740). The four-firm analysis sample is 150,442 signatures with both measures computed, from 457 accountants (Table I); the accountant-level partition (Table III, Section V-C) is fit on the 437 of these with at least ten signatures (171/112/102/52 across Firms A–D).
+
+**Table I — Detection and extraction summary.**
+
+| Quantity | Value |
+|---|---|
+| Documents with a signature page | 86,071 |
+| Detector precision / recall | 0.97–0.98 / 0.95–0.98 |
+| Detector mAP@0.50 / mAP@0.50:0.95 | 0.98–0.99 / 0.85–0.90 |
+| Signatures extracted | 182,328 (2.14 per document) |
+| VLM–detector agreement | 98.8% |
+| Signatures matched to an accountant | 168,755 (92.6%) |
+| Four-firm analysis sample | 457 accountants; 150,442 signatures |
+
+The calibrated operating point: the four cut values and their bases. The five-way rule of Section III-D uses four cut values; we state them here because two are read directly from this study's data. The low cosine cut, 0.8547, is the crossover of the same-accountant and different-accountant cosine distributions computed on the calibration cell alone (Firms B/C/D, 2013–2019, closed-world: both the source signatures and their comparison set drawn from that cell; Section IV-C). We use this closed-world value as the primary cut rather than the corpus-wide crossover, so that the one data-derived threshold in the rule is estimated only on the calibration-only Firms-B/C/D 2013–2019 cell, held out from Firm A and from post-2020 scoring. The cut is stable across scopes — 0.8547 (calibration closed-world), 0.8367 corpus-wide, 0.8489 on the all-period baseline firms, 0.8302 with the non-Big-4 firms added; it moves by at most 0.025 across all four scopes (0.018 from the corpus-wide value), so the choice of scope is immaterial and the broader-scope values stand as robustness checks (Section V-C). The high cosine cut, 0.95, is the high-similarity operating point: it sits in the region where genuine reuse concentrates — the byte-identical anchor (Section IV-C) lies at cosine 1 — and a recalibration cannot move it onto a distributional antimode because none exists (no within-population bimodality, Section V-A). The near-identical structural cut, dHash ≤ 5, is the perceptual-hash distance below which two rasters are pixel-equivalent up to mild recompression, and dHash ≤ 15 bounds the looser "structurally similar" band; both follow the standard 64-bit dHash distance scale [27]. We therefore do not re-derive these three as optimal cutoffs but characterize their chance-of-firing behavior directly (the full prior-calibration provenance is in the supplementary materials), and we make them operator-tunable in one direction: their specificity proxy at these values is read off the chance-rate calibration below, and an operator can tighten the floor by inverting the ICCR curve (for example, dHash ≤ 3). This is a conservativeness dial, not a precision–recall control: tightening raises the specificity proxy and lowers the flag count, but there is no observable recall to trade back, so loosening cannot be calibrated against a known cost. We deliver these as a concrete, calibrated operating point — in particular the high-confidence (HC) rule, cosine > 0.95 and dHash ≤ 5 — whose between-accountant coincidence behavior the calibration below makes explicit. Because the rule is calibrated on a large Chinese-signature corpus, the HC values double as a practical starting reference for practitioners working with comparable Chinese-signature image pipelines, rather than a setting to transplant unchanged.
+
+![](figures/fig3.png)
+
+*Figure 3. The two measures and the five regions, drawn as the real 2D density of all Big-4 signatures (n = 150,442; log color scale, integer dHash bins). The cosine axis is split at the low cut 0.8547 (the calibration-cell same-vs-different-accountant crossover) and the high cut 0.95; within the high-cosine band the dHash axis is split at 5 and 15. The mass concentrates in the bottom-right HC corner — high cosine with near-identical structure — and thins out as a single continuum toward lower cosine and higher dHash, with no gap separating a "reuse" cluster from a "hand-signed" one (Section V-A); note also that essentially all signatures sit above cosine ≈ 0.85, the compressed high-similarity range discussed in Section V-A.*
+
+The calibration sample itself (Firms B/C/D, 2013–2019). The chance-rate calibration that follows is computed on the clean cell only, and the reader should be able to see the calibration base directly rather than infer it from the full-period totals above. The Firms-B/C/D 2013–2019 cell contains 226 accountants, 52,071 signatures with both measures computed, and 26,042 reports; the per-comparison ICCR below is estimated from 5×10⁵ inter-CPA signature pairs sampled uniformly from this cell. Every ICCR source signature is restricted to this cell — the headline per-signature and per-document rates reproduce on the 52,071-signature 2013–2019 cell, not on the full-period BCD record (~90,000 signatures), which is used only where a robustness figure is explicitly quoted — so no post-2020 or Firm-A signature enters the calibration.
+
+How often the strict rule fires by chance (pooled). In the Firms-B/C/D 2013–2019 group, the strict (HC) rule fires by chance very rarely at every level (Table II): about 1 in 100,000 per comparison (Wilson 95% CI [4×10⁻⁶, 2.3×10⁻⁵]), 0.59% per signature ([0.45%, 0.73%]), and 1.2% per report. These are roughly ten times lower than the contaminated all-four-firm figures (1.4×10⁻⁴, 11.0%, 18.0%); the difference is exactly the within-firm matching that the clean group leaves out. So a clean group of unrelated accountants almost never produces an HC report, which makes HC a high-specificity operating point. (The per-comparison figure rests on a small number of chance hits — 5 of 5×10⁵ pairs — and is best read as an order-of-magnitude value; the per-signature and per-report figures, which are well powered, carry the weight.) A low rate is not a small number at archive scale, and we state the absolute consequence plainly: applied blindly across all 150,442 analyzable Big-4 signatures, the clean per-signature rate alone would be expected to yield about 888 HC flags by chance (95% CI [677, 1,098]), scaling further if the screen is run over the full archive. This is exactly why a single HC flag is never read in isolation: the evidential weight is carried by the firm-level contrast (Section IV-C) and accountant-level aggregation (Section IV-B), not by a raw archive-wide HC count.
+
+**Table II — Chance-firing rates (ICCR) by level and group: the strict HC rule (top two rows), with the looser MC band's per-report rate shown for contrast (bottom row).**
+
+| Group / rule | Per comparison | Per signature | Per report |
+|---|---|---|---|
+| HC — B/C/D 2013–2019 (calibration) | 1.0×10⁻⁵ [4×10⁻⁶, 2.3×10⁻⁵] | 0.59% [0.45%, 0.73%] | 1.2% |
+| HC — all four firms (contaminated) | 1.4×10⁻⁴ | 11.0% | 18.0% |
+| MC band (HC+MC) — B/C/D 2013–2019 | — | — | ≈17.5% |
+
+Each baseline firm on its own (B, C, D). Reported separately, the three baseline firms are alike and uniformly low. A logistic regression of the per-signature HC flag on firm (with Firm D as the reference) over the baseline cell puts Firms B and C within about 3.5× of each other (odds ratios 1.73 and 0.49), and none of them comes close to the high rates we see for Firm A in Section IV-C. The 2013–2019 five-way breakdown for each of Firms B/C/D (counts and within-firm percentages) is reported in Table II-b; the full-period (2013–2023) breakdown is in Table IV for reference.
+
+**Table II-b — Five-way breakdown for each baseline firm, calibration period (B/C/D, 2013–2019).**
+
+| Firm | HC | MC | HSC | UN | LH | signatures |
+|---|---|---|---|---|---|---|
+| Firm B | 29.04% | 39.31% | 0.39% | 30.91% | 0.35% | 19,677 |
+| Firm C | 21.59% | 42.09% | 0.37% | 35.53% | 0.43% | 22,449 |
+| Firm D | 22.01% | 29.67% | 0.20% | 47.35% | 0.76% | 9,945 |
+
+One point in Table II-b needs to be made explicit, because at first glance it looks like a contradiction: the within-firm HC percentages here (Firm B 29.0%, Firm C 21.6%, Firm D 22.0%) are an order of magnitude above the 0.59% chance rate of Table II, even though both are computed on the same clean calibration cell. They are not in tension, because they measure different things. The 0.59% is a *between-accountant* rate — how often the HC rule fires on the signatures of two *different* accountants — and that is the quantity the calibration uses and that stays tiny. The 21–29% figures are *within-accountant* rates — how often an accountant's own signatures fire the rule against each other — and a substantial within-accountant rate is exactly what one expects from anyone with a consistent hand or a uniform house style, before any reuse is invoked; at these firms it also carries a genuine but smaller component of image reuse that grows after 2020 (Section V-B). "Clean," in this paper, means the between-accountant coincidence is rare, not that within-accountant similarity is low. This is also why the within-accountant rate cannot be read as a false-positive rate (Section III-E), and why the contrast that isolates Firm A in Section IV-C is not "HC fires at all" but that Firm A's within-accountant rate (82%) stands far above the 21–29% of three otherwise-alike firms.
+
+### B. From Categories to Actions: Review as Exception Management
+
+The proportions first, stated plainly. Before saying what to do with each category, we show how much of the data falls into each (Table IV, full corpus): HC 49.6%, MC 26.5%, UN 23.4%, HSC 0.2%, LH 0.3%. The ambiguous middle (MC + UN) is therefore not a fringe: about half of all four-firm signatures, and 65–76% at Firms B/C/D individually, against 18.1% at Firm A. Read against the institutional background (Section III-A), this is exactly the expected shape. Firm A behaves in the screen as a reuse-dominated population — a reading consistent with the interviews and with the byte-identical evidence, though it rests on those rather than on per-signature ground truth — and the screen settles most of its signatures outright; Firms B/C/D in this period are mixed populations in which hand-signing and informal individual stamping coexist, so per-signature similarity is genuinely ambiguous there. The right response to a large middle is not to hide it but to give it a disposition path that does not require a per-signature verdict. Four moves do this. To be clear about what is established versus proposed here: the category proportions above, the per-band chance rates, and the byte-identity counts are empirical results, whereas the four moves are a *designed* operating procedure derived from the calibration — they are an argument that the workload is tractable, not a validated workflow. The protocol's end-to-end first run on a bounded, human-labeled sample, which is what would actually measure its discriminating behavior, is left to future work (Section V); we therefore present the moves as the intended use of the calibrated rule, not as evidence in their own right.
+
+Move 1 — calibrate each band's evidential weight, and demote what fails. The calibration tells us what each flag is worth. The HC band fires by chance on only about 1.2% of reports in the clean reference group, so an HC flag is close to self-certifying: it needs essentially no verification effort, and it goes straight onto the action list — findings to count, report, or investigate — rather than onto a list of flags still to be checked. The MC band fires by chance on about 17.5% of reports in the clean reference group — roughly one clean-group report in six — and, unlike HC, this rate does not drop when Firm A's accountants are excluded from the cross-accountant comparison pool (it edges up, because removing Firm A's distinctive template leaves a pool whose members resemble one another a little more at the coarse dHash ≤ 15 scale); the boundary at dHash = 15 also sits in a flat region of the sensitivity sweep, adding flagged cases without adding specificity (Section V-C). An MC flag on its own therefore carries almost no information and does not justify verification effort; it matters only in combination with other evidence. The UN band is ambiguous in the same spirit and is treated alongside MC; on the clean baseline the UN cosine band is reached by chance about 88% of the time per signature (98.2% per report), confirming that a UN flag is essentially uninformative about reuse on its own, whereas the HSC band is reached by chance only about 0.13% of the time per signature (0.25% per report) and in any case points away from reuse (style match without structural support). The HSC band is tiny (0.2%), so it warrants only a light spot-check. The LH band needs no action. Demotion, however, only says what an MC or UN flag is not — standalone evidence; what becomes of these signatures is the business of the next three moves: their information flows into the accountant-level scores (Move 2), which byte-identity hits then sharpen by proving that an accountant's stored image is in circulation (Move 3); the residual's data needs are named rather than guessed at (Move 4); and where a human does look at individual cases, the bounded protocol specified below applies.
+
+Move 2 — lift the unit of decision from the signature to the accountant. The middle categories rarely need to be resolved one signature at a time, because the operational question is almost always about an accountant or a firm, and the ambiguous signatures still carry information at that level. Three accountant-level scores — a mixture-model position score on the two-measure plane, a percentile relative to an external non-Big-4 reference population, and the accountant's own rate of replication-consistent labels — rank the 437 accountants in close agreement (Spearman ρ ≥ 0.879; reported as internal consistency among scores built on the same descriptors, not as external validation). A signature that is individually undecidable still moves its accountant's position; several hundred per-signature questions collapse into one per-accountant judgment.
+
+Move 3 — anchor with byte-identity, the one check that yields certainty. An exact byte-level comparison costs little, and what it finds is proof rather than evidence: independent hand-signing cannot produce byte-identical images, so every byte-identical pair is confirmed reuse with no human judgment required (the corpus contains 262 such signatures; Section IV-C). To be precise about where this bites: a byte-identical pair has cosine 1 and dHash 0, so these signatures sit in HC by construction — byte-identity rescues no case from the ambiguous middle. Its role is twofold. Within HC, it upgrades a subset from high-confidence candidate to logical certainty, removing even the pipeline-and-house-style caveat of Section III-D for those cases — the difference between a statistical screen and an exhibit one can act on without qualification. And at the accountant level, a byte-identical hit proves that a stored image of that accountant is in circulation, which raises the prior on the rest of that accountant's near-identical cluster — including its MC and UN members — and thereby sharpens the per-accountant judgment of Move 2. (A byte-identical pair has cosine = 1 and dHash = 0, so it falls in HC by construction; that the rule "captures" the whole byte-identical set is therefore tautological, and we do not read it as a recall measure.)
+
+Move 4 — state what the residual needs, instead of classifying it anyway. After the three moves, a residual middle remains whose mechanism the two measures genuinely cannot identify: reuse through a noisy pipeline, a very steady hand, and a homogeneous scanning infrastructure can occupy the same spot on the plane. We name the data that would resolve it — a proposed resolution path, not one executed in this study. Image-acquisition metadata is machine-readable provenance that could be extracted automatically rather than judged by eye: scanner identifiers and PDF-generator strings recorded in the files themselves, and compression markers such as JPEG quantization tables, which encode the processing history an image has been through. This adds the axis the two similarity measures lack — two near-identical images that arrived through different production pipelines are hard to explain except by reuse, while two that shared one pipeline may owe their similarity to the pipeline itself. (Whether this provenance survives the upload platform is itself an empirical question, and we checked: we verified across a stratified sample of MOPS reports (all four firms, 2014–2022) that producer/creator strings, PDF versions, and image encodings are heterogeneous report-to-report — distinct scanner models (Fuji Xerox D125, ApeosPort-III/IV/V), born-digital producers (Microsoft Word, Adobe, Acrobat Distiller), and a mix of CCITT-grayscale and JPEG-RGB encodings at differing resolutions — so the platform does not flatten uploads to a uniform template and the acquisition history is recoverable here; firms' own internal archives would retain at least as much.) A small labeled set of known hand-signed examples — certified by the firms, or accumulated case by case as a by-product of the review protocol below — would turn the chance-rate calibration into directly estimated error rates. Naming these is the honest alternative to pretending the residual can be classified from similarity alone.
+
+Where a human does look, the review follows a defined and bounded protocol. We specify the protocol here as a design deliverable of the method: the discriminating behaviors stated below are design expectations, following from the artifact properties of reused versus independently signed images, and the protocol's first execution, on a bounded sample, is listed as future work (Section VI). (1) Side-by-side overlay inspection: the reviewer is shown the flagged signature next to the same-accountant signature(s) that produced its score, with a pixel-difference overlay and an edge-aligned superposition; a reused image is expected to overlay almost exactly, whereas two independent signings show natural variation in pressure, ink, and baseline. (2) Secondary artifact checks not used by the rule — exact registration, JPEG and scan-noise fingerprints (the compression and anti-aliasing traces a reused raster carries with it), and scaling traces — are designed to separate a reused raster from a re-scanned genuine signature at low cost. (3) Document and time context: the reviewer checks whether the matched signatures come from reports of different dates or engagements (reuse across time is more telling than within a single filing) and whether the surrounding layout shows a standard template or stamp. (4) Bounded per-accountant sampling: because the operational question is usually at the accountant or firm level, the reviewer judges a bounded random sample per accountant rather than every flagged signature, keeping the effort proportional to the number of accountants, not the number of signatures. (5) Feedback into calibration: each adjudicated case yields a label — reuse, hand-signed, or undetermined — and these accumulate into the small ground-truth set the setting otherwise lacks, which can later tighten the operating point or support supervised validation. The protocol's relation to Move 4 is one of scale: steps 1–3 apply per-case versions of the same artifact evidence that Move 4 would collect corpus-wide, step 4 bounds how many cases a human ever sees, and step 5 accumulates the labeled set Move 4 asks for. What the protocol cannot do — and is not claimed to do — is resolve the residual at scale; that is exactly what the corpus-wide metadata collection of Move 4 would add.
+
+Why this is exception management rather than caseload. Where a firm's output is dominated by reuse, the high-confidence tier settles most signatures directly (82% at Firm A), and the four moves reduce the remaining 18% to per-accountant judgments and a review queue bounded by the number of accountants rather than the number of signatures — exception management at the signature level. In a mixed population the same machinery delivers the same promise one level up, at the accountant. Move 1 removes the bulk of the apparent caseload outright: an MC flag alone does not justify verification, which at Firms B/C/D takes 29–41% of signatures off the worklist before anyone is assigned to anything (the UN band carries no flag in the first place, so the demotion bites on MC alone). Move 2 positions every accountant on the replication-dominance spectrum, so attention concentrates on the few high-ranked or mixed cases rather than on tens of thousands of signatures. Move 3 supplies proof where proof exists: 117 of the 262 byte-identical signatures sit at Firms B/C/D, demonstrating that stored-image reuse is a real practice at the mixed firms too, and anchoring the accountant-level judgments there. And the staggered post-2020 adoption of formal systems gives the mixed firms a readable time axis (Section V-B). What is not delivered in a mixed population is a per-signature verdict for the ambiguous middle — a limit of identification, not of workload. Exception management therefore holds in both settings; what changes is only the level at which exceptions are defined — the signature where reuse dominates, the accountant where practices are mixed. Because the cutoffs are tunable, a reviewer who wants higher specificity can tighten them (for example, cosine > 0.98 and dHash ≤ 3), trading a lighter caseload against the risk of missing noisier reused signatures — a trade-off we cannot tune against recall, since recall is unobservable here.
+
+### C. Held-Out Benchmark: Firm A (a Known Positive)
+
+Firm A — described by the interviews as a mainly-stamping firm, and kept out of the calibration — is our main benchmark. Because the interviews already identify it as a stamping firm, it is best read as a *quasi-positive institutional benchmark*: held out from calibration, but a known positive rather than a blinded out-of-sample test. What it can confirm is that the screen's measures move as expected on a firm independently believed to reuse images; what it cannot do is stand in for a blinded evaluation against ground-truth labels, which the corpus does not provide.
+
+(1) Firm A's two measures against the baseline. Comparing Firm A's within-accountant similarities to those of Firms B/C/D (full record, 2013–2023²), Firm A's cos values are shifted toward 1.0 and its dHash distances toward 0 — the direction we would expect if a stored image is reused rather than re-signed. Concretely, Firm A's within-accountant cosine is centered at a median of 0.986 (mean 0.980) versus 0.959 (mean 0.954) for Firms B/C/D, and its smallest-dHash distance at a median of 2 (mean 2.7) versus 7 (mean 7.0); both shifts are in the reuse direction and overwhelmingly significant (Mann–Whitney U, p < 10⁻³⁰⁰ for each; two-sample Kolmogorov–Smirnov D = 0.60 for cosine and 0.57 for dHash). The decisive number is this: scored as a held-out (but not blinded) case — Firm A's signatures matched against unrelated accountants drawn from the clean 2013–2019 group — Firm A's per-signature cross-firm HC rate is 0.42% (154/36,552; Wilson 95% CI [0.36%, 0.49%]), at or below the clean reference ICCR of 0.59%. In other words, Firm A's cross-firm match rate sits at the level a clean inter-CPA comparison produces by chance — it is not elevated relative to the reference, and it is negligible beside the within-firm rate below — so the entire rise in Firm A's rate comes from matches with other Firm-A signatures, not from resemblance to other firms. The signal is inside the firm, not across firms. (Against the full-period BCD pool the same across-firm rate is 1.0%; the small difference reflects the post-2020 rise in baseline similarity of Section V-B. Both lie at the clean floor, two orders of magnitude below the within-firm rate that follows.)
+
+> ² Restricting both groups to 2013–2019 gives essentially the same picture (Firm A cosine median 0.986, dHash 2; Firms B/C/D 0.957 and 7; Mann–Whitney p < 10⁻³⁰⁰ for each), confirming the contrast is not a post-2020 artifact.
+
+Firm A's within-firm repeatability, against the other firms. On their own signatures, the HC rule fires on 82% of Firm A's, versus 24–35% for Firms B/C/D. We deliberately report these as raw within-accountant fire rates and do not divide them by the between-accountant clean floor: as Section III-E explains, that floor is the wrong null for a within-accountant question, so an "X× the floor" multiplier would overstate the gap. The firm-to-firm contrast in raw rates is what carries the result. A logistic regression of the per-signature HC flag on firm and pool size, with Firm A as the reference, gives odds ratios of 0.053, 0.010, and 0.027 for Firms B/C/D — one to two orders of magnitude lower (the odds ratio for log pool size is 4.01). Firm A stands alone, against a baseline of three firms that look alike.
+
+Four further checks confirm the contrast is not an artifact of how the comparison pools are built, of the imaging-pipeline trend, or of any single year. First, pool size. Stratifying accountants by how many signatures they contribute and comparing within each stratum, Firm A's HC rate exceeds the other firms' at every level — 66% versus 20% for the smallest pools (under 50 signatures), rising to 76–84% versus 21–29% for larger pools. Even Firm-A accountants with few signatures to match against fire the rule far more often than B/C/D accountants with the same pool size; pool size raises the rate within every firm (the log-pool-size odds ratio of 4.01), but the firm gap dwarfs it and survives at fixed pool size, which rules out the "more signatures, more chances for an extreme match" explanation. Second, dependence among an accountant's own signatures. Re-estimating the gap with the bootstrap resampled at the accountant level (179 Firm-A accountants, 280 at Firms B/C/D) rather than treating signatures as independent, the Firm-A-minus-B/C/D difference in HC rate is 53.7 percentage points with a 95% interval of [49.5, 57.5] — accountant-level clustering widens the intervals the per-signature Wilson bounds give, but leaves the contrast far too large to be explained away. Third, the time trend and pipeline shift (Section V-B). Adding year fixed effects to the logistic regression — so the firm effect is identified within year, net of the 2020–2021 imaging-pipeline transition — leaves Firms B/C/D at 0.06–0.12 times Firm A's odds of an HC flag (odds ratios 0.116, 0.061, 0.070), still an order of magnitude lower once the common time trend is absorbed. Fourth, single-year dependence. Leaving out each calendar year in turn and recomputing, the Firm-A-minus-B/C/D gap stays within 53.1–54.9 percentage points (full-sample 53.7), so neither the high-reuse digital-native years (2022–2023) nor any earlier year drives it.
+
+![](figures/fig4.png)
+
+*Figure 4. Within-accountant similarities, Firm A vs Firms B/C/D: (a) cosine, (b) dHash. Firm A's mass sits near cosine = 1 and dHash = 0 (medians 0.986 / 2) against Firms B/C/D's 0.959 / 7; dashed lines mark the cuts (cosine 0.95; dHash 5), the dotted line the LH/UN crossover (0.8547). The held-out cross-firm HC rate (0.42%) sits at/below the clean reference ICCR (0.59%), while the within-Firm-A HC rate is 82% — the signal is inside the firm (annotation below panels).*
+
+(2) Ranking accountants by similarity, in each period. Ranking every accountant in Firms A–D by a single within-accountant similarity score, separately for 2013–2019 and for 2020–2023, Firm A's accountants sit at the high-similarity (templated) end. A descriptive three-group summary of the two-measure space tells the same story: its high-cosine/low-dHash group holds 82.5% of Firm A's accountants and almost none of the others' (Table III). The period split confirms the expected pattern: Firm A's per-signature HC rate is at the top in both periods (80.3% in 2013–2019, 83.8% in 2020–2023), while Firms B/C/D move upward after 2020 as the formal systems came in — Firm B from 29.0% to 42.0%, Firm C from 21.6% to 26.7%, Firm D from 22.0% to 28.0% (see Section V-B).
+
+**Table III — Firm by descriptive-group membership (whole corpus). The "high-cosine/low-dHash group" is the templated-end cluster of the three-group (K = 3) descriptive Gaussian-mixture partition of the accountant-level two-measure plane (Section V-C); membership is the cluster of maximum posterior probability for each accountant with at least ten signatures (437 accountants: 171/112/102/52 across Firms A–D). The groups are used for description only, never as operational labels.**
+
+| Firm | Accountants | Share in the high-cosine/low-dHash group |
+|---|---|---|
+| Firm A | 171 | 82.5% |
+| Firm B | 112 | 0.0% |
+| Firm C | 102 | 1.0% |
+| Firm D | 52 | 1.9% |
+
+![](figures/fig5.png)
+
+*Figure 5. Per-accountant HC rate, ranked, one panel per period (2013–2019; 2020–2023), points colored by firm (accountants with ≥ 5 signatures in the period). Firm A (red) occupies the templated top of the ranking in both periods; Firms B/C/D rise after 2020 (HC rate B 29.0→42.0%, C 21.6→26.7%, D 22.0→28.0%; Firm A 80.3→83.8%), consistent with staggered formal-system adoption (Section V-B).*
+
+(3) Applying the calibrated rule to Firm A, 2013–2023. Taking the operating point calibrated on Firms B/C/D in 2013–2019 and applying it across Firm A's full record, 81.70% of Firm A's signatures (82% rounded) land in HC (per signature; the full five-way breakdown is in Table IV). Read together with the interview fact that Firm A mainly uses overlay stamping, the system's firm-level output matches the practice the firm itself describes. We say this carefully: it is a match at the firm level, not a label on any single signature. We do not classify the individual signatures as non-hand-signed, because for any one signature the two measures cannot separate reuse from a shared scanning pipeline or a uniform house style (Section III-D).
+
+**Table IV — Five-way breakdown by firm (whole corpus, 2013–2023; for reference; n = 150,442).**
+
+| Firm | HC | MC | HSC | UN | LH | signatures |
+|---|---|---|---|---|---|---|
+| Firm A | 81.70% | 10.76% | 0.05% | 7.35% | 0.14% | 60,448 |
+| Firm B | 34.56% | 35.88% | 0.29% | 28.95% | 0.32% | 34,248 |
+| Firm C | 23.75% | 41.44% | 0.38% | 33.97% | 0.47% | 38,613 |
+| Firm 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 |
+
+Reading the five-way mix across firms. Table IV is also the quantitative basis for the positioning in Section IV-B. At Firm A the ambiguous middle (MC + UN) is 18.1% — the screen reads this population almost cleanly, with four signatures in five settled outright. At Firms B/C/D the middle is 65–76% — the signature of a mixed population in which hand-signing and informal stamping coexist (Section III-A), where per-signature similarity is genuinely ambiguous. There the screen's deliverables move up one level (Section IV-B): the MC share (29–41% of these firms' signatures, against the 26.5% corpus-wide MC share) is demoted off the worklist, the accountant-level scores rank these firms' accountants alongside everyone else's, and the byte-identical signatures at these firms (117 of the 262) are threshold-free proof that reuse occurs there too. The per-signature mix stays ambiguous; the disposition does not.
+
+Byte-identical signatures: direct evidence of reuse. Beyond the screening numbers, 262 signatures across the four firms are byte-for-byte identical to another signature — 145 of them at Firm A, spread across about fifty partners. Identical files cannot come from independent hand-signing, so their existence is direct, hard evidence that image reuse happens and that it concentrates at Firm A. These pairs are not a bookkeeping artifact: every one of the 262 matches a signature in a *different* report PDF (none is the same file double-counted), and 170 of the 262 fall in different filing months, so duplicate filings or corrected re-submissions of one report cannot explain them. One caveat belongs with this count, developed in Section V-B: most of the 262 (232) occur in the post-2020 digital-native era, where exact reuse is both easier and perfectly preserved, so the raw count is not a clean prevalence trend; the pipeline-independent core is the 30 in the pre-2021 pure-scan era (18 at Firm A), which scanning noise alone cannot produce. Because a byte-identical pair has cosine = 1 and dHash = 0, it lands in HC by definition; the rule's "100% capture" of this set is therefore tautological, and we do not read it as a sanity check or a lower bound on recall. We use byte-identity only for what it can show directly — that reuse occurs and where it concentrates — as a prevalence signal, not a measure of detector performance.
+
+## V. Other Analyses
+
+This section gathers analyses that support the design and test its robustness: (a) the diagnostic showing the data contain no natural cutoff — the premise the whole calibration rests on; (b) how the baseline behaves after 2020; and (c) sensitivity checks.
+
+### A. Why the Data Contain No Natural Cutoff
+
+This diagnostic backs the design choice announced in Section III-D and Section III-E: that no cutoff can be read off the data, so the operating point has to be set from an outside reference. The Hartigan dip test [37] rejects a single-peak shape for both measures at the Big-4-pooled accountant level (p < 5×10⁻⁴), which might look like a clean split into two groups. But that rejection comes from two side-effects. Once we remove the differences between firms (by centering each firm on its own mean) and the effect of the hash taking only whole-number values (by adding a small jitter to dHash), the single-peak shape comes back (median p = 0.35 over jitter seeds). Tested firm by firm, each Big-4 firm is already unimodal on both axes (Firm A p_cos = 0.99, p_dHash = 0.92; B/C/D pooled p_cos = 0.998, p_dHash = 0.91), so the pooled rejection is a between-firm location-shift artifact, not within-population bimodality. A density-smoothness test in the Burgstahler–Dichev / McCrary style [38], [39] finds no real break in either measure at the Big-4 scope (Appendix A.1 shows the apparent signature-level breaks drift with histogram bin width and sit inside the high-similarity region — a resolution artifact, not an antimode). So the data hold no real gap; per-signature similarity is best read as one continuous spread of quality, not two separate classes. This is exactly why the operating point is set from an outside reference (Section III-E) rather than read off the data, and why the three groups used for description in Section IV-C are treated as a summary of composition, not as real mechanisms.
+
+A property of the cosine measure reinforces this and explains why the rule never leans on cosine alone. On these fixed-size, white-padded, ImageNet-normalized crops the within-accountant cosine is compressed into a narrow band at the top of its range: 97.7% of signatures score above 0.90, the median is 0.969, and only 0.3% fall below 0.85 (Appendix). Two signatures of the same accountant are highly cosine-similar whether the hand is steady or the image is reused, because the shared crop geometry and normalization contribute a common-structure baseline before any signature content is compared. The high cosine cut (0.95) therefore sits *inside* this saturated region — about three-quarters of signatures lie above it — so cosine on its own separates almost nothing; the structural dHash measure does the discriminating, which is why HC requires both and the cosine-only HSC band carries no evidential weight (Section III-D). We do not try to decompose this cosine baseline into its preprocessing and genuine-style parts here: doing so cleanly would mean re-extracting features with the padding and normalization ablated, which we flag as the way to quantify the preprocessing contribution and as a construct-validity check for future work.
+
+### B. Time Trend and the Firm–Pipeline Confound (Secondary)
+
+Looking only at Firms B/C/D, the strict rule's chance rate rises after 2020 (per comparison from 1.0×10⁻⁵ to 3.6×10⁻⁵; per signature from 0.59% to 1.05%), and the deployed HC rate rises in parallel (Firm B 29.0→42.0%, Firm C 21.6→26.7%, Firm D 22.0→28.0% across the two periods, Section IV-C). The rise is heterogeneous in timing rather than a common step. Tracing the yearly HC rate, Firm C's increase is concentrated in 2022 (about 18% through 2021, then ~30% in 2022 and ~40% in 2023) and Firm B's mainly in 2023 (about 33% in 2022, ~54% in 2023), while Firm D rises gradually with no visible step; Firm A, by contrast, is already high throughout the decade (80.3→83.8%) with no adoption-like jump — consistent with the interviews' account of long-standing stamping. This firm-by-firm staggering is what one would expect from progressive, independent adoption of formal signing systems (Section III-A), and it is why we limit the calibration to the pre-2020 years. Table II-c gives the full five-way breakdown by firm for the 2020–2023 deployment period, as a companion to the calibration-period Table II-b and for direct cross-checking against the proportions quoted here and in Section IV-C.
+
+**Table II-c — Five-way breakdown by firm, deployment period (Firms A–D, 2020–2023).**
+
+| Firm | HC | MC | HSC | UN | LH | signatures |
+|---|---|---|---|---|---|---|
+| Firm A | 83.84% | 9.13% | 0.04% | 6.88% | 0.11% | 23,898 |
+| Firm B | 42.01% | 31.24% | 0.16% | 26.31% | 0.28% | 14,571 |
+| Firm C | 26.74% | 40.55% | 0.40% | 31.80% | 0.51% | 16,164 |
+| Firm D | 27.98% | 28.85% | 0.24% | 42.42% | 0.51% | 7,188 |
+
+We deliberately stop short of reading this as a *detected* e-signing effect, because of a confound these data cannot break: firm identity — and period within a firm — bundles signing practice together with the entire imaging pipeline, and that pipeline demonstrably changes across the decade. We audited the production provenance of a stratified sample of 880 report PDFs (20 per firm-year) from their embedded metadata and page structure. The shift is stark (Table V): through 2020, reports are overwhelmingly plain scanned rasters — 70–85% in the early years carry no text layer at all, and their PDF metadata names the scanning hardware directly (for example "Fuji Xerox D125" and "ApeosPort-IV 7080") — whereas from 2021 plain scans collapse to about 1–2% as firms move to OCR'd and digital-native production. The two similarity measures are therefore computed on a substrate that itself transforms around 2020–2021, exactly when the baseline firms' similarity rises; firms also differ from one another in this respect (Firm A adopts digital-native output earliest, Firm C latest), though the cross-firm gap is much smaller than the temporal one. A post-2020 rise in similarity could thus come from this coincident pipeline change just as easily as from a change in how signatures are applied (Section III-D), and with no labels and no externally-dated adoption events the two are not separable here.
+
+**Table V — Imaging-pipeline audit: production type by year (stratified sample, 880 PDFs, 20 per firm-year). "Scanned" = no extractable text layer; "digital-native" = text-based PDF with embedded image objects; the remainder are scanned-then-OCR'd.**
+
+| Year | Scanned % | Digital-native % | Year | Scanned % | Digital-native % |
+|---|---|---|---|---|---|
+| 2013 | 82 | 0 | 2019 | 56 | 0 |
+| 2014 | 76 | 0 | 2020 | 52 | 0 |
+| 2015 | 85 | 0 | 2021 | 1 | 7 |
+| 2016 | 70 | 2 | 2022 | 2 | 16 |
+| 2017 | 50 | 0 | 2023 | 2 | 30 |
+| 2018 | 55 | 0 | | | |
+
+This same transition qualifies the byte-identity evidence (Section IV-C), which we flag rather than let the raw count mislead. Of the 262 byte-identical signatures, 232 fall in the digital-native era (2021–2023), where embedding a discrete signature image makes exact reuse both easy to do and perfectly preserved — so the post-2020 surge in byte-identical pairs is inflated by detectability and should not be read as a purely behavioral increase. The pipeline-robust core is the 30 byte-identical signatures in the pre-2021 pure-scan era, 18 of them at Firm A: two independently hand-signed pages, separately scanned, cannot yield byte-identical crops, so these are direct evidence of digital image reuse that predates the digital-native transition and concentrates at Firm A. This is also why Firm A's elevation, present throughout the scanned years, cannot be an artifact of digital-native embedding.
+
+The clean way to separate them would be an event study that aligns each firm to its own externally-documented e-signing adoption date and absorbs static firm differences and common-year shocks with firm and year fixed effects; a within-firm jump locked to each firm's own adoption month would be evidence for signing practice over static pipeline. We do not possess those adoption dates — and inferring them from the very HC series we would then test would be circular (Section III-E) — so we flag this event study as the natural next step rather than a result we can claim here. For this paper the trend serves only its narrower purpose: it justifies the pre-2020 calibration window and stands as a robustness check, not as a causal finding.
+
+### C. Sensitivity and Robustness
+
+We summarize the robustness checks here; full detail is in the supplementary materials.
+
+How sensitive the operating point is. Right around the HC cutoff the per-signature firing rate changes quickly — its local slope is about 25× the median across a cosine sweep and about 3.8× across a dHash sweep — which confirms that the HC point is a chosen, specificity-anchored operating point rather than a natural gap.
+
+A single slope understates how the rule behaves, so we map the full surface rather than defend one cut. Figure 6 plots, over the entire (cosine cut × dHash cut) plane, the clean-group flag rate (panel a) and the Firm A − B/C/D flag-rate contrast (panel b), and neither view favors the chosen cut by construction. First, the surfaces are smooth: there is no cliff at (0.95, dHash ≤ 5), so the operating point is a readable choice on a continuous trade-off rather than a discovered boundary (Section V-A), and an operator who wants a tighter floor can move toward higher cosine and lower dHash and read the consequence off the surface. Second, the firm contrast is not an artifact of the threshold: it exceeds 45 percentage points across a broad region of low-dHash, high-cosine cuts and in fact grows as the cut tightens (for example 58 pp at cosine 0.97, dHash ≤ 3), so the deliberately looser HC point trades a few points of contrast for catching more reuse, not the reverse. The same surface makes the weakness of the cosine-only direction explicit: extending the structural cut to the MC bound (dHash ≤ 15) roughly halves the contrast (to about 27 pp) while sharply inflating the clean-group flag rate. That is precisely why the MC band is only advisory and the cosine-only HSC band carries no weight (Section III-D): the partition is not drawn to flatter the narrative, and the surface shows directly where each band earns its keep and where it does not.
+
+![](figures/fig6.png)
+
+*Figure 6. Sensitivity surface of the deployed rule over the two-measure threshold plane (Big-4, n = 150,442). (a) Clean-group (B/C/D) flag rate at each (cosine cut, dHash cut); the chosen HC operating point (star) sits in a low-rate, high-specificity region with no cliff. (b) Firm A minus B/C/D flag-rate contrast (percentage points); the contrast exceeds 45 pp across a broad low-dHash, high-cosine band and weakens toward the MC bound (dHash ≤ 15, dotted), so the operating point is not a cherry-picked threshold and the MC band is visibly the less discriminating region.* The MC/HSC boundary at dHash = 15 sits in a flat (saturating) region, where moving the line adds flagged cases without adding specificity; this is a further reason to treat the MC band as advisory (Section IV-B).
+
+Leaving out one firm at a time. A two-group fit is unstable across firms — its boundary is basically a "Firm A versus the rest" divider — while a three-group fit keeps a stable shape (its low-cosine/high-dHash group drifts by at most 0.005 in cosine) but a membership that shifts with the mix of firms (by up to 12.8 percentage points). So we use the groups only as descriptions, never as operational labels.
+
+Crossover scope. The low cosine cut is the same-vs-different-accountant cosine crossover; recomputing it across scopes moves it by at most 0.025 — 0.8547 on the calibration cell (the primary value; Section IV-A), 0.8367 corpus-wide, 0.8489 on the all-period baseline firms, 0.8302 with the non-Big-4 firms added — and because the cut affects only the UN/LH boundary, switching among these scopes changes no HC/MC/HSC result and shifts the UN/LH split by at most 0.4 percentage points per firm. We use the calibration-cell value as primary for held-out discipline and report the others as robustness.
+
+The same-pair variant. A reader may worry that the deployed rule is a *derived* statistic rather than an observation: the cosine maximum and the dHash minimum are each taken over the accountant's pool and can originate from different partner signatures, so the high-confidence region might in principle be assembled from two unrelated extrema. We therefore recompute the rule under the strict *same-pair* construction, where a single partner signature must satisfy both inequalities at once (Section III-D), and report it in the main text rather than the supplement. Two views agree. First, the within-firm concentration of cross-accountant matches is higher under same-pair (97.0–99.96% across the four firms) than under the deployed any-pair rule (76.7–98.8%). Second, and more directly, the per-signature HC flag rate — the quantity the any-pair concern targets — behaves the same way (Table VI): requiring one partner to satisfy both inequalities lowers every firm's rate, as expected, but it *widens* the firm gap rather than narrowing it. Firm A still fires on a majority of its own signatures (57.3%) while the baseline firms fall to 5–9%, so the Firm-A-to-baseline ratio rises from about 2.4–3.4× under any-pair to about 6.4–10.8× under same-pair. The high-confidence region is therefore not an artifact of combining extrema from different partner signatures; pushed to the stricter event, the structure gets stronger.
+
+**Table VI — HC flag rate by firm under the deployed any-pair rule and the strict same-pair rule.**
+
+| Firm | Signatures | Any-pair HC | Same-pair HC |
+|---|---|---|---|
+| Firm A | 60,448 | 81.7% | 57.3% |
+| Firm B | 34,248 | 34.6% | 9.0% |
+| Firm C | 38,613 | 23.7% | 5.3% |
+| Firm D | 17,133 | 24.5% | 7.7% |
+| All Big-4 | 150,442 | 49.6% | 27.3% |
+
+Each gate adds specificity. On the all-four-firm pool the cosine gate alone fires per comparison at 6.0×10⁻⁴; adding the structural gate multiplies this by 0.234 (the conditional ICCR of dHash ≤ 5 given cos > 0.95), giving the joint 1.4×10⁻⁴. Each axis contributes specificity beyond the other — quantitative support for the two-gate design over either measure alone (Section I, Section III-D).
+
+Which network we use. We compare ResNet-50 against VGG-16 and EfficientNet-B0 under the same preprocessing and L2 normalization (Appendix A; supplementary backbone-ablation table). EfficientNet-B0 gives the largest intra/inter separation (Cohen's d = 0.707) but also the widest descriptor spread (intra std 0.123 vs ResNet-50's 0.098); VGG-16 is worst on every key metric despite its larger 4096-dim features. ResNet-50 is the best overall balance: its Cohen's d (0.669) is competitive, its tighter distributions give more stable per-signature behavior, it yields the highest Firm A all-pairs 1st-percentile similarity (0.543), and its 2048-dim features are a practical compromise for processing 182K+ signatures. The comparison supports ResNet-50.
+
+### D. Threats to Validity
+
+For the reader's convenience we collect the main threats to validity in one place, each with a pointer to where it is treated and, where relevant, the direction of its bias. They are consequences of working without labels, and we state them as limitations rather than dissolve them.
+
+1. *No signature-level ground truth.* The archive labels no signature as hand-signed or reused, so we report no recall, precision, ROC-AUC, or false-rejection rate, and every rate is a chance rate, not an error rate (Section III-D, Section VI).
+2. *Wrong null for the reuse question.* The ICCR is a between-accountant coincidence rate; the within-accountant false-positive rate the question needs is not estimable and the ICCR is not even a bound on it, so "X× the floor" comparisons are avoided as anti-conservative (Section III-E, Section IV-C).
+3. *Reference-group contamination / circular selection.* The clean floor is conditional on the reference group truly being clean; undetected reuse there would only raise the floor, biasing the Firm-A contrast conservatively, and the floor does not hinge on any single baseline firm (Section III-E).
+4. *Pool-size and extremal dependence.* The rule takes a maximum over a pool, so larger pools mechanically raise fire rates; the firm contrast nonetheless holds within every pool-size stratum and under accountant-clustered resampling (Section IV-C).
+5. *Firm–pipeline confound.* Firm identity bundles signing practice with the imaging pipeline (crop geometry and reuse rates differ by firm), and internal timing cannot separate the two without externally-dated adoption events; a fixed-effects event study is the natural next step (Section V-B).
+6. *Preprocessing and construct validity.* Padding and ImageNet normalization compress cosine into a narrow high band, so cosine alone discriminates little and the rule relies on the structural measure; a padding/normalization ablation is needed to quantify the preprocessing contribution (Section V-A).
+7. *Generalizability.* Calibration is on a Chinese-signature corpus from one jurisdiction with a specific pipeline; the operating point is a starting reference for comparable pipelines, not a transplantable constant, and requires recalibration elsewhere (Section III-D, Section VI).
+8. *Non-reproducible corroboration and an unrun protocol.* The interviews are self-reported and not reproducible, so agreement with them shows consistency with domain knowledge, not measured accuracy (Section III-A); and the review protocol of Section IV-B is a designed procedure whose validating first run remains future work.
+
+## VI. Conclusion
+
+We have presented a label-free, anchor-calibrated way to screen for non-hand-signed signatures in large numbers of audit reports. It has three working parts — a pipeline that takes raw PDFs through page-finding, detection, feature extraction, and a two-measure similarity step; a pair of measures that separate style consistency from image reproduction; and, in place of a natural cutoff we do not have and labeled data we cannot get, a calibration based on how often the rule fires by chance in a clean reference group. That calibration yields both a between-accountant specificity proxy and a concrete operating point: the high-confidence rule almost never fires by chance on the clean group, so it is a usable, highly specific screen, with a defined, bounded human-review protocol (Section IV-B) for the advisory and uncertain cases. Operationally the screen earns its keep in two ways: run over an archive, it discovers where reuse concentrates; and it keeps human review at the scale of exceptions in both kinds of population — settling most signatures directly where reuse dominates, and, where practices are mixed, demoting the low-specificity band, ranking accountants, and confirming the byte-identical cases, withholding only the per-signature verdict for the ambiguous middle. We report the category proportions that make that distinction concrete. Because it is calibrated on a large Chinese-signature corpus and uses script-agnostic image descriptors, the rule offers a practical starting reference for comparable Chinese-signature pipelines and, in principle, an approach portable to other scripts — subject in each case to recalibration on the new setting. Held out as a known-positive benchmark, one firm stands alone in how alike its own signatures are, its output matches the stamping practice the firm itself describes, and byte-identical signatures give direct evidence that reuse happens and concentrates there.
+
+The limits are built into working without labels, and we have stated them alongside the design. There is no signature-level ground truth, so we report no false-rejection rate, recall, ROC-AUC, or precision; every rate we give is a between-accountant chance rate read as a proxy for specificity, not a true false-acceptance rate, and not even a bound on the within-accountant false-positive rate the reuse question would need (Section III-E). The contrast between firms is something the method can see, not a finding about why the signatures look alike: for any single signature, the two measures cannot separate reuse from a shared scanning pipeline or a uniform house style, and for Firms B/C/D we make no claim about firm practice at all. Whether firm-level signing patterns matter for audit quality is a question for a dedicated companion study — one this screening points toward, together with the low-presence character of proxy-executed stamping shown in the behavioral literature, but one that similarity alone cannot settle.
+
+Four directions follow. First, a set-level reading of each accountant: judging the shape of an accountant's whole signature set — a tight cluster that recurs near-identically across reports and years (the signature of a stored image) versus a dispersed cloud (the signature of a hand) — instead of per-signature extrema. This would collapse much of the remaining middle into a few per-accountant cluster decisions, and it is the natural tool for separating the mixed signers of the baseline firms, whose sets may contain both a tight recurring sub-cluster and a dispersed remainder if both practices are present. We view this as the highest-value methodological extension, while noting honestly that it narrows but does not remove the fundamental ambiguity: a very steady hand and a noisy reused image can still meet in the middle of any set-level statistic. A first-pass probe on the calibration cell is consistent with this caution — across the 206 of the 226 Firms-B/C/D 2013–2019 accountants with enough signatures for a set-level shape to be estimated, the within-accountant similarity forms a continuum that piles up just below the high-similarity cut rather than splitting into a tight reused cluster and a dispersed hand-signed cloud (no accountant shows a tight-versus-remainder cosine gap above 0.10), so the no-natural-cutoff structure of Section V-A recurs at the accountant level; we therefore treat set-level adjudication as a research direction rather than a ready robustness result. Second, executing the review protocol of Section IV-B on a bounded sample — its first run — would both test the protocol's expected discriminating behavior and accumulate the small human-labeled set that permits supervised validation and direct error rates. Third, image-acquisition metadata (scanner identifiers, PDF-generator fingerprints, compression markers) adds a provenance axis that could help resolve the pipeline-versus-reuse ambiguity similarity alone cannot; we confirmed this metadata survives in the present corpus rather than being flattened by the platform, though its discriminative power remains to be validated (Section IV-B, Move 4). Fourth, the audit-quality question itself: whether firm-level signing patterns correlate with audit outcomes, for which this screening supplies the measurement layer.
+
+## Appendix A. Supplementary Diagnostic Detail
+
+### A.1. BD/McCrary Bin-Width Sensitivity (Signature Level)
+
+The main text (Section III-D, Section V-A) 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 dHash direction.
+
+**Table A.I. BD/McCrary bin-width sensitivity (two-sided α = 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 (sig-level) | 60,448 | 1 | 2.0 | −4.69 | +10.01 |
+| Firm A dHash (sig-level) | 60,448 | 2 | no transition | — | — |
+| Firm A dHash (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 (sig-level) | 168,740 | 1 | 2.0 | −6.22 | +4.89 |
+| Full-sample dHash (sig-level) | 168,740 | 2 | 10.0 | −7.35 | +3.83 |
+| Full-sample dHash (sig-level) | 168,740 | 3 | 9.0 | −11.05 | +45.39 |
+
+Two patterns are visible. First, the procedure consistently identifies a "transition" under every bin width, but the location 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 bin width grows from 1 to 3), and the Z statistics inflate superlinearly with bin width because wider bins aggregate more mass and shrink the per-bin standard error on a very large sample. Both features are characteristic of a histogram-resolution artifact rather than a genuine density discontinuity. Second, the candidate transitions all locate inside the high-similarity region (cosine ≥ 0.975, dHash ≤ 10) rather than at a between-mode boundary. Taken together, the signature-level BD/McCrary transitions are not a threshold in the usual sense — they are histogram-resolution-dependent local density anomalies inside the high-similarity descriptor region rather than between modes — which supports using BD/McCrary as a density-smoothness diagnostic, not a threshold estimator (Section V-A).
+
+### A.2. Diagnostic Summary
+
+The unsupervised-diagnostic strategy is a set of complementary checks, each addressing one specific failure mode of an unsupervised screening classifier under an explicitly disclosed untested assumption.
+
+**Table A.II. Diagnostics, failure mode addressed, and disclosed untested assumption (abridged).**
+
+| Diagnostic | Failure mode addressed | Disclosed untested assumption |
+|---|---|---|
+| Composition decomposition (Section V-A) | Whether descriptor multimodality is within-population (mechanism) or between-group (composition + integer artifact); p_median = 0.35 under joint firm-mean centering + integer-tie jitter | Integer-tie jitter and firm-mean centering are unbiased over the descriptor support |
+| Per-comparison ICCR (Section IV-A) | Pair-level specificity proxy under a random-pair negative anchor, on the BCD baseline | Inter-CPA pairs are negative; addressed by anchoring on B/C/D and holding Firm A out |
+| Pool-normalised per-signature ICCR (Section IV-A) | Deployed-rule specificity proxy at per-signature unit, accounting for pool size | As above + pool replacement preserves the negative-anchor property |
+| Document-level ICCR (Section IV-A) | Operational alarm-rate proxy at per-document unit (HC and HC+MC) | As above |
+| Firm-heterogeneity logistic regression (Section IV-C) | Multiplicative effect of firm membership on per-signature rate, controlling for pool size | Observations clustered by CPA/firm; cluster-robust SEs are a future check |
+| Cross-firm hit matrix (Section IV-C, Section V-C) | Concentration of inter-CPA collisions within source firm | Concentration depends on deployed-rule semantics (same-pair 97.0–99.96% vs any-pair 76.7–98.8%) |
+| Alert-rate sensitivity sweep (Section V-C) | Local sensitivity of the deployed rule to threshold perturbation | Gradient comparison is descriptive, not a formal plateau test |
+| Convergent score Spearman ranking (Section IV-B) | Internal consistency of three feature-derived per-CPA scores | Scores share inputs; not statistically independent |
+| Pixel-identical positive capture (Section IV-C) | Prevalence evidence that reuse occurs and where it concentrates | Anchor is tautologically captured by any reasonable threshold; not read as a recall or performance measure |
+
+## Appendix B. Reproducibility Materials
+
+The full table-to-script provenance mapping, script source code, and report artifacts for every numerical table and figure in this paper are provided in the supplementary materials. Scripts run deterministically under fixed random seeds documented there (the inter-CPA candidate sampler uses seed 42 and a retry-loop matching the canonical samplers; CPA-block bootstraps use 1,000 replicates); reviewer reproduction should re-emit artifacts from the listed scripts rather than rely on any local path layout. The calibration baseline (BCD 2013–2019), the contamination-comparison scope (all-Big-4), the Firm-A out-of-sample scoring, and the five-way classification are all emitted by the same canonical pipeline so that the headline numbers in Tables I, II, II-b, and IV reproduce bit-for-bit.
+
+## References
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+
+## Declarations
+
+**Conflict of interest.** The authors declare no conflict of interest with Firm A, Firm B, Firm C, or Firm D, or with any other entity referenced in this work.
+
+**Data availability.** All audit reports analyzed in this study were obtained from the Market Observation Post System (MOPS) operated by the Taiwan Stock Exchange Corporation, a publicly accessible regulatory disclosure platform. The CPA registry used to map signatures to certifying CPAs is publicly available. The reproducibility scripts and trained model weights are provided in the supplementary materials; signature-image release is subject to the firm-anonymization constraints of Section III-A (a de-identified subset and the per-table provenance mapping are included, with the full image set available to reviewers under the platform's public-data terms).

paper/v13_build/scripts/close_items.py

@@ -0,0 +1,79 @@
+import sqlite3, numpy as np
+from collections import defaultdict
+from scipy.stats import gaussian_kde
+DB='/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+FIRM_A='勤業眾信聯合'; BIG4=('勤業眾信聯合','安侯建業聯合','資誠聯合','安永聯合')
+SEED=42; POP=np.array([bin(i).count('1') for i in range(256)],dtype=np.uint8)
+def load():
+    c=sqlite3.connect(f'file:{DB}?mode=ro',uri=True)
+    r=c.execute("""SELECT s.assigned_accountant,a.firm,s.source_pdf,s.feature_vector,s.dhash_vector,
+      CAST(substr(s.year_month,1,4) AS INT) FROM signatures s JOIN accountants a ON s.assigned_accountant=a.name
+      WHERE s.assigned_accountant IS NOT NULL AND a.firm IS NOT NULL AND s.feature_vector IS NOT NULL AND s.dhash_vector IS NOT NULL""").fetchall()
+    c.close(); return r
+def crossover(keep,label):
+    feats=np.stack([np.frombuffer(r[3],np.float32) for r in keep]).astype(np.float32)
+    feats/=np.clip(np.linalg.norm(feats,axis=1,keepdims=True),1e-9,None)
+    cpas=np.array([r[0] for r in keep]); by=defaultdict(list)
+    for i,c in enumerate(cpas): by[c].append(i)
+    by={c:np.array(v) for c,v in by.items() if len(v)>=3}; accts=list(by.keys())
+    pw=np.array([len(by[c])*(len(by[c])-1)/2 for c in accts],float); pw/=pw.sum()
+    rng=np.random.default_rng(SEED); M=100_000
+    intra=np.empty(M,np.float32); ci=rng.choice(len(accts),M,p=pw)
+    for t in range(M):
+        a,b=rng.choice(by[accts[ci[t]]],2,replace=False); intra[t]=feats[a]@feats[b]
+    inter=np.empty(M,np.float32)
+    for t in range(M):
+        i,j=rng.choice(len(accts),2,replace=False); inter[t]=feats[rng.choice(by[accts[i]])]@feats[rng.choice(by[accts[j]])]
+    xs=np.linspace(0.3,1.0,10000); diff=gaussian_kde(intra)(xs)-gaussian_kde(inter)(xs)
+    cr=[float(x) for x in xs[np.where(np.diff(np.sign(diff)))[0]] if 0.6<x<0.99]
+    print(f'  [{label}] crossover {[f"{x:.4f}" for x in cr]} (n={len(keep)}, accts>=3={len(accts)})')
+def percomp_bands(keep,label,M=500_000):
+    feats=np.stack([np.frombuffer(r[3],np.float32) for r in keep]).astype(np.float32)
+    feats/=np.clip(np.linalg.norm(feats,axis=1,keepdims=True),1e-9,None)
+    dh=np.stack([np.frombuffer(r[4],np.uint8) for r in keep]); cpas=np.array([r[0] for r in keep])
+    by=defaultdict(list)
+    for i,c in enumerate(cpas): by[c].append(i)
+    accts=[c for c,v in by.items() if len(v)>=1]; rng=np.random.default_rng(SEED)
+    n=len(keep); ii=rng.integers(0,n,M*2); jj=rng.integers(0,n,M*2)
+    keepm=cpas[ii]!=cpas[jj]; ii=ii[keepm][:M]; jj=jj[keepm][:M]
+    cos=np.einsum('ij,ij->i',feats[ii],feats[jj]); d=POP[dh[ii]^dh[jj]].sum(1)
+    hc=(cos>0.95)&(d<=5); mc=(cos>0.95)&(d>5)&(d<=15); hsc=(cos>0.95)&(d>15)
+    un=(cos>0.837)&(cos<=0.95); lh=cos<=0.837
+    print(f'  [{label}] per-COMPARISON ICCR (M={len(ii)}): HC {hc.mean():.6f}  MC {mc.mean():.6f}  HSC {hsc.mean():.6f}  UN {un.mean():.4f}  LH {lh.mean():.4f}')
+def persig_perdoc_bands(keep,label):
+    n=len(keep); feats=np.stack([np.frombuffer(r[3],np.float32) for r in keep]).astype(np.float32)
+    feats/=np.clip(np.linalg.norm(feats,axis=1,keepdims=True),1e-9,None)
+    dh=np.stack([np.frombuffer(r[4],np.uint8) for r in keep]); cpas=np.array([r[0] for r in keep]); docs=np.array([r[2] for r in keep])
+    ci=defaultdict(list)
+    for i,c in enumerate(cpas): ci[c].append(i)
+    ci={c:np.array(v) for c,v in ci.items()}; ps={c:len(v)-1 for c,v in ci.items()}
+    allidx=np.arange(n); rng=np.random.default_rng(SEED); mc=np.zeros(n,np.float32); md=np.full(n,64,np.int32)
+    for si in range(n):
+        p=ps[cpas[si]]
+        if p<=0: continue
+        same=ci[cpas[si]]; need=p; cand=[]; att=0
+        while need>0 and att<10:
+            dr=rng.choice(n,size=need*2,replace=True); ok=dr[~np.isin(dr,same)]; cand.extend(ok[:need].tolist()); need-=len(ok[:need]); att+=1
+        cand=np.array(cand[:p],dtype=np.int64)
+        mc[si]=(feats[cand]@feats[si]).max(); md[si]=int(POP[dh[cand]^dh[si]].sum(1).min())
+    un=(mc>0.837)&(mc<=0.95); hsc=(mc>0.95)&(md>15)
+    # per-doc: any signature in band
+    dd=defaultdict(list)
+    for i in range(n): dd[docs[i]].append(i)
+    docs_un=np.mean([un[v].any() for v in dd.values()]); docs_hsc=np.mean([hsc[v].any() for v in dd.values()])
+    print(f'  [{label}] per-SIGNATURE ICCR: UN {un.mean():.4f}  HSC {hsc.mean():.6f}')
+    print(f'  [{label}] per-REPORT ICCR:    UN {docs_un:.4f}  HSC {docs_hsc:.6f}  (n_doc={len(dd)})')
+
+rows=load()
+bcd_all=[r for r in rows if r[1] in BIG4 and r[1]!=FIRM_A]
+bcd_19=[r for r in bcd_all if 2013<=r[5]<=2019]
+print("=== ITEM 11: KDE crossover (verify corpus 0.837 / BCD-all 0.8489, then closed-world 2013-2019) ===")
+crossover(rows,'corpus-wide (verify ~0.8367)')
+crossover(bcd_all,'BCD-only ALL period (verify 0.8489)')
+crossover(bcd_19,'BCD 2013-2019 CLOSED-WORLD (NEW primary candidate)')
+print("\n=== ITEM 3: UN / HSC full ICCR on BCD 2013-2019 ===")
+percomp_bands(bcd_19,'BCD 2013-2019')
+persig_perdoc_bands(bcd_19,'BCD 2013-2019')
+print("\n=== ITEM 12: n reconciliation ===")
+print(f"  BCD full-period (2013-2023) signatures = {len(bcd_all)}  <- Script53 logged n=89,994")
+print(f"  BCD 2013-2019 signatures              = {len(bcd_19)}  <- headline ICCR base (reproduces 0.0059)")

paper/v13_build/scripts/f5_fe_loyo.py

@@ -0,0 +1,63 @@
+"""F5 robustness: firm+year fixed-effects logistic regression and leave-one-year-out.
+Complements the pool-size stratification and accountant-clustered bootstrap (Section IV-C).
+Uses numpy+scipy only (no statsmodels). Reproduces from signature_analysis.db.
+"""
+import sqlite3, numpy as np
+from scipy.optimize import minimize
+
+DB = "/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db"
+BIG4 = ('勤業眾信聯合', '資誠聯合', '安侯建業聯合', '安永聯合')
+FM = {'勤業眾信聯合': 'A', '安侯建業聯合': 'B', '資誠聯合': 'C', '安永聯合': 'D'}
+
+con = sqlite3.connect(DB); cur = con.cursor()
+cur.execute(f"""
+SELECT s.excel_firm, CAST(substr(s.year_month,1,4) AS INT) yr,
+  (CASE WHEN s.max_similarity_to_same_accountant>0.95 AND s.min_dhash_independent<=5 THEN 1 ELSE 0 END) hc,
+  p.psize
+FROM signatures s
+JOIN (SELECT accountant_id, COUNT(*) psize FROM signatures
+      WHERE is_valid=1 AND excel_firm IN ({','.join('?'*4)})
+        AND max_similarity_to_same_accountant IS NOT NULL AND min_dhash_independent IS NOT NULL
+      GROUP BY accountant_id) p ON s.accountant_id=p.accountant_id
+WHERE s.is_valid=1 AND s.excel_firm IN ({','.join('?'*4)})
+  AND s.max_similarity_to_same_accountant IS NOT NULL AND s.min_dhash_independent IS NOT NULL
+  AND s.year_month GLOB '2[0-9][0-9][0-9][0-9][0-9]'
+""", BIG4 + BIG4)
+rows = cur.fetchall(); con.close()
+firm = np.array([FM[r[0]] for r in rows]); yr = np.array([r[1] for r in rows])
+hc = np.array([r[2] for r in rows], float); pool = np.array([r[3] for r in rows], float)
+n = len(hc); years = sorted(set(yr.tolist()))
+
+# --- firm + year FE logistic (Firm A & first year = reference) ---
+cols = [np.ones(n)]; names = ['const']
+for f in ['B', 'C', 'D']:
+    cols.append((firm == f).astype(float)); names.append(f'firm_{f}')
+for y in years[1:]:
+    cols.append((yr == y).astype(float)); names.append(f'yr_{y}')
+lp = np.log(pool); lp = (lp - lp.mean()) / lp.std()
+cols.append(lp); names.append('logpool_z')
+X = np.column_stack(cols)
+
+def nll(b):
+    z = X @ b
+    return -np.sum(hc * z - np.logaddexp(0, z)) + 1e-6 * np.sum(b * b)
+def grad(b):
+    p = 1 / (1 + np.exp(-(X @ b)))
+    return -X.T @ (hc - p) + 2e-6 * b
+b = minimize(nll, np.zeros(X.shape[1]), jac=grad, method='L-BFGS-B').x
+print("Firm+Year FE logistic (Firm A & first year = ref):")
+for nm, bi in zip(names, b):
+    if nm.startswith('firm') or nm == 'logpool_z':
+        print(f"  {nm:11} coef={bi:7.3f}  OR={np.exp(bi):.4f}")
+
+# --- leave-one-year-out firm contrast ---
+grp = np.where(firm == 'A', 'A', 'BCD')
+def rate(mask, g):
+    m = mask & (grp == g); return 100 * hc[m].mean()
+print("\nLeave-one-year-out (Firm A minus B/C/D HC gap):")
+gaps = []
+for y in years:
+    keep = (yr != y); a = rate(keep, 'A'); bb = rate(keep, 'BCD'); gaps.append(a - bb)
+    print(f"  drop {y}: A={a:.1f}%  BCD={bb:.1f}%  gap={a-bb:.1f}pp")
+print(f"  full-sample gap={rate(np.ones(n, bool),'A')-rate(np.ones(n, bool),'BCD'):.1f}pp; "
+      f"LOYO range=[{min(gaps):.1f}, {max(gaps):.1f}]pp")

paper/v13_build/scripts/harvest_period.py

@@ -0,0 +1,70 @@
+import sqlite3
+from collections import defaultdict, Counter
+import numpy as np
+DB='/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+FIRM_A='勤業眾信聯合'; BIG4=('勤業眾信聯合','安侯建業聯合','資誠聯合','安永聯合')
+ALIAS={'勤業眾信聯合':'A','安侯建業聯合':'B','資誠聯合':'C','安永聯合':'D'}
+SEED=42; POP=np.array([bin(i).count('1') for i in range(256)],dtype=np.uint8)
+def wilson(k,n,z=1.96):
+    if n==0: return (None,None)
+    p=k/n; d=1+z*z/n; c=(p+z*z/(2*n))/d; h=z*np.sqrt(p*(1-p)/n+z*z/(4*n*n))/d
+    return (max(0,c-h),min(1,c+h))
+def load():
+    c=sqlite3.connect(f'file:{DB}?mode=ro',uri=True); cur=c.cursor()
+    cur.execute("""SELECT s.assigned_accountant,a.firm,s.source_pdf,s.feature_vector,s.dhash_vector,
+        CAST(substr(s.year_month,1,4) AS INT) FROM signatures s JOIN accountants a ON s.assigned_accountant=a.name
+        WHERE s.assigned_accountant IS NOT NULL AND a.firm IS NOT NULL AND s.feature_vector IS NOT NULL AND s.dhash_vector IS NOT NULL""")
+    r=cur.fetchall(); c.close(); return r
+def canonical_sampler(rng,n,n_pool,same_cpa,all_idx):
+    need=n_pool; cand=[]; att=0
+    while need>0 and att<10:
+        draw=rng.choice(n,size=need*2,replace=True); ok=draw[~np.isin(draw,same_cpa)]
+        cand.extend(ok[:need].tolist()); need-=len(ok[:need]); att+=1
+    if need>0:
+        pm=np.ones(n,bool); pm[same_cpa]=False
+        cand.extend(rng.choice(all_idx[pm],size=need,replace=False).tolist())
+    return np.array(cand[:n_pool],dtype=np.int64)
+def simulate(keep):
+    n=len(keep); feats=np.stack([np.frombuffer(r[3],np.float32) for r in keep]).astype(np.float32)
+    nr=np.linalg.norm(feats,axis=1,keepdims=True); nr[nr==0]=1; feats=feats/nr
+    dh=np.stack([np.frombuffer(r[4],np.uint8) for r in keep]); cpas=np.array([r[0] for r in keep])
+    cpa_idx=defaultdict(list)
+    for i,c in enumerate(cpas): cpa_idx[c].append(i)
+    cpa_idx={c:np.array(v) for c,v in cpa_idx.items()}; ps={c:len(v)-1 for c,v in cpa_idx.items()}
+    all_idx=np.arange(n); rng=np.random.default_rng(SEED)
+    mc=np.zeros(n,np.float32); md=np.full(n,64,np.int32)
+    for si in range(n):
+        p=ps[cpas[si]]
+        if p<=0: continue
+        cand=canonical_sampler(rng,n,p,cpa_idx[cpas[si]],all_idx)
+        mc[si]=(feats[cand]@feats[si]).max(); md[si]=int(POP[dh[cand]^dh[si]].sum(axis=1).min())
+    return mc,md
+def iccr(keep,label):
+    mc,md=simulate(keep); n=len(keep)
+    hc=(mc>0.95)&(md<=5); d2=(mc>0.95)&(md<=15)
+    un=(mc>0.837)&(mc<=0.95); hsc=(mc>0.95)&(md>15)
+    print(f"\n== {label} (n_sig={n:,}) ==")
+    for nm,a in [('HC',hc),('HC+MC',d2),('UN-band',un),('HSC-band',hsc)]:
+        k=int(a.sum()); lo,hi=wilson(k,n); print(f"  ICCR per-sig {nm}: {k/n:.6f} ({k}/{n}) [{lo:.5f},{hi:.5f}]")
+def a_oos(rows,label):
+    A=[r for r in rows if r[1]==FIRM_A]; BCD=[r for r in rows if r[1] in BIG4 and r[1]!=FIRM_A]
+    bf=np.stack([np.frombuffer(r[3],np.float32) for r in BCD]).astype(np.float32)
+    bn=np.linalg.norm(bf,axis=1,keepdims=True); bn[bn==0]=1; bf=bf/bn
+    bdh=np.stack([np.frombuffer(r[4],np.uint8) for r in BCD]); nb=bf.shape[0]
+    ac=defaultdict(list)
+    for i,r in enumerate(A): ac[r[0]].append(i)
+    ps={c:len(v)-1 for c,v in ac.items()}; rng=np.random.default_rng(SEED); hc=np.zeros(len(A),bool)
+    for i,r in enumerate(A):
+        p=ps[r[0]]
+        if p<=0: continue
+        cand=rng.choice(nb,size=p,replace=True); sf=np.frombuffer(r[3],np.float32).astype(np.float32); sf=sf/max(np.linalg.norm(sf),1e-9)
+        mc=(bf[cand]@sf).max(); mdv=int(POP[bdh[cand]^np.frombuffer(r[4],np.uint8)].sum(axis=1).min())
+        hc[i]=(mc>0.95)and(mdv<=5)
+    k=int(hc.sum()); n=len(A); lo,hi=wilson(k,n)
+    print(f"\n== Firm A OOS vs {label} BCD pool == per-sig HC: {k/n:.6f} ({k}/{n}) [{lo:.6f},{hi:.6f}]")
+rows=load()
+bcd_all=[r for r in rows if r[1] in BIG4 and r[1]!=FIRM_A]
+bcd_19=[r for r in bcd_all if 2013<=r[5]<=2019]
+iccr(bcd_19,'BCD 2013-2019 (verify per-sig HC~0.0059)')
+a_oos([r for r in rows if 2013<=r[5]<=2019],'2013-2019')
+a_oos(rows,'full-period')

paper/v13_build/scripts/make_fig3_density.py

@@ -0,0 +1,84 @@
+"""Figure 3 (real data version): 2D density of the two measures over the five-region scheme.
+Replaces the earlier schematic with the actual distribution, with axis ticks and the rule cuts.
+Reproduces from signature_analysis.db; Big-4, is_valid=1, both measures present."""
+import matplotlib
+matplotlib.use('Agg')
+import matplotlib.pyplot as plt
+from matplotlib.colors import LogNorm
+from matplotlib.patches import Rectangle
+import numpy as np
+import sqlite3
+
+DB = "/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db"
+BIG4 = ('勤業眾信聯合', '資誠聯合', '安侯建業聯合', '安永聯合')
+
+con = sqlite3.connect(DB)
+cur = con.cursor()
+cur.execute(f"""
+SELECT s.max_similarity_to_same_accountant, s.min_dhash_independent
+FROM signatures s JOIN accountants a ON s.assigned_accountant=a.name
+WHERE s.max_similarity_to_same_accountant IS NOT NULL
+  AND s.min_dhash_independent IS NOT NULL
+  AND a.firm IN ({','.join(['?']*4)})
+""", BIG4)
+rows = cur.fetchall()
+con.close()
+cos = np.array([r[0] for r in rows], dtype=float)
+dh = np.array([r[1] for r in rows], dtype=float)
+n = len(cos)
+
+LO, HI = 0.8547, 0.95
+DH1, DH2 = 5, 15
+xmin, xmax = 0.70, 1.002
+ymin, ymax = -0.5, 30
+ycap = 30  # display cap; values above are piled into the top row for visibility
+
+dh_disp = np.minimum(dh, ycap - 0.5)
+
+fig, ax = plt.subplots(figsize=(5.6, 4.4))
+
+# faint region tint behind the density
+ax.add_patch(Rectangle((xmin, ymin), LO - xmin, ymax - ymin, facecolor='#bdc3c7', alpha=0.12, zorder=0))
+ax.add_patch(Rectangle((LO, ymin), HI - LO, ymax - ymin, facecolor='#f7dc6f', alpha=0.12, zorder=0))
+ax.add_patch(Rectangle((HI, ymin), xmax - HI, DH1 - ymin, facecolor='#cb4335', alpha=0.14, zorder=0))
+ax.add_patch(Rectangle((HI, DH1), xmax - HI, DH2 - DH1, facecolor='#eb984e', alpha=0.14, zorder=0))
+ax.add_patch(Rectangle((HI, DH2), xmax - HI, ymax - DH2, facecolor='#aed6f1', alpha=0.14, zorder=0))
+
+# real 2D density (log counts)
+xedges = np.linspace(xmin, xmax, 90)
+yedges = np.arange(-0.5, ycap + 0.5, 1.0)  # integer dHash bins
+H, xe, ye = np.histogram2d(cos, dh_disp, bins=[xedges, yedges])
+pcm = ax.pcolormesh(xe, ye, H.T, norm=LogNorm(vmin=1, vmax=H.max()),
+                    cmap='viridis', zorder=1, shading='flat')
+cb = fig.colorbar(pcm, ax=ax, pad=0.02)
+cb.set_label('signatures per cell (log scale)', fontsize=8)
+cb.ax.tick_params(labelsize=7)
+
+# cut lines
+ax.axvline(LO, color='gray', ls=':', lw=1.1, zorder=3)
+ax.axvline(HI, color='black', ls='--', lw=1.1, zorder=3)
+ax.plot([HI, xmax], [DH1, DH1], 'k--', lw=0.9, zorder=3)
+ax.plot([HI, xmax], [DH2, DH2], 'k--', lw=0.9, zorder=3)
+
+# region labels
+ax.text((xmin + LO) / 2, 24, 'LH', ha='center', fontsize=10, weight='bold', color='#34495e', zorder=4)
+ax.text((LO + HI) / 2, 24, 'UN', ha='center', fontsize=10, weight='bold', color='#7d6608', zorder=4)
+ax.text((HI + xmax) / 2, 2.2, 'HC', ha='center', fontsize=10, weight='bold', color='#cb4335', zorder=4)
+ax.text((HI + xmax) / 2, 9.7, 'MC', ha='center', fontsize=10, weight='bold', color='#a04000', zorder=4)
+ax.text((HI + xmax) / 2, 24, 'HSC', ha='center', fontsize=9, weight='bold', color='#21618c', zorder=4)
+
+ax.set_xlim(xmin, xmax)
+ax.set_ylim(ymin, ymax)
+ax.set_xticks([0.70, 0.75, 0.80, 0.8547, 0.90, 0.95, 1.00])
+ax.set_xticklabels(['0.70', '0.75', '0.80', '0.855', '0.90', '0.95', '1.00'], fontsize=7.5)
+ax.set_yticks([0, 5, 10, 15, 20, 25, 30])
+ax.set_yticklabels(['0', '5', '10', '15', '20', '25', '≥30'], fontsize=7.5)
+ax.set_xlabel('cosine similarity to same accountant (style)', fontsize=9)
+ax.set_ylabel('min dHash distance (structure)', fontsize=9)
+ax.set_title(f'Figure 3. Two-measure plane: real density over the five regions (Big-4, n={n:,})',
+             fontsize=8.5)
+fig.tight_layout()
+out = '/Volumes/NV2/pdf_recognize/paper/v13_build/figures/fig3.png'
+fig.savefig(out, dpi=200, bbox_inches='tight')
+plt.close(fig)
+print(f'fig3 density OK: n={n:,}, dHash>=30 piled: {(dh>=ycap).sum()}, written {out}')

paper/v13_build/scripts/make_fig6_sensitivity.py

@@ -0,0 +1,62 @@
+"""Figure 6: two-measure sensitivity surface over the (cosine cut x dHash cut) plane.
+Panel A: clean-group (B/C/D) flag rate -- how permissive the operating point is.
+Panel B: Firm A minus B/C/D flag-rate contrast (pp) -- discrimination across the plane.
+Shows the chosen HC point (0.95, dHash<=5) is not a cherry-picked threshold and exposes
+the weaker MC band (dHash<=15). Reproduces from signature_analysis.db (DB columns only).
+"""
+import matplotlib
+matplotlib.use('Agg')
+import matplotlib.pyplot as plt
+import numpy as np
+import sqlite3
+
+DB = "/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db"
+BIG4 = ('勤業眾信聯合', '資誠聯合', '安侯建業聯合', '安永聯合')
+
+con = sqlite3.connect(DB); cur = con.cursor()
+cur.execute(f"""SELECT CASE WHEN a.firm='勤業眾信聯合' THEN 1 ELSE 0 END isA,
+  s.max_similarity_to_same_accountant c, s.min_dhash_independent d
+FROM signatures s JOIN accountants a ON s.assigned_accountant=a.name
+WHERE a.firm IN ({','.join('?'*4)})
+  AND s.max_similarity_to_same_accountant IS NOT NULL AND s.min_dhash_independent IS NOT NULL""", BIG4)
+rows = cur.fetchall(); con.close()
+isA = np.array([r[0] for r in rows], bool)
+c = np.array([r[1] for r in rows]); d = np.array([r[2] for r in rows])
+cA, dA = c[isA], d[isA]; cB, dB = c[~isA], d[~isA]
+
+cos_cuts = np.arange(0.85, 0.9901, 0.0025)
+dh_cuts = np.arange(0, 21, 1)
+A = np.zeros((len(dh_cuts), len(cos_cuts)))
+B = np.zeros_like(A)
+for j, cc in enumerate(cos_cuts):
+    for i, dd in enumerate(dh_cuts):
+        A[i, j] = 100 * np.mean((cA > cc) & (dA <= dd))
+        B[i, j] = 100 * np.mean((cB > cc) & (dB <= dd))
+contrast = A - B
+
+extent = [cos_cuts[0], cos_cuts[-1], dh_cuts[0], dh_cuts[-1]]
+fig, axes = plt.subplots(1, 2, figsize=(10.5, 4.3))
+
+for ax, Z, title, cmap, lab in [
+    (axes[0], B, '(a) Clean group (B/C/D) flag rate', 'viridis', 'flag rate (%)'),
+    (axes[1], contrast, '(b) Firm A − B/C/D contrast', 'magma', 'contrast (pp)')]:
+    im = ax.imshow(Z, origin='lower', aspect='auto', extent=extent, cmap=cmap)
+    cb = fig.colorbar(im, ax=ax, pad=0.02); cb.set_label(lab, fontsize=8); cb.ax.tick_params(labelsize=7)
+    # operating points
+    ax.scatter([0.95], [5], marker='*', s=180, color='white', edgecolor='black', zorder=5,
+               label='HC operating point (0.95, dHash≤5)')
+    ax.axhline(15, color='white', ls=':', lw=1.0)
+    ax.text(0.853, 15.4, 'MC upper bound (dHash≤15)', color='white', fontsize=6.5, va='bottom')
+    ax.set_xlabel('cosine cut', fontsize=9)
+    ax.set_ylabel('dHash cut (≤)', fontsize=9)
+    ax.set_title(title, fontsize=9)
+    ax.tick_params(labelsize=7.5)
+    ax.legend(loc='lower left', fontsize=6.5, framealpha=0.85)
+
+fig.suptitle('Figure 6. Sensitivity surface of the deployed rule over the two-measure threshold plane (Big-4, n=%d).' % len(c),
+             fontsize=9, y=1.02)
+fig.tight_layout()
+out = '/Volumes/NV2/pdf_recognize/paper/v13_build/figures/fig6.png'
+fig.savefig(out, dpi=200, bbox_inches='tight')
+plt.close(fig)
+print(f"fig6 OK n={len(c)}; HC(0.95,5) contrast={contrast[5, np.argmin(abs(cos_cuts-0.95))]:.1f}pp; written {out}")

paper/v13_build/scripts/make_figs.py

@@ -0,0 +1,58 @@
+import sqlite3, numpy as np
+import matplotlib
+matplotlib.use('Agg')
+import matplotlib.pyplot as plt
+DB='/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+ALIAS={'勤業眾信聯合':'A','安侯建業聯合':'B','資誠聯合':'C','安永聯合':'D'}
+COL={'A':'#c0392b','B':'#2980b9','C':'#27ae60','D':'#8e44ad'}
+c=sqlite3.connect(f'file:{DB}?mode=ro',uri=True)
+rows=c.execute("""SELECT a.firm, s.max_similarity_to_same_accountant, s.min_dhash_independent,
+  s.assigned_accountant, CAST(substr(s.year_month,1,4) AS INT)
+  FROM signatures s JOIN accountants a ON s.assigned_accountant=a.name
+  WHERE s.max_similarity_to_same_accountant IS NOT NULL AND s.min_dhash_independent IS NOT NULL
+    AND a.firm IN ('勤業眾信聯合','安侯建業聯合','資誠聯合','安永聯合')""").fetchall()
+firm=np.array([ALIAS[r[0]] for r in rows]); cos=np.array([r[1] for r in rows],float)
+dh=np.array([r[2] for r in rows],float); acc=np.array([r[3] for r in rows]); yr=np.array([r[4] for r in rows])
+A=firm=='A'; BCD=np.isin(firm,['B','C','D'])
+
+# ---- Figure 4: two panels, Firm A vs BCD ----
+fig,ax=plt.subplots(1,2,figsize=(9,3.4))
+ax[0].hist(cos[A],bins=np.linspace(0.7,1.0,60),density=True,alpha=0.6,color='#c0392b',label='Firm A')
+ax[0].hist(cos[BCD],bins=np.linspace(0.7,1.0,60),density=True,alpha=0.5,color='#34495e',label='Firms B/C/D')
+ax[0].axvline(0.95,ls='--',c='k',lw=0.8); ax[0].axvline(0.8547,ls=':',c='gray',lw=0.8)
+ax[0].set_title('(a) Within-accountant cosine',fontsize=10)
+ax[0].set_xlabel('max cosine to same accountant'); ax[0].set_ylabel('density')
+ax[0].text(0.952,ax[0].get_ylim()[1]*0.9,'0.95',fontsize=7); ax[0].legend(fontsize=8,frameon=False)
+ax[0].annotate('A median 0.986',(0.986,0),(0.80,ax[0].get_ylim()[1]*0.55),fontsize=7,color='#c0392b',arrowprops=dict(arrowstyle='->',color='#c0392b',lw=0.7))
+ax[0].annotate('B/C/D median 0.959',(0.959,0),(0.72,ax[0].get_ylim()[1]*0.35),fontsize=7,color='#34495e',arrowprops=dict(arrowstyle='->',color='#34495e',lw=0.7))
+bins=np.arange(0,21)-0.5
+ax[1].hist(np.clip(dh[A],0,20),bins=bins,density=True,alpha=0.6,color='#c0392b',label='Firm A')
+ax[1].hist(np.clip(dh[BCD],0,20),bins=bins,density=True,alpha=0.5,color='#34495e',label='Firms B/C/D')
+ax[1].axvline(5,ls='--',c='k',lw=0.8)
+ax[1].set_title('(b) Within-accountant dHash',fontsize=10)
+ax[1].set_xlabel('min dHash to same accountant'); ax[1].set_ylabel('density')
+ax[1].text(5.1,ax[1].get_ylim()[1]*0.9,'5',fontsize=7); ax[1].legend(fontsize=8,frameon=False)
+ax[1].text(0.50,0.62,'A median 2 / B,C,D median 7',transform=ax[1].transAxes,fontsize=7)
+fig.text(0.5,-0.02,'Cross-firm held-out HC rate 0.42% sits at/below the clean reference ICCR 0.59%; within-Firm-A HC rate is 82%.',ha='center',fontsize=7,style='italic')
+fig.tight_layout(); fig.savefig('/tmp/fig4.png',dpi=200,bbox_inches='tight'); plt.close(fig)
+
+# ---- Figure 5: per-accountant HC rate, ranked, per period ----
+def hc_by_acc(mask):
+    out={}
+    a=acc[mask]; h=((cos[mask]>0.95)&(dh[mask]<=5)).astype(float); f=firm[mask]
+    for ai in np.unique(a):
+        m=a==ai
+        if m.sum()>=5: out[ai]=(h[m].mean(),f[m][0])
+    return out
+fig,ax=plt.subplots(1,2,figsize=(9,3.4),sharey=True)
+for j,(lo,hi,ttl) in enumerate([(2013,2019,'(a) 2013–2019'),(2020,2023,'(b) 2020–2023')]):
+    d=hc_by_acc(BCD|A if False else ((yr>=lo)&(yr<=hi)))
+    items=sorted(d.items(),key=lambda kv:-kv[1][0])
+    xs=np.arange(len(items)); ys=[v[0]*100 for _,v in items]; cs=[COL[v[1]] for _,v in items]
+    ax[j].scatter(xs,ys,c=cs,s=10)
+    ax[j].set_title(ttl,fontsize=10); ax[j].set_xlabel('accountant rank'); 
+    if j==0: ax[j].set_ylabel('per-accountant HC rate (%)')
+from matplotlib.lines import Line2D
+ax[1].legend([Line2D([0],[0],marker='o',ls='',color=COL[k]) for k in 'ABCD'],['Firm A','Firm B','Firm C','Firm D'],fontsize=7,frameon=False,loc='upper right')
+fig.tight_layout(); fig.savefig('/tmp/fig5.png',dpi=200,bbox_inches='tight'); plt.close(fig)
+print('figs OK', __import__('os').path.getsize('/tmp/fig4.png'), __import__('os').path.getsize('/tmp/fig5.png'))

paper/v13_build/scripts/make_figs123.py

@@ -0,0 +1,75 @@
+import matplotlib
+matplotlib.use('Agg')
+import matplotlib.pyplot as plt
+from matplotlib.patches import FancyBboxPatch, FancyArrowPatch, Rectangle
+import numpy as np
+
+# ============ Figure 1: data split grid ============
+fig, ax = plt.subplots(figsize=(7, 3.2))
+firms = ['Firm A', 'Firm B', 'Firm C', 'Firm D']
+periods = ['2013–2019', '2020–2023']
+# role per (row firm, col period)
+def role(f, p):
+    if f == 'Firm A':
+        return ('Held-out test 1\n(Firm A, full record)', '#c0392b')
+    if p == '2013–2019':
+        return ('Calibration\n(clean reference)', '#27ae60')
+    return ('Held-out test 2\n(secondary)', '#2980b9')
+for i, f in enumerate(firms):
+    for j, p in enumerate(periods):
+        txt, col = role(f, p)
+        ax.add_patch(Rectangle((j, len(firms)-1-i), 1, 1, facecolor=col, alpha=0.30, edgecolor='black', lw=1))
+        ax.text(j+0.5, len(firms)-1-i+0.5, txt, ha='center', va='center', fontsize=6.5)
+ax.set_xlim(0, 2); ax.set_ylim(0, 4)
+ax.set_xticks([0.5, 1.5]); ax.set_xticklabels(periods, fontsize=9)
+ax.set_yticks([3.5, 2.5, 1.5, 0.5]); ax.set_yticklabels(firms, fontsize=9)
+ax.tick_params(length=0)
+for s in ax.spines.values(): s.set_visible(False)
+ax.set_title('Figure 1. Data split: calibrate on the clean cell, test everything else', fontsize=9)
+fig.tight_layout(); fig.savefig('/tmp/fig1.png', dpi=200, bbox_inches='tight'); plt.close(fig)
+
+# ============ Figure 2: pipeline ============
+fig, ax = plt.subplots(figsize=(9, 2.5))
+steps = ['Raw PDF\nreport', 'Find signature\npage (VLM)', 'Detect signatures\n(YOLOv11)\n+ red-stamp removal',
+         'Feature extraction\n(ResNet-50, 2048-d)', 'Two similarities\ncosine (style)\nmin dHash (structure)', 'Five-way\nlabel']
+n = len(steps); w = 1.0/n
+cols = ['#ecf0f1', '#d6eaf8', '#d5f5e3', '#fcf3cf', '#fadbd8', '#e8daef']
+for i, (s, c) in enumerate(zip(steps, cols)):
+    x = i*w + 0.01
+    ax.add_patch(FancyBboxPatch((x, 0.30), w-0.02, 0.40, boxstyle='round,pad=0.005,rounding_size=0.02',
+                                facecolor=c, edgecolor='black', lw=1, transform=ax.transAxes))
+    ax.text(x+(w-0.02)/2, 0.50, s, ha='center', va='center', fontsize=6.8, transform=ax.transAxes)
+    if i < n-1:
+        ax.add_patch(FancyArrowPatch((x+w-0.012, 0.50), (x+w+0.002, 0.50), transform=ax.transAxes,
+                                     arrowstyle='-|>', mutation_scale=10, lw=1.2, color='black'))
+ax.axis('off')
+ax.set_title('Figure 2. The screening pipeline', fontsize=9, y=0.92)
+fig.savefig('/tmp/fig2.png', dpi=200, bbox_inches='tight'); plt.close(fig)
+
+# ============ Figure 3: two-measure plane, five regions ============
+fig, ax = plt.subplots(figsize=(5.2, 4.2))
+LO, HI = 0.8547, 0.95
+DH1, DH2 = 5, 15
+xmin, xmax = 0.70, 1.005
+ymin, ymax = -1, 30
+# LH (cos<=LO): whole column
+ax.add_patch(Rectangle((xmin, ymin), LO-xmin, ymax-ymin, facecolor='#bdc3c7', alpha=0.5))
+# UN (LO<cos<=HI)
+ax.add_patch(Rectangle((LO, ymin), HI-LO, ymax-ymin, facecolor='#f7dc6f', alpha=0.5))
+# high-cosine band subdivided by dHash
+ax.add_patch(Rectangle((HI, ymin), xmax-HI, DH1-ymin, facecolor='#cb4335', alpha=0.55))   # HC dHash<=5
+ax.add_patch(Rectangle((HI, DH1), xmax-HI, DH2-DH1, facecolor='#eb984e', alpha=0.55))      # MC 5<dHash<=15
+ax.add_patch(Rectangle((HI, DH2), xmax-HI, ymax-DH2, facecolor='#aed6f1', alpha=0.6))      # HSC dHash>15
+ax.axvline(LO, color='gray', ls=':', lw=1); ax.axvline(HI, color='black', ls='--', lw=1)
+ax.plot([HI, xmax], [DH1, DH1], 'k--', lw=0.8); ax.plot([HI, xmax], [DH2, DH2], 'k--', lw=0.8)
+ax.text((xmin+LO)/2, 22, 'LH', ha='center', fontsize=11, weight='bold')
+ax.text((LO+HI)/2, 22, 'UN', ha='center', fontsize=11, weight='bold')
+ax.text((HI+xmax)/2, 2, 'HC', ha='center', fontsize=11, weight='bold', color='white')
+ax.text((HI+xmax)/2, 9.5, 'MC', ha='center', fontsize=11, weight='bold')
+ax.text((HI+xmax)/2, 22, 'HSC', ha='center', fontsize=10, weight='bold')
+ax.text(LO, ymin-1.5, '0.8547', ha='center', fontsize=7); ax.text(HI, ymin-1.5, '0.95', ha='center', fontsize=7)
+ax.set_xlim(xmin, xmax); ax.set_ylim(ymin, ymax)
+ax.set_xlabel('cosine similarity (style)'); ax.set_ylabel('dHash distance (structure)')
+ax.set_title('Figure 3. The two measures and the five regions', fontsize=9)
+fig.tight_layout(); fig.savefig('/tmp/fig3.png', dpi=200, bbox_inches='tight'); plt.close(fig)
+print('figs 1/2/3 OK')

paper/v13_build/scripts/pipeline_audit.py

@@ -0,0 +1,63 @@
+"""Imaging-pipeline audit (Table V) + byte-identity era split (Section V-B).
+Classifies a stratified sample of report PDFs as scanned / OCR'd / digital-native
+from embedded metadata + extractable-text heuristic, and tabulates by year and firm.
+Also reports the scan-era vs digital-era split of the 262 byte-identical signatures.
+
+Requires: PyMuPDF (fitz); signature_analysis.db; original PDFs under total-pdf/.
+"""
+import fitz, os, glob, sqlite3
+from collections import defaultdict
+
+fitz.TOOLS.mupdf_display_errors(False)
+DB = "/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db"
+PDF_ROOT = "/Volumes/NV2/PDF-Processing/total-pdf"
+BIG4 = ('勤業眾信聯合', '資誠聯合', '安侯建業聯合', '安永聯合')
+FMAP = {'勤業眾信聯合': 'A', '安侯建業聯合': 'B', '資誠聯合': 'C', '安永聯合': 'D'}
+
+con = sqlite3.connect(DB); cur = con.cursor()
+
+# --- stratified sample: 20 distinct PDFs per firm-year ---
+cur.execute(f"""
+WITH d AS (SELECT DISTINCT excel_firm, substr(year_month,1,4) yr, source_pdf,
+  ROW_NUMBER() OVER (PARTITION BY excel_firm, substr(year_month,1,4) ORDER BY source_pdf) rn
+  FROM signatures WHERE excel_firm IN ({','.join(['?']*4)}) AND source_pdf IS NOT NULL)
+SELECT excel_firm, yr, source_pdf FROM d WHERE rn<=20 ORDER BY yr""", BIG4)
+rows = cur.fetchall()
+idx = {os.path.basename(p): p for p in glob.glob(PDF_ROOT + '/*/*.pdf')}
+
+def classify(path):
+    try:
+        doc = fitz.open(path)
+    except Exception:
+        return None
+    text = sum(len(doc[i].get_text().strip()) for i in range(min(len(doc), 4)))
+    doc.close()
+    return 'DIGITAL' if text > 2000 else ('OCR' if text > 200 else 'SCAN')
+
+byyear = defaultdict(lambda: defaultdict(int))
+for firm, yr, fn in rows:
+    p = idx.get(fn)
+    if not p:
+        continue
+    k = classify(p)
+    if k:
+        byyear[yr][k] += 1
+
+print("year | n | scan% | ocr% | digital%")
+for yr in sorted(byyear):
+    d = byyear[yr]; n = sum(d.values())
+    print(f"{yr} | {n} | {100*d['SCAN']//n} | {100*d['OCR']//n} | {100*d['DIGITAL']//n}")
+
+# --- byte-identity era split ---
+cur.execute(f"""
+SELECT CASE WHEN year_month<'202101' THEN 'scan-era' ELSE 'digital-era' END era,
+  CASE excel_firm WHEN '勤業眾信聯合' THEN 'A' WHEN '安侯建業聯合' THEN 'B'
+                  WHEN '資誠聯合' THEN 'C' WHEN '安永聯合' THEN 'D' END firm,
+  COUNT(*) n
+FROM signatures WHERE is_valid=1 AND pixel_identical_to_closest=1
+  AND excel_firm IN ({','.join(['?']*4)})
+GROUP BY era, firm ORDER BY era, firm""", BIG4)
+print("\nbyte-identical by era x firm:")
+for era, firm, n in cur.fetchall():
+    print(f"  {era} | {firm} | {n}")
+con.close()

paper/v13_build/scripts/samepair_hc.py

@@ -0,0 +1,28 @@
+"""Table VI: HC flag rate by firm under any-pair (deployed) vs strict same-pair rule.
+same-pair dHash = Hamming distance between a signature's dHash and its cosine-closest
+same-accountant partner (closest_match_file). Reproduces from signature_analysis.db."""
+import sqlite3
+from collections import defaultdict
+DB="/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db"
+BIG4=('勤業眾信聯合','資誠聯合','安侯建業聯合','安永聯合')
+FM={'勤業眾信聯合':'A','安侯建業聯合':'B','資誠聯合':'C','安永聯合':'D'}
+con=sqlite3.connect(DB);cur=con.cursor()
+cur.execute("SELECT image_filename, dhash_vector FROM signatures WHERE dhash_vector IS NOT NULL")
+dh={fn:bytes(b) for fn,b in cur.fetchall()}
+ham=lambda a,b: bin(int.from_bytes(a,'big')^int.from_bytes(b,'big')).count('1')
+cur.execute(f"""SELECT excel_firm,max_similarity_to_same_accountant,min_dhash_independent,closest_match_file,image_filename
+ FROM signatures WHERE is_valid=1 AND max_similarity_to_same_accountant IS NOT NULL
+   AND min_dhash_independent IS NOT NULL AND excel_firm IN ({','.join('?'*4)})""",BIG4)
+st=defaultdict(lambda:[0,0,0])
+for firm,cos,mindh,cmf,imf in cur.fetchall():
+    f=FM[firm]; st[f][0]+=1
+    st[f][1]+= (cos>0.95 and mindh<=5)
+    sp=ham(dh[imf],dh[cmf]) if (cmf in dh and imf in dh) else 99
+    st[f][2]+= (cos>0.95 and sp<=5)
+con.close()
+print(f"{'firm':5}{'n':>8}{'any-pair%':>11}{'same-pair%':>12}")
+T=[0,0,0]
+for f in 'ABCD':
+    n,a,s=st[f]; T=[T[0]+n,T[1]+a,T[2]+s]
+    print(f"{f:5}{n:>8}{100*a/n:>10.1f}%{100*s/n:>11.1f}%")
+n,a,s=T; print(f"{'all':5}{n:>8}{100*a/n:>10.1f}%{100*s/n:>11.1f}%")

paper/v13_build/scripts/spike_setlevel.py

@@ -0,0 +1,49 @@
+import sqlite3, numpy as np
+DB='/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+BCD=('安侯建業聯合','資誠聯合','安永聯合')
+c=sqlite3.connect(f'file:{DB}?mode=ro',uri=True)
+rows=c.execute("""SELECT s.assigned_accountant, s.max_similarity_to_same_accountant, s.min_dhash_independent
+  FROM signatures s JOIN accountants a ON s.assigned_accountant=a.name
+  WHERE a.firm IN ('安侯建業聯合','資誠聯合','安永聯合')
+    AND CAST(substr(s.year_month,1,4) AS INT) BETWEEN 2013 AND 2019
+    AND s.max_similarity_to_same_accountant IS NOT NULL AND s.min_dhash_independent IS NOT NULL""").fetchall()
+from collections import defaultdict
+by=defaultdict(list)
+for a,cos,dh in rows: by[a].append((cos,dh))
+accs={a:np.array(v) for a,v in by.items() if len(v)>=15}
+print(f"BCD 2013-2019: {len(accs)} accountants with >=15 signatures (of {len(by)} total)")
+
+rep=[]; tight=[]; rem_med=[]; klass=[]
+for a,v in accs.items():
+    cos=v[:,0]; dh=v[:,1]
+    hc=(cos>0.95)&(dh<=5)
+    rf=hc.mean(); tf=(cos>0.95).mean()
+    isolated=cos[cos<=0.95]
+    rm=np.median(isolated) if len(isolated)>=3 else np.nan
+    rep.append(rf); tight.append(tf); rem_med.append(rm)
+    klass.append('pure-hand' if rf<0.10 else ('pure-stamp' if rf>0.90 else 'mixed'))
+rep=np.array(rep); tight=np.array(tight); rem_med=np.array(rem_med); klass=np.array(klass)
+
+import collections
+print("\n=== Per-accountant replication-fraction (HC share) distribution ===")
+for lo,hi in [(0,0.1),(0.1,0.3),(0.3,0.5),(0.5,0.7),(0.7,0.9),(0.9,1.01)]:
+    n=((rep>=lo)&(rep<hi)).sum(); print(f"  rep_frac [{lo:.1f},{hi:.1f}): {n:3d} accountants")
+print("  class counts:", dict(collections.Counter(klass)))
+
+mixed=klass=='mixed'
+print(f"\n=== MIXED accountants (n={mixed.sum()}): is the non-tight remainder dispersed (separable)? ===")
+rm_mixed=rem_med[mixed & ~np.isnan(rem_med)]
+print(f"  remainder (cos<=0.95) median cosine across mixed accountants: median={np.median(rm_mixed):.3f}, IQR[{np.percentile(rm_mixed,25):.3f},{np.percentile(rm_mixed,75):.3f}]")
+print(f"  fraction of mixed accountants whose remainder median < 0.90 (clearly dispersed): {(rm_mixed<0.90).mean():.2f}")
+print(f"  fraction with remainder median < 0.85 (very dispersed): {(rm_mixed<0.85).mean():.2f}")
+# gap between tight group (cos>0.95) and remainder: per mixed accountant
+gaps=[]
+for a,v in accs.items():
+    cos=v[:,0]
+    t=cos[cos>0.95]; r=cos[cos<=0.95]
+    if len(t)>=3 and len(r)>=3:
+        gaps.append(np.median(t)-np.median(r))
+gaps=np.array(gaps)
+print(f"\n=== Tight-vs-remainder cosine gap (all accountants with both parts, n={len(gaps)}) ===")
+print(f"  median gap = {np.median(gaps):.3f}  (large gap => two-component structure is real & separable)")
+print(f"  fraction with gap > 0.10: {(gaps>0.10).mean():.2f}")

paper/v4/paper_a_methodology_v4_section_iii.md

@@ -56,7 +56,7 @@ This section characterises the joint distribution of accountant-level descriptor
 
 *Within-firm signature-level dip (Scripts 39b, 39c).* Repeating the dip test at the signature level inside each individual Big-4 firm (Script 39b) and inside each individual non-Big-4 firm with $\geq 500$ signatures (Script 39c) yields a consistent picture. The cosine marginal *fails* to reject unimodality in every single firm tested — all four Big-4 firms ($p_{\text{cos}} \in \{0.176, 0.991, 0.551, 0.976\}$ for Firms A through D; Script 39b) and ten non-Big-4 firms with $\geq 500$ signatures ($p_{\text{cos}} \in [0.59, 0.99]$; Script 39c). The raw dHash marginal *does* reject unimodality in every firm tested ($p < 5 \times 10^{-4}$ in all $14$ firms), but the raw dHash values are integer-valued in $\{0, 1, \ldots, 64\}$, leaving open the possibility of an integer-tie artefact.
 
-*Integer-jitter robustness (Scripts 39d, 39e).* Adding independent uniform jitter $\sim \mathrm{U}[-0.5, +0.5]$ to break exact dHash ties and re-running the dip test on the perturbed signature cloud (5 seeds, $n_{\text{boot}} = 2000$; Script 39d) eliminates the dHash within-firm rejection in every Big-4 firm tested (Firm A jittered $p_{\text{median}} = 0.999$; B $0.996$; C $0.999$; D $0.9995$; $0$/$5$ seeds reject at $\alpha = 0.05$ in any firm). All ten non-Big-4 firms similarly fail to reject after jitter ($p \in [0.71, 1.00]$). The pooled-Big-4 dHash dip *does* survive jitter alone ($p_{\text{median}} = 0$, $5$/$5$ seeds reject), but Firm A's mean dHash ($2.73$) is substantially below Firms B/C/D's ($6.46$, $7.39$, $7.21$) — a between-firm location shift. Script 39e applies a 2 \times 2 factorial correction (firm-mean centring $\times$ integer jitter) on the Big-4 pooled dHash:
+*Integer-jitter robustness (Scripts 39d, 39e).* Adding independent uniform jitter $\sim \mathrm{U}[-0.5, +0.5]$ to break exact dHash ties and re-running the dip test on the perturbed signature cloud (5 seeds, $n_{\text{boot}} = 2000$; Script 39d) eliminates the dHash within-firm rejection in every Big-4 firm tested (Firm A jittered $p_{\text{median}} = 0.999$; B $0.996$; C $0.999$; D $0.9995$; $0$/$5$ seeds reject at $\alpha = 0.05$ in any firm). A codex-verified read-only spike applying the same jitter procedure to the ten non-Big-4 firms with $\geq 500$ signatures (Script 39c substrate) likewise yields no rejection ($0$/$10$ firms reject at $\alpha = 0.05$; per-firm median-$p$ range $[0.38, 1.00]$). The pooled-Big-4 dHash dip *does* survive jitter alone ($p_{\text{median}} = 0$, $5$/$5$ seeds reject), but Firm A's mean dHash ($2.73$) is substantially below Firms B/C/D's ($6.46$, $7.39$, $7.21$) — a between-firm location shift. Script 39e applies a 2 \times 2 factorial correction (firm-mean centring $\times$ integer jitter) on the Big-4 pooled dHash:
 
 | Condition | Firm-mean centred | Integer jitter | Median dip $p$ |  Reject at $\alpha = 0.05$ |
 |---|---|---|---|---|
@@ -96,7 +96,7 @@ $\text{BIC}(K{=}3) = -1111.93$, lower than $K{=}2$ by $3.48$ (mild numerical pre
 - Firm C: $23.5\%$ C1, $75.5\%$ C2, $1.0\%$ C3
 - Firm D: $11.5\%$ C1, $\sim 84\%$ C2, $\sim 4.5\%$ C3
 
-Firm A accounts for $141$ of the $143$ C3-assigned CPAs; Firm C accounts for $24$ of the $40$ C1-assigned CPAs. The K=3 partition is therefore well-described as a firm-compositional decomposition: C3 is essentially "Firm A and any non-Firm-A CPA whose mean descriptors happen to land in the high-cos / low-dHash corner"; C1 is essentially "non-Firm-A CPAs whose mean descriptors land in the low-cos / high-dHash corner." The composition contrast that K=3 captures at the accountant level reappears at the deployment level in the cross-firm hit matrix of §III-L.4 (Script 44): nearly all (98%) of the inter-CPA-anchor hits for a Firm A source signature have a Firm A candidate, and the same within-firm concentration holds for Firms B, C, D individually. The K=3 partition and the cross-firm hit matrix therefore describe the same underlying firm-compositional structure at two different units of analysis.
+Firm A accounts for $141$ of the $143$ C3-assigned CPAs; Firm C accounts for $24$ of the $40$ C1-assigned CPAs. The K=3 partition is therefore well-described as a firm-compositional decomposition: C3 is essentially "Firm A and any non-Firm-A CPA whose mean descriptors happen to land in the high-cos / low-dHash corner"; C1 is essentially "non-Firm-A CPAs whose mean descriptors land in the low-cos / high-dHash corner." The composition contrast that K=3 captures at the accountant level reappears at the deployment level in the cross-firm hit matrix of §III-L.4 (Script 44): under the deployed any-pair rule, within-firm collision concentration is $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms). The K=3 partition and the cross-firm hit matrix therefore describe the same underlying firm-compositional structure at two different units of analysis.
 
 **Leave-one-firm-out stability (Scripts 36, 37).** Leave-one-firm-out cross-validation shows that K=2 is unstable across folds: holding Firm A out gives a fold rule cos $> 0.938$ AND dHash $\leq 8.79$, while holding any single non-Firm-A Big-4 firm out gives a fold rule near cos $> 0.975$ AND dHash $\leq 3.76$ (Script 36). The maximum absolute deviation of the four fold cosine crossings from their across-fold mean is $0.028$ (the corresponding pairwise across-fold range is $0.0376$, from $0.9380$ for the held-out-Firm-A fold to $0.9756$ for the held-out-Firm-D fold; Script 36 stability summary). The $0.028$ value is $5.6\times$ the report's $0.005$ across-fold stability tolerance. K=3 in contrast has a *reproducible component shape*: across the four folds the C1 cosine mean varies by at most $0.005$, the C1 dHash mean by at most $0.96$, and the C1 weight by at most $0.023$ (Script 37). K=3 hard-posterior membership for the held-out firm is composition-sensitive — for Firm C the held-out C1 rate is $36.3\%$ vs the full-Big-4 baseline of $23.5\%$, an absolute difference of $12.8$ pp; for Firm A the held-out C1 rate is $4.7\%$ vs baseline $0.0\%$; the report's own legend classifies this pattern as `P2_PARTIAL` ("the C1 cluster exists but membership is not well-predicted by the held-out fit"). We accordingly do not use K=3 hard-posterior membership as an operational label.
 
@@ -122,13 +122,13 @@ Pairwise Spearman rank correlations among the three scores, $n = 437$ Big-4 CPAs
 
 | Pair | Spearman $\rho$ | $p$-value |
 |---|---|---|
-| Score 1 vs Score 3 | $+0.963$ | $< 10^{-248}$ |
-| Score 2 vs Score 3 | $+0.889$ | $< 10^{-149}$ |
-| Score 1 vs Score 2 | $+0.879$ | $< 10^{-142}$ |
+| Score 1 vs Score 3 | $+0.9627$ | $< 10^{-248}$ |
+| Score 2 vs Score 3 | $+0.8890$ | $< 10^{-149}$ |
+| Score 1 vs Score 2 | $+0.8794$ | $< 10^{-142}$ |
 
 We read this as the strongest internal-consistency signal in v4.0: three different summarisations of the same descriptor pair agree on the per-CPA descriptor-position ranking with $\rho > 0.87$. The three scores agree on placing Firm A as the most replication-dominated descriptor position and the three non-Firm-A Big-4 firms further from the templated end, but they do not all rank the non-Firm-A firms identically: the K=3 posterior P(C1) and the box-rule less-replication-dominated rate (Scores 1 and 3) place Firm C at the less-replication-dominated end of Big-4 (mean P(C1) $= 0.311$; mean box-rule less-replication-dominated rate $= 0.790$), while the reverse-anchor cosine percentile (Score 2) places Firm D fractionally higher than Firm C (mean reverse-anchor score $-0.7125$ vs Firm C $-0.7672$, with higher value indicating deeper into the reference left tail). The mean values for Firms B and D sit between Firms A and C on Scores 1 and 3 (Script 38 per-firm summary). We do not claim this constitutes external validation of any operational classifier; the inherited box rule is calibrated separately (§III-L), and the convergence above shows that a mixture-derived score and a reverse-anchor score concur with the box rule's per-CPA-aggregated outputs on the directional ordering, with a modest disagreement at the less-replication-dominated end between the three non-A Big-4 firms.
 
-**2. Per-signature consistency (Script 39).** Per-CPA aggregation could in principle reflect averaging across within-CPA heterogeneity rather than coherent within-CPA behaviour. We test this by repeating the K=3 fit at the signature level — fitting a fresh K=3 GMM to the 150,442 Big-4 signature-level $(\text{cos}, \text{dHash}_{\text{indep}})$ points (Script 39) — and comparing labels. The per-CPA and per-signature K=3 fits recover a broadly similar three-component ordering; per-CPA C1 is at $\overline{\text{cos}} = 0.946$, $\overline{\text{dHash}} = 9.17$ vs per-signature C1 at $\overline{\text{cos}} = 0.928$, $\overline{\text{dHash}} = 9.75$ (an absolute cosine drift of $0.018$). Cohen $\kappa$ on the binary collapse (replicated vs not-replicated):
+**2. Per-signature consistency (Script 39).** Per-CPA aggregation could in principle reflect averaging across within-CPA heterogeneity rather than coherent within-CPA behaviour. We test this by repeating the K=3 fit at the signature level — fitting a fresh K=3 GMM to the 150,442 Big-4 signature-level $(\text{cos}, \text{dHash}_{\text{indep}})$ points (Script 39) — and comparing labels. The per-CPA and per-signature K=3 fits recover a broadly similar three-component ordering; per-CPA C1 is at $\overline{\text{cos}} = 0.946$, $\overline{\text{dHash}} = 9.17$ vs per-signature C1 at $\overline{\text{cos}} = 0.928$, $\overline{\text{dHash}} = 9.75$ (an absolute cosine drift of $0.018$). Cohen $\kappa$ on the binary collapse (replication-dominated vs less-replication-dominated):
 
 | Pair | Cohen $\kappa$ |
 |---|---|
@@ -146,9 +146,9 @@ The Script 39 report verdict is `SIG_CONVERGENCE_MODERATE`. The $\kappa = 0.870$
 
 **4. Positive-anchor miss rate on byte-identical signatures (Script 40).** The corpus provides one hard ground-truth subset: 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 Big-4 byte-identical subset comprises $n = 262$ signatures ($145 / 8 / 107 / 2$ across Firms A through D; Script 40).
 
-We report each candidate classifier's *positive-anchor miss rate* — the fraction of byte-identical signatures classified as belonging to the less-replication-dominated descriptor positions. This is a one-sided check against a conservative positive subset, **not a paired specificity metric in the usual two-class sense**; we do not report a paired negative-anchor metric here because no signature-level hand-signed ground truth exists. The corresponding signature-level inter-CPA negative-anchor ICCR evidence is developed in §III-L.1 (Big-4 sample) and the v3.x §IV-I corpus-wide version (reported under prior "FAR" terminology):
+We report each candidate check's *positive-anchor miss rate* — the fraction of byte-identical signatures classified as belonging to the less-replication-dominated descriptor positions. This is a one-sided check against a conservative positive subset, **not a paired specificity metric in the usual two-class sense**; we do not report a paired negative-anchor metric here because no signature-level hand-signed ground truth exists. The corresponding signature-level inter-CPA negative-anchor ICCR evidence is developed in §III-L.1 (Big-4 v4 sample) and the inherited corpus-wide v3.x version cited at §IV-I (reported under prior "FAR" terminology):
 
-| Candidate classifier | Pixel-identity miss rate (Wilson 95% CI) |
+| Candidate check | Pixel-identity miss rate (Wilson 95% CI) |
 |---|---|
 | Inherited Paper A binary high-confidence box rule (cos $> 0.95$ AND dHash $\leq 5$) | $0\%$ $[0\%, 1.45\%]$ |
 | K=3 per-CPA hard label (C3 high-cos / low-dHash corner; descriptive only) | $0\%$ $[0\%, 1.45\%]$ |
@@ -280,9 +280,9 @@ The per-decile per-firm breakdown (Script 44) confirms the pattern: within every
 
 For the same-pair joint event (a single candidate satisfying both $\text{cos} > 0.95$ and $\text{dHash} \leq 5$), the candidate firm is even more strongly concentrated within the source firm: Firm A source $\to$ Firm A candidate in $11{,}314$ of $11{,}319$ same-pair hits ($99.96\%$); Firm B source $\to$ Firm B candidate in $85$ of $87$ ($97.7\%$); Firm C source $\to$ Firm C candidate in $54$ of $55$ ($98.2\%$); Firm D source $\to$ Firm D candidate in $64$ of $66$ ($97.0\%$).
 
-**Interpretation.** The cross-firm hit matrix shows that nearly all inter-CPA collisions under the deployed rule originate from candidates within the source firm (different CPA, same firm). This pattern is consistent with — but not by itself diagnostic of — firm-specific template, stamp, or document-production reuse: within-firm scanning workflows, common form templates, and shared report-generation infrastructure could produce visually similar signature crops across different CPAs within the same firm. The byte-level evidence of v3.x §IV-F.1 (Firm A's $145$ pixel-identical signatures across $\sim 50$ distinct certifying partners) provides direct evidence that firm-level template reuse does occur at Firm A; the broader inter-CPA collision pattern in §III-L.4 is consistent with that mechanism extending in milder form to Firms B/C/D. We report this as "inter-CPA collision concentration is within-firm" — a descriptive observation about deployed-rule behaviour — and refrain from inferring that the within-firm hits constitute deliberate or systematic template sharing.
+**Interpretation.** Under the deployed any-pair rule, the within-firm collision concentration is $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D — Firm A's pattern is markedly more within-firm-concentrated than the other three firms', though every Big-4 firm still has more than three quarters of its any-pair collisions falling on candidates within the same firm. The stricter same-pair joint event — a single candidate satisfying both cos $> 0.95$ and dHash $\leq 5$ — saturates at $97.0$–$99.96\%$ within-firm across all four firms. This pattern is consistent with — but not by itself diagnostic of — firm-specific template, stamp, or document-production reuse: within-firm scanning workflows, common form templates, and shared report-generation infrastructure could produce visually similar signature crops across different CPAs within the same firm. The byte-level evidence of v3.x §IV-F.1 (Firm A's $145$ pixel-identical signatures across $\sim 50$ distinct certifying partners) provides direct evidence that firm-level template reuse does occur at Firm A; the broader inter-CPA collision pattern in §III-L.4 is consistent with that mechanism extending in milder form to Firms B/C/D. We report this as "inter-CPA collision concentration is within-firm" — a descriptive observation about deployed-rule behaviour — and refrain from inferring that the within-firm hits constitute deliberate or systematic template sharing.
 
-This connects back to §III-J: the K=3 firm-composition contrast at the accountant level (Firm A dominating C3; Firm C dominating C1) reappears at the deployment level in the cross-firm hit matrix, where nearly all collisions are within-firm. The K=3 partition and the cross-firm hit matrix describe the same underlying firm-compositional structure at two different units of analysis.
+This connects back to §III-J: the K=3 firm-composition contrast at the accountant level (Firm A dominating C3; Firm C dominating C1) reappears at the deployment level in the cross-firm hit matrix, where the within-firm collision concentration is the dominant pattern at all four Big-4 firms — most strongly at Firm A ($98.8\%$ any-pair, $99.96\%$ same-pair) and at materially lower but still majority levels at Firms B/C/D ($76.7$–$83.7\%$ any-pair; $97.0$–$98.2\%$ same-pair). The K=3 partition and the cross-firm hit matrix describe the same underlying firm-compositional structure at two different units of analysis.
 
 ### L.5. Alert-rate sensitivity around inherited thresholds (Script 46)
 
@@ -313,15 +313,18 @@ The K=3 mixture of §III-J is reported in §IV as an accountant-level descriptiv
 
 The v4.0 corpus lacks signature-level ground-truth replication labels: no signature is annotated as definitively hand-signed or definitively templated. The conservative positive anchor (pixel-identical same-CPA signatures; §III-K.4 and v3.x §IV-F.1) is by construction near $\text{cos} = 1$ and $\text{dHash} = 0$, providing a tautological capture-check rather than a sensitivity estimate for the non-byte-identical replicated class. The corpus therefore does not admit standard supervised classifier validation: we cannot report False Rejection Rate, sensitivity, recall, Equal Error Rate, ROC-AUC, or precision against ground truth.
 
-In place of supervised validation, v4.0 adopts a **multi-tool collection of partial-evidence diagnostics**, each with an explicitly disclosed assumption:
+In place of supervised validation, v4.0 adopts a **multi-tool collection of partial-evidence diagnostics** (Table XXVII), each with an explicitly disclosed assumption:
+
+**Table XXVII.** Ten-tool unsupervised-validation collection with disclosed untested assumptions.
 
 | Tool | What it measures | 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 inflated; cluster-robust analysis is a future check |
-| Cross-firm hit matrix (§III-L.4; Script 44) | Concentration of inter-CPA collisions within source firm | None — direct descriptive observation |
+| 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 footnote) |
 | 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; v3.x; Script 40) | Trivial sanity check on the conservative positive anchor | Anchor is tautologically captured by any reasonable threshold |
@@ -376,7 +379,7 @@ The table below lists the principal numerical claims and their data-source scrip
 | Within-firm signature-level dip $p_{\text{cos}}$ Big-4 (A/B/C/D) | $0.176 / 0.991 / 0.551 / 0.976$ | Script 39b per-firm | direct, $n_{\text{boot}} = 2000$ |
 | Within-firm signature-level dip $p_{\text{cos}}$ non-Big-4 (10 firms, range) | $[0.59, 0.99]$ | Script 39c per-firm | direct, firms with $\geq 500$ signatures |
 | Within-firm jittered-dHash dip $p$ Big-4 (5 seeds, median) A/B/C/D | $0.999 / 0.996 / 0.999 / 0.9995$ | Script 39d multi-seed | uniform jitter $[-0.5, +0.5]$ |
-| Within-firm jittered-dHash dip $p$ non-Big-4 (5 seeds, range across 10 firms) | $[0.71, 1.00]$ | Script 39d / 39c | uniform jitter $[-0.5, +0.5]$ |
+| Within-firm jittered-dHash dip $p$ non-Big-4 (5 seeds, range across 10 firms) | $[0.38, 1.00]$ ($0/10$ firms reject) | codex-verified read-only spike on Script 39c substrate | uniform jitter $[-0.5, +0.5]$ |
 | Big-4 pooled dHash dip $p$ raw / jittered (seed median) | $< 5 \times 10^{-4}$ / $< 5 \times 10^{-4}$ | Script 39d | jitter alone does not eliminate Big-4 pooled rejection |
 | Big-4 pooled dHash dip $p$ firm-centred + jittered (5-seed median) | $0.35$ | Script 39e 2×2 factorial | both corrections eliminate rejection ($0/5$ seeds at $\alpha = 0.05$) |
 | Big-4 firm-centred signature-level cos dip $p$ | $0.597$ | codex round-30 verification on Script 43 substrate | independent verification |

paper/v4/paper_a_prose_v4_phase4.md

@@ -8,7 +8,7 @@
 
 > *IEEE Access target: <= 250 words, single paragraph.*
 
-Regulations require Certified Public Accountants (CPAs) to attest each audit report with a signature, but digitization makes reusing a stored signature image across reports — through administrative stamping or firm-level electronic signing — technically trivial and visually invisible, undermining individualized attestation. We build an end-to-end pipeline detecting such *non-hand-signed* signatures at scale: a Vision-Language Model identifies signature pages, YOLOv11 localizes signatures, ResNet-50 supplies deep features, and a dual-descriptor layer combines cosine similarity with an independent-minimum perceptual hash (dHash) to separate *style consistency* from *image reproduction*. Applied to 90,282 Taiwan audit reports (2013–2023), the pipeline yields 182,328 signatures from 758 CPAs; primary analyses are scoped to the Big-4 sub-corpus (437 CPAs; 150,442 signatures). Distributional diagnostics show that the apparent multimodality of the descriptor distribution dissolves under joint firm-mean centring and integer-tie jitter ($p$ rises to $0.35$), so no within-population bimodal antimode anchors the operational thresholds. We instead adopt an anchor-based inter-CPA coincidence-rate (ICCR) calibration at three units: per-comparison ($0.0006$ at cos$>0.95$; $0.0013$ at dHash$\leq 5$; $0.00014$ jointly), pool-normalised per-signature ($0.11$ under the deployed any-pair high-confidence rule), and per-document ($0.34$ for the operational HC+MC alarm). Firm heterogeneity is decisive: Firm A's per-document HC+MC alarm rate is $0.62$ versus $0.09$–$0.16$ at Firms B/C/D after pool-size adjustment, with $98$–$100\%$ of inter-CPA collisions concentrated within the source firm — consistent with firm-level template-like reuse. We position the system as a specificity-proxy-anchored screening framework with human-in-the-loop review, not as a validated forensic detector; no calibrated error rates are reportable without signature-level ground truth.
+Regulations require Certified Public Accountants (CPAs) to attest each audit report with a signature, but digitization makes reusing a stored signature image across reports — through administrative stamping or firm-level electronic signing — undermining individualized attestation. We build an end-to-end pipeline detecting such *non-hand-signed* signatures at scale: a Vision-Language Model identifies signature pages, YOLOv11 localizes signatures, ResNet-50 supplies deep features, and a dual-descriptor layer combines cosine similarity with an independent-minimum perceptual hash (dHash) to separate *style consistency* from *image reproduction*. Applied to 90,282 Taiwan audit reports (2013–2023), the pipeline yields 182,328 signatures from 758 CPAs; primary analyses are scoped to the Big-4 sub-corpus (437 CPAs; 150,442 signatures). Distributional diagnostics show that the apparent multimodality of the descriptor distribution dissolves under joint firm-mean centring and integer-tie jitter ($p$ rises to $0.35$), so no within-population bimodal antimode anchors the operational thresholds. We instead adopt an anchor-based inter-CPA coincidence-rate (ICCR) calibration at three units: per-comparison ($0.0006$ at cos$>0.95$; $0.0013$ at dHash$\leq 5$; $0.00014$ jointly), pool-normalised per-signature ($0.11$ under the deployed any-pair high-confidence rule), and per-document ($0.34$ for the operational HC+MC alarm). Firm heterogeneity is decisive: Firm A's per-document HC+MC alarm rate is $0.62$ versus $0.09$–$0.16$ at Firms B/C/D after pool-size adjustment, and under the deployed any-pair rule $77$–$99\%$ of inter-CPA collisions concentrate within the source firm — consistent with firm-level template-like reuse. We position the system as a specificity-proxy-anchored screening framework with human-in-the-loop review, not as a validated forensic detector; no calibrated error rates are reportable without signature-level ground truth.
 
 ---
 
@@ -28,11 +28,11 @@ Despite the significance of the problem for audit quality and regulatory oversig
 
 In this paper we present a fully automated, end-to-end pipeline for detecting non-hand-signed CPA signatures in audit reports at scale, together with a multi-tool validation framework that explicitly discloses the unsupervised setting's limits. The pipeline processes raw PDF documents through (1) signature page identification with a Vision-Language Model; (2) signature region detection with a trained YOLOv11 object detector; (3) deep feature extraction via a pre-trained ResNet-50; (4) dual-descriptor similarity (cosine + independent-minimum dHash); (5) anchor-based threshold calibration at three units of analysis (per-comparison, pool-normalised per-signature, per-document) against an inter-CPA negative-anchor coincidence-rate proxy (§III-L); (6) firm-stratified per-rule reporting and a within-firm cross-CPA hit-matrix analysis (§III-L.4); (7) a composition decomposition that establishes the absence of a within-population bimodal antimode in the descriptor distributions (§III-I.4); and (8) a multi-tool unsupervised validation strategy with disclosed assumption-violation analysis (§III-M).
 
-The methodological reframing relative to earlier versions of this work is central to our v4.0 contribution. Earlier work in this lineage adopted a distributional path to thresholds — fitting accountant-level finite-mixture models and treating their marginal crossings as data-derived "natural" thresholds. v4.0 reports a composition decomposition diagnostic (§III-I.4) that overturns this reading: the apparent multimodality of the Big-4 accountant-level distribution is fully explained by between-firm location-shift effects (Firm A's mean dHash of $2.73$ versus Firms B/C/D's $6.46$, $7.39$, $7.21$) and integer mass-point artefacts on the integer-valued dHash axis. Once both confounds are removed (firm-mean centring plus uniform integer jitter), the Big-4 pooled dHash dip test yields $p_{\text{median}} = 0.35$ across five jitter seeds, eliminating the rejection. Within-firm signature-level cosine and jittered-dHash dip tests fail to reject in every individual Big-4 firm and in every individual mid/small firm with $\geq 500$ signatures (10 firms tested in Script 39c). The descriptor distributions therefore contain no within-population bimodal antimode that could anchor an operational threshold.
+The methodological reframing relative to earlier versions of this work is central to our v4.0 contribution. Earlier work in this lineage adopted a distributional path to thresholds — fitting accountant-level finite-mixture models and treating their marginal crossings as data-derived "natural" thresholds. v4.0 reports a composition decomposition diagnostic (§III-I.4) that overturns this reading: the apparent multimodality of the Big-4 accountant-level distribution is fully explained by between-firm location-shift effects (Firm A's mean dHash of $2.73$ versus Firms B/C/D's $6.46$, $7.39$, $7.21$) and integer mass-point artefacts on the integer-valued dHash axis. Once both confounds are removed (firm-mean centring plus uniform integer jitter), the Big-4 pooled dHash dip test yields $p_{\text{median}} = 0.35$ across five jitter seeds, eliminating the rejection. Within-firm signature-level cosine dip tests fail to reject in every individual Big-4 firm and in every individual mid/small firm with $\geq 500$ signatures (10 firms tested in Script 39c), and the corresponding within-firm jittered-dHash dip tests likewise fail to reject in all four Big-4 firms (Script 39d) and across a codex-verified read-only spike on the same ten mid/small firms ($0/10$ reject; §III-I.4). The descriptor distributions therefore contain no within-population bimodal antimode that could anchor an operational threshold.
 
 In place of distributional anchoring, v4.0 adopts an anchor-based inter-CPA coincidence-rate (ICCR) calibration. At the per-comparison unit, the inherited cos$>0.95$ operating point yields ICCR $= 0.00060$ on a $5 \times 10^5$-pair Big-4 sample (replicating v3.x's reported per-comparison rate of $0.0005$ under prior "FAR" terminology); the dHash$\leq 5$ structural cutoff yields ICCR $= 0.00129$ (v4 new); the joint rule cos$>0.95$ AND dHash$\leq 5$ yields joint ICCR $= 0.00014$ (any-pair semantics, matching the deployed extrema rule). At the pool-normalised per-signature unit, the same rule's effective coincidence rate is materially higher because the deployed classifier takes max-cosine and min-dHash over a same-CPA pool: pooled Big-4 any-pair ICCR is $0.1102$ (Wilson 95% CI $[0.1086, 0.1118]$; CPA-block bootstrap 95% $[0.0908, 0.1330]$). At the per-document unit, the operational HC$+$MC alarm fires on $33.75\%$ of Big-4 documents under the inter-CPA candidate-pool counterfactual.
 
-The pooled per-signature and per-document rates conceal striking firm heterogeneity. A logistic regression of the per-signature hit indicator on firm dummies (Firm A reference) and centred log pool size yields odds ratios of $0.053$ (Firm B), $0.010$ (Firm C), and $0.027$ (Firm D) — Firms B/C/D are an order of magnitude below Firm A even after controlling for the pool-size confound (Script 44). Cross-firm hit matrix analysis shows that $98$–$100\%$ of inter-CPA collisions originate from candidates within the source firm (different CPA, same firm), consistent with firm-specific template, stamp, or document-production reuse mechanisms — though not by itself diagnostic of deliberate sharing. We retain the inherited Paper A v3.x five-way box rule as the operational classifier; v4.0's contribution is to characterise its multi-level coincidence behaviour against the inter-CPA negative anchor rather than to derive new thresholds.
+The pooled per-signature and per-document rates conceal striking firm heterogeneity. A logistic regression of the per-signature hit indicator on firm dummies (Firm A reference) and centred log pool size yields odds ratios of $0.053$ (Firm B), $0.010$ (Firm C), and $0.027$ (Firm D) — Firms B/C/D are an order of magnitude below Firm A even after controlling for the pool-size confound (Script 44). Cross-firm hit matrix analysis under the deployed any-pair rule shows within-firm collision concentrations of $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D (Table XXV; the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms). The pattern is consistent with firm-specific template, stamp, or document-production reuse mechanisms — though not by itself diagnostic of deliberate sharing. We retain the inherited Paper A v3.x five-way box rule as the operational classifier; v4.0's contribution is to characterise its multi-level coincidence behaviour against the inter-CPA negative anchor rather than to derive new thresholds.
 
 Three feature-derived scores converge on the per-CPA descriptor-position ranking with Spearman $\rho \geq 0.879$ (Script 38): the K=3 mixture posterior (now interpreted as a firm-compositional position score, not a mechanism cluster posterior; §III-J), a reverse-anchor cosine percentile relative to a strictly-out-of-target non-Big-4 reference, and the inherited box-rule less-replication-dominated rate. The three scores are deterministic functions of the same per-CPA descriptor pair, so the convergence is documented as internal consistency among feature-derived ranks rather than external validation. Hard ground truth for the *replicated* class is provided by 262 byte-identical signatures in the Big-4 subset (Firm A 145, Firm B 8, Firm C 107, Firm D 2), against which all three candidate checks achieve $0\%$ positive-anchor miss rate (Wilson 95% upper bound $1.45\%$). For the box rule this result is close to tautological at byte-identity; we discuss the conservative-subset caveat in §V-G.
 
@@ -50,11 +50,11 @@ The contributions of this paper are:
 
 5. **Anchor-based multi-level inter-CPA coincidence-rate calibration.** We characterise the deployed five-way classifier at three units of analysis: per-comparison ICCR (cos$>0.95$: $0.0006$; dHash$\leq 5$: $0.0013$; joint: $0.00014$), pool-normalised per-signature ICCR ($0.11$ for the deployed any-pair high-confidence rule), and per-document ICCR ($0.34$ for the operational HC$+$MC alarm). We adopt "inter-CPA coincidence rate" as the metric name throughout and reserve "False Acceptance Rate" for terminology that requires ground-truth negative labels, which the corpus does not provide.
 
-6. **Firm heterogeneity quantification and within-firm cross-CPA collision concentration.** Per-firm rates differ by an order of magnitude after pool-size adjustment (Firm A's per-document HC$+$MC alarm at $0.62$ versus Firms B/C/D at $0.09$–$0.16$). Cross-firm hit matrix analysis shows that $98$–$100\%$ of inter-CPA collisions originate from candidates within the source firm, consistent with firm-specific template, stamp, or document-production reuse mechanisms — a descriptive finding about deployed-rule behaviour, not a claim of deliberate template sharing.
+6. **Firm heterogeneity quantification and within-firm cross-CPA collision concentration.** Per-firm rates differ by an order of magnitude after pool-size adjustment (Firm A's per-document HC$+$MC alarm at $0.62$ versus Firms B/C/D at $0.09$–$0.16$). Cross-firm hit matrix analysis shows within-firm collision concentrations of $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D under the deployed any-pair rule (the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms); the pattern is consistent with firm-specific template, stamp, or document-production reuse mechanisms — a descriptive finding about deployed-rule behaviour, not a claim of deliberate template sharing.
 
 7. **K=3 as descriptive firm-compositional partition; three-score convergent internal consistency.** We fit a K=3 Gaussian mixture as a descriptive partition of the Big-4 accountant-level distribution (no longer interpreted as three mechanism clusters). Three feature-derived scores agree on the per-CPA descriptor-position ranking at Spearman $\rho \geq 0.879$; we report this as internal consistency rather than external validation, given that the scores share the underlying descriptor pair.
 
-8. **Annotation-free positive-anchor validation and unsupervised validation ceiling.** We achieve $0\%$ positive-anchor miss rate (Wilson 95% upper bound $1.45\%$) on 262 byte-identical Big-4 signatures, with the conservative-subset caveat that byte-identical pairs are by construction near cos$=1$ and dHash$=0$. We frame the overall validation strategy as a multi-tool collection of nine partial-evidence diagnostics, each with an explicitly disclosed untested assumption; their conjunction constitutes the unsupervised validation ceiling achievable on this corpus. We do not claim a validated forensic detector; we position the system as a specificity-proxy-anchored screening framework with human-in-the-loop review.
+8. **Annotation-free positive-anchor validation and unsupervised validation ceiling.** We achieve $0\%$ positive-anchor miss rate (Wilson 95% upper bound $1.45\%$) on 262 byte-identical Big-4 signatures, with the conservative-subset caveat that byte-identical pairs are by construction near cos$=1$ and dHash$=0$. We frame the overall validation strategy as a multi-tool collection of ten partial-evidence diagnostics (§III-M Table XXVII), each with an explicitly disclosed untested assumption; their conjunction constitutes the unsupervised validation ceiling achievable on this corpus. We do not claim a validated forensic detector; we position the system as a specificity-proxy-anchored screening framework with human-in-the-loop review.
 
 The remainder of the paper is organized as follows. Section II reviews related work on signature verification, document forensics, perceptual hashing, and the statistical methods used. Section III describes the proposed methodology. Section IV presents the experimental results — distributional characterisation, mixture fits, convergent internal-consistency checks, leave-one-firm-out reproducibility, pixel-identity validation, and full-dataset robustness. Section V discusses the implications and limitations. Section VI concludes with directions for future work.
 
@@ -64,7 +64,7 @@ The remainder of the paper is organized as follows. Section II reviews related w
 
 > *Note for the Phase 4 review pass: §II is inherited substantively unchanged from v3.20.0 §II in the master manuscript, with one new paragraph added below. The unchanged content is not reproduced in this Phase 4 file; readers reviewing this draft should consult `paper/paper_a_related_work_v3.md` for the v3.20.0 §II text covering offline signature verification, near-duplicate detection, copy-move forgery detection, perceptual hashing, deep-feature similarity, and the statistical methods adopted (Hartigan dip test, finite mixture EM, Burgstahler-Dichev / McCrary density-smoothness diagnostic). The paragraph below is the only v4.0-specific §II addition.*
 
-**Addition for v4.0: leave-one-firm-out cross-validation in a small-cluster scope.** Cross-validation methodology in the leave-one-out tradition has been developed extensively in statistics since Stone [42] and Geisser [43], and modern surveys including Vehtari et al. [44] discuss its application to mixture models. In document-forensics calibration the technique has been used selectively, typically with the individual document or signature as the hold-out unit. Our application in §III-K differs in two respects from the standard usage: (i) the hold-out unit is the *firm* (not the individual CPA or signature), so the analysis directly probes cross-firm reproducibility of the fitted mixture rather than within-firm sampling variance; and (ii) the held-out predictions are interpreted as a *composition-sensitivity band* on the candidate mixture boundary, not as a sufficiency claim for the inherited five-way operational classifier (which is calibrated separately; §III-L). We treat LOOO drift as descriptive information about how the mixture characterisation moves when training composition changes, not as a pass/fail test for the operational classifier. Numerical references [42]–[44] are placeholders in this draft and will be replaced with the project's preferred references at copy-edit time.
+**Addition for v4.0: leave-one-firm-out cross-validation in a small-cluster scope.** Cross-validation methodology in the leave-one-out tradition has been developed extensively in statistics since Stone [42] and Geisser [43], and modern surveys including Vehtari et al. [44] discuss its application to mixture models. In document-forensics calibration the technique has been used selectively, typically with the individual document or signature as the hold-out unit. Our application in §III-K differs in two respects from the standard usage: (i) the hold-out unit is the *firm* (not the individual CPA or signature), so the analysis directly probes cross-firm reproducibility of the fitted mixture rather than within-firm sampling variance; and (ii) the held-out predictions are interpreted as a *composition-sensitivity band* on the candidate mixture boundary, not as a sufficiency claim for the inherited five-way operational classifier (which is calibrated separately; §III-L). We treat LOOO drift as descriptive information about how the mixture characterisation moves when training composition changes, not as a pass/fail test for the operational classifier.
 
 ---
 
@@ -78,13 +78,13 @@ Non-hand-signing differs from forgery in that the questioned signature is produc
 
 A central empirical finding of v3.x was that *per-signature* similarity does not admit a clean two-mechanism mixture: dip-test fails to reject unimodality at the signature level for Firm A, BIC prefers a 3-component fit, and BD/McCrary candidate transitions lie inside the high-similarity mode rather than between modes. v4.0 strengthens and extends this signature-level reading.
 
-The Big-4 accountant-level descriptor distribution does reject unimodality on both marginals at $p < 5 \times 10^{-4}$ (Script 34). v4.0's composition decomposition (§III-I.4; Scripts 39b–39e) shows that this rejection is fully attributable to two non-mechanistic sources: (a) between-firm location-shift effects on both axes — Firm A's mean dHash of $2.73$ versus Firms B/C/D's $6.46$, $7.39$, $7.21$ creates a multi-peaked pooled distribution that any single firm's distribution lacks — and (b) integer mass-point artefacts on the integer-valued dHash axis, which inflate the dip statistic against a continuous-density null. A 2×2 factorial diagnostic applied to the Big-4 pooled dHash (firm-mean centring × uniform integer jitter $[-0.5, +0.5]$, 5 jitter seeds) shows that the dip test fails to reject ($p_{\text{median}} = 0.35$, 0/5 seeds reject) when *both* corrections are applied; either correction alone leaves the rejection in place. Within-firm signature-level cosine and jittered-dHash dip tests fail to reject in every individual Big-4 firm and in every individual non-Big-4 firm with $\geq 500$ signatures (10 firms tested). The descriptor distributions therefore lack a within-population bimodal antimode that could anchor an operational threshold. The K=2 / K=3 mixture fits are retained in §III-J as descriptive partitions of the joint Big-4 distribution that reflect firm-compositional structure, not as inferential evidence for two or three latent mechanism modes.
+The Big-4 accountant-level descriptor distribution does reject unimodality on both marginals at $p < 5 \times 10^{-4}$ (Script 34). v4.0's composition decomposition (§III-I.4; Scripts 39b–39e) shows that this rejection is fully attributable to two non-mechanistic sources: (a) between-firm location-shift effects on both axes — Firm A's mean dHash of $2.73$ versus Firms B/C/D's $6.46$, $7.39$, $7.21$ creates a multi-peaked pooled distribution that any single firm's distribution lacks — and (b) integer mass-point artefacts on the integer-valued dHash axis, which inflate the dip statistic against a continuous-density null. A 2×2 factorial diagnostic applied to the Big-4 pooled dHash (firm-mean centring × uniform integer jitter $[-0.5, +0.5]$, 5 jitter seeds) shows that the dip test fails to reject ($p_{\text{median}} = 0.35$, 0/5 seeds reject) when *both* corrections are applied; either correction alone leaves the rejection in place. Within-firm signature-level cosine and jittered-dHash dip tests fail to reject in every individual Big-4 firm and in every individual non-Big-4 firm with $\geq 500$ signatures tested (cosine: Scripts 39b/39c; jittered-dHash: Script 39d for Big-4 plus codex-verified read-only spike for the ten non-Big-4 firms; see §III-I.4). The descriptor distributions therefore lack a within-population bimodal antimode that could anchor an operational threshold. The K=2 / K=3 mixture fits are retained in §III-J as descriptive partitions of the joint Big-4 distribution that reflect firm-compositional structure, not as inferential evidence for two or three latent mechanism modes.
 
 ## C. Firm A as the Templated End of Big-4 (Case Study, Not Calibration Anchor)
 
 Firm A is empirically the firm whose CPAs are most concentrated in the high-cosine, low-dHash corner of the Big-4 descriptor plane. In the Big-4 K=3 hard-posterior assignment (now interpreted as a firm-compositional position assignment; §III-J), Firm A accounts for $0\%$ of C1 (low-cos / high-dHash position) and $82.5\%$ of C3 (high-cos / low-dHash position); the opposite pattern holds at Firm C, which has the highest C1 concentration at $23.5\%$. Firm A also accounts for 145 of the 262 byte-identical signatures in the Big-4 byte-identical anchor of §IV-H (with Firm B 8, Firm C 107, Firm D 2). The additional v3.x finding that the 145 Firm A pixel-identical signatures span 50 distinct Firm A partners (of 180 registered), with 35 byte-identical matches across different fiscal years, is inherited from v3.20.0 §IV-F.1 / Script 28 / Appendix B byte-decomposition output and was not regenerated in v4.0's spike scripts; we retain those numbers by reference.
 
-In v4.0 we treat Firm A as a *templated-end case study* rather than as the calibration anchor for the operational threshold. Firm A enters the Big-4 anchor-based ICCR calibration on equal footing with the other three Big-4 firms (§III-L). The cross-firm hit matrix of §III-L.4 strengthens this framing: $98$–$100\%$ of inter-CPA collisions originate from candidates within the source firm, regardless of which Big-4 firm is the source. Firm A's high per-document HC$+$MC alarm rate of $0.62$ (versus Firms B/C/D's $0.09$–$0.16$) reflects high inter-CPA collision concentration under the deployed rule on real same-CPA pools, consistent with firm-specific template, stamp, or document-production reuse — though the inter-CPA-anchor analysis alone is not diagnostic of deliberate template sharing. The byte-level evidence of v3.x §IV-F.1 (Firm A's 145 pixel-identical signatures across $\sim 50$ distinct partners) provides direct evidence that firm-level template reuse does occur at Firm A; the within-firm collision pattern at all four Big-4 firms is consistent with that mechanism extending in milder form to Firms B/C/D.
+In v4.0 we treat Firm A as a *templated-end case study* rather than as the calibration anchor for the operational threshold. Firm A enters the Big-4 anchor-based ICCR calibration on equal footing with the other three Big-4 firms (§III-L). The cross-firm hit matrix of §III-L.4 strengthens this framing: under the deployed any-pair rule, within-firm collision concentration is $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms). Firm A's high per-document HC$+$MC alarm rate of $0.62$ (versus Firms B/C/D's $0.09$–$0.16$) reflects high inter-CPA collision concentration under the deployed rule on real same-CPA pools, consistent with firm-specific template, stamp, or document-production reuse — though the inter-CPA-anchor analysis alone is not diagnostic of deliberate template sharing. The byte-level evidence of v3.x §IV-F.1 (Firm A's 145 pixel-identical signatures across $\sim 50$ distinct partners) provides direct evidence that firm-level template reuse does occur at Firm A; the within-firm collision pattern at all four Big-4 firms is consistent with that mechanism extending in milder form to Firms B/C/D.
 
 ## D. K=2 / K=3 as Descriptive Firm-Compositional Partitions
 
@@ -104,15 +104,15 @@ Two additional findings refine the calibration story. First, the per-pair condit
 
 ## G. Pixel-Identity as a Hard Positive Anchor; Inherited Inter-CPA Negative Anchor Reframed as Coincidence Rate
 
-The only hard ground-truth subset in the corpus is pixel-identical signatures: those whose nearest same-CPA match is byte-identical after crop and normalisation. Independent hand-signing cannot produce byte-identical images, so these signatures are conservative-subset ground truth for the *replicated* class. On the Big-4 subset ($n = 262$ pixel-identical signatures), all three candidate classifiers — the inherited box rule, the K=3 hard label, and the reverse-anchor metric with a prevalence-calibrated cut — achieve $0\%$ positive-anchor miss rate (Wilson 95% upper bound $1.45\%$). We caution that this result is necessary but not sufficient: for the box rule it is close to tautological, because byte-identical neighbours have cosine $\approx 1$ and dHash $\approx 0$, well inside the rule's high-confidence region. The corresponding signature-level *negative* anchor evidence is developed in §III-L.1 above (v4 spike: cos$>0.95$ per-comparison ICCR $= 0.00060$, replicating v3.20.0's reported $0.0005$ under prior "FAR" terminology). We frame the per-comparison rate as a specificity proxy under the assumption that inter-CPA pairs constitute a clean negative anchor, and we document in §III-L.4 that this assumption is partially violated by within-firm cross-CPA template-like collision structures.
+The only hard ground-truth subset in the corpus is pixel-identical signatures: those whose nearest same-CPA match is byte-identical after crop and normalisation. Independent hand-signing cannot produce byte-identical images, so these signatures are conservative-subset ground truth for the *replicated* class. On the Big-4 subset ($n = 262$ pixel-identical signatures), all three candidate checks — the inherited box rule, the K=3 hard label, and the reverse-anchor metric with a prevalence-calibrated cut — achieve $0\%$ positive-anchor miss rate (Wilson 95% upper bound $1.45\%$). We caution that this result is necessary but not sufficient: for the box rule it is close to tautological, because byte-identical neighbours have cosine $\approx 1$ and dHash $\approx 0$, well inside the rule's high-confidence region. The corresponding signature-level *negative* anchor evidence is developed in §III-L.1 above (v4 spike: cos$>0.95$ per-comparison ICCR $= 0.00060$, replicating v3.20.0's reported $0.0005$ under prior "FAR" terminology). We frame the per-comparison rate as a specificity proxy under the assumption that inter-CPA pairs constitute a clean negative anchor, and we document in §III-L.4 that this assumption is partially violated by within-firm cross-CPA template-like collision structures.
 
-## G. Limitations
+## H. Limitations
 
 Several limitations should be transparent. The first nine are v4.0-specific; the last five are inherited from v3.20.0 §V-G and still apply to the v4.0 pipeline.
 
 *No signature-level ground truth; no true error rates reportable.* The corpus does not contain labelled hand-signed or replicated classes at the signature level. We therefore cannot report False Rejection Rate, sensitivity, recall, Equal Error Rate, ROC-AUC, precision, or positive predictive value against ground truth. All quantitative rates reported in §III-L are inter-CPA negative-anchor coincidence rates (ICCRs) under the assumption that inter-CPA pairs constitute a clean negative anchor; this is a specificity proxy, not a calibrated specificity (§III-M).
 
-*Inter-CPA negative-anchor assumption is partially violated.* The cross-firm hit matrix of §III-L.4 shows that $98$–$100\%$ of inter-CPA collisions under the deployed rule originate from candidates within the source firm, consistent with firm-specific template, stamp, or document-production reuse. The inter-CPA-as-negative assumption is therefore not exactly satisfied — some inter-CPA pairs may share firm-level templates rather than being independent random matches. Our reported per-comparison ICCRs are best read as specificity-proxy rates under a partially-violated assumption, not as calibrated FARs.
+*Inter-CPA negative-anchor assumption is partially violated and the violation is firm-dependent.* The cross-firm hit matrix of §III-L.4 shows that under the deployed any-pair rule, within-firm collision concentration is $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms), consistent with firm-specific template, stamp, or document-production reuse. The inter-CPA-as-negative assumption is therefore not exactly satisfied — some inter-CPA pairs may share firm-level templates rather than being independent random matches. Our reported per-comparison ICCRs are best read as specificity-proxy rates under a partially-violated assumption, not as calibrated FARs. Because the violation is firm-dependent, Firm A's per-firm ICCR is more contaminated by within-firm sharing than Firms B/C/D's; the per-firm B/C/D rates of $0.09$–$0.16$ are therefore closer to a clean specificity estimate than the pooled rate, and the Firm A vs Firms B/C/D contrast reflects both genuine firm heterogeneity and a firm-dependent proxy-contamination gradient.
 
 *Scope.* The v4.0 primary analyses are scoped to the Big-4 sub-corpus. We did not perform the full per-signature pool-normalised ICCR analysis at the full $n = 686$ scope; the §IV-K full-dataset Spearman re-run shows the K=3 $+$ box-rule rank-convergence is preserved at $n = 686$ but does not validate the Big-4 operational ICCRs, the LOOO firm-fold structure, or the five-way operational classifier at the broader scope.
 
@@ -144,9 +144,9 @@ Several limitations should be transparent. The first nine are v4.0-specific; the
 
 We present a fully automated pipeline for detecting non-hand-signed CPA signatures in Taiwan-listed financial audit reports and a multi-tool framework for characterising and disclosing its operational behaviour at the Big-4 sub-corpus scope. The pipeline processes raw PDFs through VLM-based page identification, YOLO-based signature detection, ResNet-50 feature extraction, and dual-descriptor (cosine + independent-minimum dHash) similarity computation. The operational output is an inherited Paper A five-way per-signature classifier with worst-case document-level aggregation (§III-L). Applied to 90,282 audit reports filed between 2013 and 2023, the pipeline extracts 182,328 signatures from 758 CPAs, with the Big-4 sub-corpus (437 CPAs at accountant level; 150,442–150,453 signatures at signature level) as the primary analytical population.
 
-Our central methodological contributions are: (1) a composition decomposition (Scripts 39b–39e) that establishes the absence of a within-population bimodal antimode in the Big-4 descriptor distribution: the apparent multimodality dissolves under joint firm-mean centring and integer-tie jitter ($p_{\text{median}} = 0.35$), so distributional "natural-threshold" framings of the inherited operating points are not empirically supported; (2) an anchor-based inter-CPA coincidence-rate (ICCR) calibration at three units of analysis — per-comparison ($0.0006$ at cos$>0.95$; $0.0013$ at dHash$\leq 5$; $0.00014$ jointly), pool-normalised per-signature ($0.11$ for the deployed any-pair HC rule), and per-document ($0.34$ for the operational HC$+$MC alarm) — with explicit terminological replacement of "FAR" by "ICCR" given the unsupervised setting; (3) firm heterogeneity quantification: logistic regression with pool-size adjustment gives odds ratios $0.053$, $0.010$, $0.027$ for Firms B/C/D relative to Firm A reference, indicating a large multiplicative effect that pool-size differences do not explain; (4) cross-firm hit matrix evidence that $98$–$100\%$ of inter-CPA collisions under the deployed rule originate from candidates within the source firm, consistent with firm-specific template, stamp, or document-production reuse mechanisms; (5) K=3 mixture demoted from "three mechanism clusters" to a descriptive firm-compositional partition; (6) three feature-derived scores converging on the per-CPA descriptor-position ranking at Spearman $\rho \geq 0.879$, reported as internal consistency rather than external validation; (7) $0\%$ positive-anchor miss rate on 262 byte-identical Big-4 signatures with the conservative-subset caveat; and (8) a nine-tool unsupervised-validation collection (§III-M) that explicitly discloses each tool's untested assumption and positions the system as an anchor-calibrated screening framework with human-in-the-loop review, not as a validated forensic detector.
+Our central methodological contributions are: (1) a composition decomposition (Scripts 39b–39e) that establishes the absence of a within-population bimodal antimode in the Big-4 descriptor distribution: the apparent multimodality dissolves under joint firm-mean centring and integer-tie jitter ($p_{\text{median}} = 0.35$), so distributional "natural-threshold" framings of the inherited operating points are not empirically supported; (2) an anchor-based inter-CPA coincidence-rate (ICCR) calibration at three units of analysis — per-comparison ($0.0006$ at cos$>0.95$; $0.0013$ at dHash$\leq 5$; $0.00014$ jointly), pool-normalised per-signature ($0.11$ for the deployed any-pair HC rule), and per-document ($0.34$ for the operational HC$+$MC alarm) — with explicit terminological replacement of "FAR" by "ICCR" given the unsupervised setting; (3) firm heterogeneity quantification: logistic regression with pool-size adjustment gives odds ratios $0.053$, $0.010$, $0.027$ for Firms B/C/D relative to Firm A reference, indicating a large multiplicative effect that pool-size differences do not explain; (4) cross-firm hit matrix evidence that under the deployed any-pair rule, within-firm collision concentration is $98.8\%$ at Firm A and $76.7$–$83.7\%$ at Firms B/C/D (the stricter same-pair joint event saturates at $97.0$–$99.96\%$ within-firm across all four firms), consistent with firm-specific template, stamp, or document-production reuse mechanisms; (5) K=3 mixture demoted from "three mechanism clusters" to a descriptive firm-compositional partition; (6) three feature-derived scores converging on the per-CPA descriptor-position ranking at Spearman $\rho \geq 0.879$, reported as internal consistency rather than external validation; (7) $0\%$ positive-anchor miss rate on 262 byte-identical Big-4 signatures with the conservative-subset caveat; and (8) a ten-tool unsupervised-validation collection (§III-M Table XXVII) that explicitly discloses each tool's untested assumption and positions the system as an anchor-calibrated screening framework with human-in-the-loop review, not as a validated forensic detector.
 
-Future work falls in four directions. *First*, a small-scale human-rated validation set would enable direct ROC optimisation and provide signature-level ground truth that v4.0 fundamentally lacks; without such ground truth, no true error rates can be reported. *Second*, the within-firm collision concentration documented in §III-L.4 (98–100% same-firm partners) invites a separate study to distinguish deliberate template sharing from passive firm-level production artefacts (shared scanners, common form templates, identical report-generation infrastructure) — a question the inter-CPA-anchor analysis alone cannot resolve. *Third*, the descriptive Firm A versus Firms B/C/D contrast (per-document HC$+$MC alarm $0.62$ vs $0.09$–$0.16$) — together with v3.x's byte-level evidence of 145 pixel-identical signatures across $\sim 50$ distinct Firm A partners — invites a companion analysis examining whether such firm-level signing patterns correlate with established audit-quality measures. *Fourth*, generalisation to mid- and small-firm contexts requires extending the anchor-based ICCR framework to scopes where firm-level LOOO folds are not available; the §III-I.4 composition diagnostics already document that the absence of within-population bimodality is corpus-universal, so the v4.0 calibration approach in principle generalises, but a full extension with cluster-robust uncertainty quantification is left as future work.
+Future work falls in four directions. *First*, a small-scale human-rated validation set would enable direct ROC optimisation and provide signature-level ground truth that v4.0 fundamentally lacks; without such ground truth, no true error rates can be reported. *Second*, the within-firm collision concentration documented in §III-L.4 (any-pair $76.7$–$98.8\%$ across Big-4; same-pair joint $97.0$–$99.96\%$) invites a separate study to distinguish deliberate template sharing from passive firm-level production artefacts (shared scanners, common form templates, identical report-generation infrastructure) — a question the inter-CPA-anchor analysis alone cannot resolve. *Third*, the descriptive Firm A versus Firms B/C/D contrast (per-document HC$+$MC alarm $0.62$ vs $0.09$–$0.16$) — together with v3.x's byte-level evidence of 145 pixel-identical signatures across $\sim 50$ distinct Firm A partners — invites a companion analysis examining whether such firm-level signing patterns correlate with established audit-quality measures. *Fourth*, generalisation to mid- and small-firm contexts requires extending the anchor-based ICCR framework to scopes where firm-level LOOO folds are not available; the §III-I.4 composition diagnostics already document that the absence of within-population bimodality is corpus-universal, so the v4.0 calibration approach in principle generalises, but a full extension with cluster-robust uncertainty quantification is left as future work.
 
 ---
 

paper/v4/paper_a_results_v4_section_iv.md

@@ -82,26 +82,26 @@ This section reports the empirical evidence for §III-K's three-score internal-c
 
 | Score pair | Spearman $\rho$ | $p$-value |
 |---|---|---|
-| K=3 P(C1) vs Paper A box-rule hand-leaning rate | $+0.9627$ | $< 10^{-248}$ |
-| Reverse-anchor cosine percentile vs Paper A box-rule hand-leaning rate | $+0.8890$ | $< 10^{-149}$ |
+| 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 hand-leaning rate |
+| 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 = more hand-leaning relative to the non-Big-4 reference.)
+(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 more hand-leaning. The K=3 posterior P(C1) and the box-rule hand-leaning rate (Score 1 and Score 3) place Firm C at the most-hand-leaning 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.
+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, replicated vs not-replicated), $n = 150{,}442$ Big-4 signatures.
+**Table XI.** Per-signature Cohen $\kappa$ (binary collapse, replication-dominated vs less-replication-dominated), $n = 150{,}442$ Big-4 signatures.
 
 | Pair | Cohen $\kappa$ |
 |---|---|
@@ -144,10 +144,10 @@ This section reports the only hard-ground-truth subset analysis available in the
 
 **Table XIV.** Positive-anchor miss rate, $n = 262$ Big-4 byte-identical signatures.
 
-| Classifier | Misclassified as hand-leaning | Miss rate | Wilson 95% CI |
+| 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 = replicated; descriptive) | $0 / 262$ | $0\%$ | $[0\%, 1.45\%]$ |
+| 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.
@@ -158,7 +158,7 @@ We caution that for the Paper A box rule this result is close to tautological (b
 
 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 Table XVI.
+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
 
@@ -174,7 +174,7 @@ This section reports the §III-L five-way per-signature + document-level worst-c
 | 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.)
+(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).**
 
@@ -185,11 +185,11 @@ This section reports the §III-L five-way per-signature + document-level worst-c
 | 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-replicated 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%).
+(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 XV-B.** Document-level worst-case category counts, Big-4 sub-corpus, $n = 75{,}233$ unique PDFs.
+**Table XIX.** Document-level worst-case category counts, Big-4 sub-corpus, $n = 75{,}233$ unique PDFs.
 
 | Category | Long name | $n$ documents | % |
 |---|---|---|---|
@@ -212,38 +212,38 @@ This section reports the §III-L five-way per-signature + document-level worst-c
 
 (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 hand-leaning ranking (e.g., Firm D's MC fraction is lower than Firm B's while Firm D's reverse-anchor score ranks it as more hand-leaning than Firm B).
+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 (hand-leaning) | C2 (mixed) | C3 (replicated) | C1 % | C3 % |
+| 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 replicated component (no Firm A CPAs in C1); Firm C has the highest hand-leaning 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).
+(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 XV-B above.**
+**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 hand-leaning 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.
+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 hand-leaning | 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 mixed | 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 replicated | 0.9826 / 2.41 / 0.321 | 0.9826 / 2.40 / 0.205 | $\lvert\Delta\rvert$ cos 0.000, dHash 0.01, wt 0.117 |
+| 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 hand-leaning rate, Big-4 sub-corpus vs full dataset.
+**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 hand-leaning rate) | $p$-value |
+| 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 | $+0.9558$ | $< 10^{-300}$ |
@@ -251,7 +251,7 @@ This section reports the v4.0 reproducibility cross-check at the full accountant
 
 (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 hand-leaning rate are preserved at the full scope. Component centres shift modestly: C3 (replicated) is essentially unchanged in centre but loses weight $0.117$ as the full population includes more non-templated CPAs (mid/small firms); C1 (hand-leaning) 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 hand-leaning CPAs 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).
+**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)
 
@@ -263,7 +263,7 @@ This section consolidates the v4-new empirical results that support the §III-L
 
 ### M.1 Composition decomposition (Scripts 39b–39e)
 
-**Table XIX.** Within-firm and between-firm decomposition of the Big-4 accountant-level dip-test rejection.
+**Table XX.** Within-firm and between-firm decomposition of the Big-4 accountant-level dip-test rejection.
 
 | Diagnostic | Scope | Statistic | Implication |
 |---|---|---|---|
@@ -277,7 +277,7 @@ This section consolidates the v4-new empirical results that support the §III-L
 
 ### M.2 Anchor-based inter-CPA pair-level ICCR (Script 40b)
 
-**Table XX.** Big-4 inter-CPA per-comparison ICCR sweep, $n = 5 \times 10^5$ pairs (Big-4 scope; v4 new).
+**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 |
 |---|---|---|
@@ -297,7 +297,7 @@ The cos $> 0.95$ row replicates v3.20.0 §IV-F.1 Table X (v3 reported $0.0005$ u
 
 ### M.3 Pool-normalised per-signature ICCR (Script 43)
 
-**Table XXI.** 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 $n_{\text{boot}} = 1000$.
+**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 |
 |---|---|---|---|
@@ -314,7 +314,7 @@ Decile trend is broadly monotone in pool size with two minor reversals (decile 5
 
 ### M.4 Document-level ICCR under three alarm definitions (Script 45)
 
-**Table XXII.** Document-level inter-CPA ICCR by alarm definition; $n_{\text{docs}} = 75{,}233$.
+**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 |
 |---|---|---|---|
@@ -322,11 +322,11 @@ Decile trend is broadly monotone in pool size with two minor reversals (decile 5
 | 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 $0.0863$ ($n = 8{,}379$).
+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 XXIII.** Logistic regression of per-signature any-pair HC hit indicator on firm dummies and centred log pool size (Firm A reference).
+**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 |
 |---|---|---|
@@ -337,7 +337,7 @@ Per-firm D2 document-level ICCR: Firm A $0.6201$ ($n = 30{,}226$); Firm B $0.160
 
 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 XXIV.** Cross-firm hit matrix among Big-4 source signatures with any-pair HC hit; max-cosine partner firm (counts).
+**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 |
 |---|---|---|---|---|---|---|
@@ -350,7 +350,7 @@ Same-pair joint hits (single candidate satisfying both cos $> 0.95$ AND dHash $\
 
 ### M.6 Alert-rate sensitivity around inherited HC threshold (Script 46)
 
-**Table XXV.** Local-gradient / median-gradient ratio at inherited thresholds (descriptive plateau diagnostic).
+**Table XXVI.** Local-gradient / median-gradient ratio at inherited thresholds (descriptive plateau diagnostic).
 
 | Threshold | Local / median gradient ratio | Interpretation |
 |---|---|---|

paper/變更摘要_給Jimmy_20260623.md

@@ -0,0 +1,38 @@
+# Paper A v13 修訂摘要（給 Jimmy）— 2026-06-23
+
+**現行版本**：rev9.1（`Paper_full_v13_filled_rev9_20260623.docx` / PDF 同名）。源檔 `paper/v13_build/paper_v13_filled.md`，已 commit+push 至 `paper-a-v4-big4`。
+
+**一句話**：用三份 AI 審稿（Gemini 3.1 + ChatGPT 5.5 + Opus 4.8 融合，共 29 點；另加兩輪 ChatGPT 5.5 敵意審稿）逐點修訂。所有新數字皆由資料庫實算、可重現，**未杜撰任何數據**。
+
+---
+
+## 一、核心宣稱「誠實化」（最重要，請確認你能接受降溫幅度）
+
+- **撤回「139×」式比較**。原本「Firm A 觸發率 = 巧合率的 139 倍」是把「同一人重複」除以「不同人互撞」，方向高估。改為只報原始比率（A 82% vs B/C/D 24–35%），不再乘除。
+- **specificity → specificity proxy**：我們沒有 labeled negatives，明說 ICCR 是「不同會計師間的巧合率」，**不是**真正的偽陽率、連上下界都不是。
+- **新增 §III-F「What HC Means and Does Not Mean」**：白紙黑字寫「**HC 不是 reuse 標籤**」——HC 只代表「同一會計師極端重複、且在不相關會計師間罕見」。reuse 是其中一種詮釋，Firm A 由 byte-identical + 訪談另外支撐；**B/C/D 完全不作 reuse 宣稱**。
+- **Firm A 改稱 known-positive / quasi-positive benchmark**，不再包裝成嚴格 blinded test（因為訪談早已知道它是 stamping firm）。
+- **訪談降為 contextual / corroborative**，明說「非 validation、不可獨立重製」。
+
+## 二、新增的實證強化（讓宣稱更站得住）
+
+- **Table VI（any-pair vs same-pair）**：審稿質疑「cosine 與 dHash 來自不同配對、HC 是拼湊的」。實算反駁——改用嚴格 same-pair（同一配對需同時滿足兩條件），Firm A 仍 **57.3%**、B/C/D 跌到 5–9%，**A/其他的比值反而從 2.4–3.4× 升到 6.4–10.8×**。
+- **F5 穩健性四連檢**（§IV-C）：pool-size 分層、會計師層 bootstrap（差距 53.7pp [49.5, 57.5]）、firm+year 固定效果、逐年剔除（53.1–54.9pp）——firm 差異穿透所有控制。
+- **Table V — 影像管線審計（880 份 PDF）**：純掃描比例 2013 **82%** → 2021 **崩到 1%**，metadata 直接點名掃描機型（Fuji Xerox D125 等）。同時：(a) 坐實「事務所＝影像管線」混淆是真的；(b) 但 Firm A 在純掃描年代就已高 HC，反證其訊號**不是**數位化 artifact。
+- **byte-identical 分期**：262 筆中 30 筆在掃描年代（18 在 Firm A，掃描雜訊無法偽造 → 重用鐵證）、232 筆在數位年代（誠實標註此暴增含「可偵測性」成分）。
+- 其餘：Figure 3 換真實密度圖、新增 Figure 6 閾值敏感度面、G5 補「全庫期望巧合 HC ≈ 888 件」。
+
+## 三、需要你知道／拍板的三個判斷
+
+1. **Framing 採「rebalance 不 relabel」**：審稿建議把全文改寫成「無標籤校準方法（審計只是 case study）」。我**沒有照做最強版**——把方法升為主貢獻、但**保留審計發現當 headline**，且**不宣稱 "general framework"**（避免「一個案例憑什麼叫 general」的新攻擊）。請確認你同意這個定位。
+2. **不做 within-CPA「真親簽」對照組**：理想上該比「真人親簽的 HC 誤觸率」，但我們沒有標記的親簽資料、公開資料集又是不同族群——硬借會引入新假設。已在 §III-E 主動說明「考慮過、刻意不做」。
+3. **PDF 管線發現是雙刃**：它讓「事務所混淆」更嚴重（我們誠實寫出），但同時用掃描年代證據鞏固核心。兩面都寫了，沒挑對我們有利的講。
+
+## 四、剩餘待辦（你／投稿前）
+
+- 作者、機構、DOI、biography placeholder（double-blind，投稿前補）
+- IEEE 模板最終排版；related work 可再收斂；表格 II-b/IV 是否整併
+- 人工 review protocol 首次執行（未做，已列 future work）
+
+---
+*完整逐點對照見 `paper/fusion_review_todo.md`（29/29 已處理）。所有分析腳本在 `paper/v13_build/scripts/`，可一鍵重現。*

signature_analysis/45_firm_year_trends.py

@@ -0,0 +1,60 @@
+#!/usr/bin/env python3
+"""Firm x year descriptor trends (B-gate diagnostic).
+
+Plots per-firm yearly mean cosine, mean dHash, and HC-box hit share to test
+whether Firms B/C/D show a 2020 structural break converging toward Firm A.
+Read-only against the production DB.
+"""
+import sqlite3
+import matplotlib.pyplot as plt
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+FIRMS = [('勤業眾信聯合', 'Firm A (Deloitte)', '#d62728'),
+         ('安侯建業聯合', 'Firm B (KPMG)', '#1f77b4'),
+         ('資誠聯合', 'Firm C (PwC)', '#2ca02c'),
+         ('安永聯合', 'Firm D (EY)', '#ff7f0e')]
+
+conn = sqlite3.connect(f'file:{DB}?mode=ro', uri=True)
+cur = conn.cursor()
+
+
+def series(firm_zh):
+    cur.execute("""
+        SELECT CAST(substr(s.year_month,1,4) AS INT) AS yr,
+               AVG(s.max_similarity_to_same_accountant),
+               AVG(s.min_dhash_independent),
+               AVG(CASE WHEN s.max_similarity_to_same_accountant>0.95
+                         AND s.min_dhash_independent<=5 THEN 1.0 ELSE 0.0 END),
+               COUNT(*)
+        FROM signatures s JOIN accountants a ON s.assigned_accountant=a.name
+        WHERE a.firm=? AND s.year_month IS NOT NULL
+          AND s.max_similarity_to_same_accountant IS NOT NULL
+          AND s.min_dhash_independent IS NOT NULL
+        GROUP BY yr ORDER BY yr""", (firm_zh,))
+    return cur.fetchall()
+
+
+fig, axes = plt.subplots(1, 3, figsize=(16, 4.8))
+for firm_zh, label, color in FIRMS:
+    rows = series(firm_zh)
+    yrs = [r[0] for r in rows]
+    axes[0].plot(yrs, [r[1] for r in rows], 'o-', color=color, label=label)
+    axes[1].plot(yrs, [r[2] for r in rows], 'o-', color=color, label=label)
+    axes[2].plot(yrs, [r[3] for r in rows], 'o-', color=color, label=label)
+
+for ax in axes:
+    ax.axvline(2020, ls='--', color='grey', alpha=0.6)
+    ax.text(2020.05, ax.get_ylim()[0], ' 2020', color='grey', fontsize=8, va='bottom')
+    ax.set_xlabel('Fiscal year')
+    ax.grid(alpha=0.3)
+axes[0].set_title('Mean best-match cosine'); axes[0].axhline(0.95, ls=':', color='k', alpha=0.4)
+axes[1].set_title('Mean independent-min dHash'); axes[1].axhline(5, ls=':', color='k', alpha=0.4)
+axes[2].set_title('HC-box share (cos>0.95 & dHash$\\leq$5)')
+axes[0].legend(fontsize=8, loc='lower right')
+fig.suptitle('Big-4 descriptor trends 2013–2023 (2023 = partial, to Apr)  —  no 2020 break, no convergence to A',
+             fontsize=11)
+fig.tight_layout()
+out = '/Volumes/NV2/pdf_recognize/signature_analysis/firm_year_trends.png'
+fig.savefig(out, dpi=130, bbox_inches='tight')
+print('saved', out)
+conn.close()

signature_analysis/46_bcd_only_iccr.py

@@ -0,0 +1,87 @@
+#!/usr/bin/env python3
+"""Script 46: BCD-only (exclude Firm A) per-comparison ICCR recompute.
+
+Replicates 40b's inter-CPA negative-anchor pair sampling (N=500k, seed=42)
+but compares three negative-anchor pool compositions:
+  - ABCD      : all Big-4 (current paper baseline)
+  - BCD       : Big-4 excluding Firm A (normative-baseline proposal)
+  - BCD+nonB4 : BCD plus all non-Big-4 firms
+Reports marginal cos>0.95, dHash<=5, and the joint HC rule cos>0.95 & dHash<=5.
+Read-only.
+"""
+import sqlite3
+from collections import defaultdict
+import numpy as np
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+N_PAIRS = 500_000
+SEED = 42
+FIRM_A = '勤業眾信聯合'
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+
+
+def hamming(a, b):
+    return (int.from_bytes(a, 'big') ^ int.from_bytes(b, 'big')).bit_count()
+
+
+def load():
+    conn = sqlite3.connect(f'file:{DB}?mode=ro', uri=True)
+    cur = conn.cursor()
+    cur.execute("""
+        SELECT s.assigned_accountant, a.firm, s.feature_vector, s.dhash_vector
+        FROM signatures s JOIN accountants a ON s.assigned_accountant=a.name
+        WHERE s.assigned_accountant IS NOT NULL AND a.firm IS NOT NULL
+          AND s.feature_vector IS NOT NULL AND s.dhash_vector IS NOT NULL""")
+    rows = cur.fetchall()
+    conn.close()
+    return rows
+
+
+def wilson(k, n, z=1.96):
+    if n == 0:
+        return (None, None)
+    p = k / n
+    d = 1 + z*z/n
+    c = (p + z*z/(2*n)) / d
+    h = z*np.sqrt(p*(1-p)/n + z*z/(4*n*n)) / d
+    return (max(0.0, c-h), min(1.0, c+h))
+
+
+def iccr(rows, label):
+    by = defaultdict(list)
+    for acct, firm, fv, dh in rows:
+        by[acct].append((fv, dh))
+    accts = list(by.keys())
+    feats = {a: np.stack([np.frombuffer(r[0], dtype=np.float32) for r in by[a]]) for a in accts}
+    dhs = {a: [r[1] for r in by[a]] for a in accts}
+    rng = np.random.default_rng(SEED)
+    cos = np.empty(N_PAIRS, np.float32)
+    dv = np.empty(N_PAIRS, np.int32)
+    na = len(accts)
+    for t in range(N_PAIRS):
+        i, j = rng.choice(na, 2, replace=False)
+        a1, a2 = accts[i], accts[j]
+        k1 = int(rng.integers(0, len(by[a1])))
+        k2 = int(rng.integers(0, len(by[a2])))
+        cos[t] = float(feats[a1][k1] @ feats[a2][k2])
+        dv[t] = hamming(dhs[a1][k1], dhs[a2][k2])
+    n = N_PAIRS
+    m_cos = int((cos > 0.95).sum())
+    m_dh = int((dv <= 5).sum())
+    joint = int(((cos > 0.95) & (dv <= 5)).sum())
+    jlo, jhi = wilson(joint, n)
+    print(f'\n== {label} ==')
+    print(f'  signatures={len(rows):,}  accountants={na}  pairs={n:,}')
+    print(f'  cos>0.95         ICCR = {m_cos/n:.5f}  ({m_cos})')
+    print(f'  dHash<=5         ICCR = {m_dh/n:.5f}  ({m_dh})')
+    print(f'  JOINT (HC rule)  ICCR = {joint/n:.6f}  ({joint})  Wilson95% [{jlo:.6f},{jhi:.6f}]')
+    return joint/n
+
+
+rows = load()
+abcd = [r for r in rows if r[1] in BIG4]
+bcd = [r for r in rows if r[1] in BIG4 and r[1] != FIRM_A]
+bcd_non = [r for r in rows if r[1] != FIRM_A]
+iccr(abcd, 'ABCD (current paper baseline)')
+iccr(bcd, 'BCD only (exclude Firm A)')
+iccr(bcd_non, 'BCD + non-Big-4')

signature_analysis/47_bcd_only_full_recompute.py

@@ -0,0 +1,160 @@
+#!/usr/bin/env python3
+"""Script 47: BCD-only recompute of (1) KDE crossover, (2) per-signature
+pool-normalized any-pair ICCR (cos>0.95 & dHash<=5), (3) per-document HC+MC
+inter-CPA ICCR (cos>0.95 & dHash<=15), each for ABCD vs BCD-only negative-anchor
+pools. Replicates Scripts 10/43/44 methodology. Document-level subsampling used
+for the pool simulation (exact same-CPA pool sizes retained). Read-only.
+"""
+import sqlite3
+from collections import defaultdict
+import numpy as np
+from scipy.stats import gaussian_kde
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+FIRM_A = '勤業眾信聯合'
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+ALIAS = {'勤業眾信聯合': 'A', '安侯建業聯合': 'B', '資誠聯合': 'C', '安永聯合': 'D'}
+SEED = 42
+N_INTRA = 200_000
+N_INTER = 500_000
+N_DOC_SUBSAMPLE = 9000   # documents processed in pool simulation per scope
+
+
+def load():
+    conn = sqlite3.connect(f'file:{DB}?mode=ro', uri=True)
+    cur = conn.cursor()
+    cur.execute("""
+        SELECT s.signature_id, s.assigned_accountant, a.firm, s.source_pdf,
+               s.feature_vector, s.dhash_vector
+        FROM signatures s JOIN accountants a ON s.assigned_accountant=a.name
+        WHERE s.assigned_accountant IS NOT NULL AND a.firm IN (?,?,?,?)
+          AND s.feature_vector IS NOT NULL AND s.dhash_vector IS NOT NULL""", BIG4)
+    rows = cur.fetchall()
+    conn.close()
+    return rows
+
+
+def hamming1(q, c):
+    return (int.from_bytes(q, 'big') ^ int.from_bytes(c, 'big')).bit_count()
+
+
+def wilson(k, n, z=1.96):
+    if n == 0:
+        return (None, None)
+    p = k/n; d = 1+z*z/n
+    c = (p+z*z/(2*n))/d
+    h = z*np.sqrt(p*(1-p)/n+z*z/(4*n*n))/d
+    return (max(0.0, c-h), min(1.0, c+h))
+
+
+def kde_crossover(feats, cpas, label):
+    by = defaultdict(list)
+    for i, c in enumerate(cpas):
+        by[c].append(i)
+    by = {c: np.array(v) for c, v in by.items() if len(v) >= 2}
+    accts = list(by.keys())
+    rng = np.random.default_rng(SEED)
+    # intra: two sigs from same random CPA
+    intra = np.empty(N_INTRA, np.float32)
+    ks = rng.integers(0, len(accts), N_INTRA)
+    for t in range(N_INTRA):
+        idx = by[accts[ks[t]]]
+        a, b = rng.choice(idx, 2, replace=False)
+        intra[t] = feats[a] @ feats[b]
+    # inter: two sigs from different CPAs
+    inter = np.empty(N_INTER, np.float32)
+    for t in range(N_INTER):
+        i, j = rng.choice(len(accts), 2, replace=False)
+        a = rng.choice(by[accts[i]]); b = rng.choice(by[accts[j]])
+        inter[t] = feats[a] @ feats[b]
+    xs = np.linspace(0.3, 1.0, 10000)
+    ki = gaussian_kde(intra[:100000]); ke = gaussian_kde(inter[:100000])
+    diff = ki(xs) - ke(xs)
+    cross = xs[np.where(np.diff(np.sign(diff)))[0]]
+    cross = [float(x) for x in cross if 0.6 < x < 0.99]
+    print(f'  [{label}] intra mean={intra.mean():.4f} inter mean={inter.mean():.4f}'
+          f'  KDE crossover(s): {[f"{x:.4f}" for x in cross]}')
+    return cross
+
+
+def pool_sim(rows, scope_firms, label):
+    """Per-signature & per-document inter-CPA any-pair ICCR over a doc subsample."""
+    keep = [r for r in rows if ALIAS[r[2]] in scope_firms]
+    feats = np.stack([np.frombuffer(r[4], np.float32) for r in keep]).astype(np.float32)
+    feats /= np.clip(np.linalg.norm(feats, axis=1, keepdims=True), 1e-9, None)
+    cpas = [r[1] for r in keep]
+    firms = [ALIAS[r[2]] for r in keep]
+    docs = [r[3] for r in keep]
+    dh = [r[5] for r in keep]
+    n = len(keep)
+    cpa_idx = defaultdict(list)
+    for i, c in enumerate(cpas):
+        cpa_idx[c].append(i)
+    cpa_idx = {c: np.array(v) for c, v in cpa_idx.items()}
+    pool_size = {c: len(v)-1 for c, v in cpa_idx.items()}
+    doc_idx = defaultdict(list)
+    for i, d in enumerate(docs):
+        doc_idx[d].append(i)
+    rng = np.random.default_rng(SEED)
+    all_docs = list(doc_idx.keys())
+    sub = rng.choice(len(all_docs), min(N_DOC_SUBSAMPLE, len(all_docs)), replace=False)
+    sel_docs = [all_docs[i] for i in sub]
+
+    sig_hc = []        # per-signature: any-pair cos>0.95 & dh<=5
+    sig_firm = []
+    doc_hcmc = {}      # per-document worst-case: any sig with cos>0.95 & dh<=15
+    doc_firm = {}
+    for d in sel_docs:
+        dhit = False
+        for si in doc_idx[d]:
+            c = cpas[si]; npool = pool_size[c]
+            if npool <= 0:
+                sig_hc.append(False); sig_firm.append(firms[si]); continue
+            same = cpa_idx[c]
+            draw = rng.choice(n, size=min(npool*2+10, n), replace=True)
+            cand = draw[~np.isin(draw, same)][:npool]
+            cosv = feats[cand] @ feats[si]
+            dhv = np.fromiter((hamming1(dh[si], dh[c2]) for c2 in cand), np.int32, len(cand))
+            cg = cosv > 0.95
+            hc = bool((cg & (dhv <= 5)).any())
+            hcmc = bool((cg & (dhv <= 15)).any())
+            sig_hc.append(hc); sig_firm.append(firms[si])
+            if hcmc:
+                dhit = True
+        doc_hcmc[d] = dhit
+        doc_firm[d] = firms[doc_idx[d][0]]
+
+    sig_hc = np.array(sig_hc); sig_firm = np.array(sig_firm)
+    k = int(sig_hc.sum()); m = len(sig_hc)
+    lo, hi = wilson(k, m)
+    print(f'\n  [{label}] per-SIGNATURE any-pair HC ICCR (cos>0.95 & dh<=5): '
+          f'{k/m:.4f}  ({k}/{m})  Wilson95% [{lo:.4f},{hi:.4f}]')
+    for f in sorted(set(sig_firm)):
+        msk = sig_firm == f
+        kk = int(sig_hc[msk].sum()); mm = int(msk.sum())
+        print(f'      Firm {f}: {kk/mm:.4f}  ({kk}/{mm})')
+    dvals = np.array(list(doc_hcmc.values())); dfirm = np.array(list(doc_firm.values()))
+    dk = int(dvals.sum()); dm = len(dvals)
+    dlo, dhi = wilson(dk, dm)
+    print(f'  [{label}] per-DOCUMENT HC+MC ICCR (cos>0.95 & dh<=15): '
+          f'{dk/dm:.4f}  ({dk}/{dm})  Wilson95% [{dlo:.4f},{dhi:.4f}]')
+    for f in sorted(set(dfirm)):
+        msk = dfirm == f
+        kk = int(dvals[msk].sum()); mm = int(msk.sum())
+        print(f'      Firm {f}: {kk/mm:.4f}  ({kk}/{mm})')
+
+
+rows = load()
+allf = np.stack([np.frombuffer(r[4], np.float32) for r in rows]).astype(np.float32)
+allf /= np.clip(np.linalg.norm(allf, axis=1, keepdims=True), 1e-9, None)
+allc = [r[1] for r in rows]
+abcd_mask = [True]*len(rows)
+bcd_mask = [r[2] != FIRM_A for r in rows]
+
+print('=== (1) KDE crossover (intra vs inter cosine) ===')
+kde_crossover(allf, allc, 'ABCD')
+kde_crossover(allf[bcd_mask], [allc[i] for i in range(len(rows)) if bcd_mask[i]], 'BCD-only')
+
+print('\n=== (2)(3) per-signature & per-document inter-CPA ICCR ===')
+pool_sim(rows, {'A', 'B', 'C', 'D'}, 'ABCD (reproduce)')
+pool_sim(rows, {'B', 'C', 'D'}, 'BCD-only')

signature_analysis/48_bcd_only_exact.py

@@ -0,0 +1,141 @@
+#!/usr/bin/env python3
+"""Script 48: full-fidelity (no subsample) BCD-only recompute of per-signature
+and per-document inter-CPA any-pair ICCR, plus corpus-style KDE crossover.
+Vectorized popcount. Scopes: ABCD, BCD-only, BCD+non-Big-4. Read-only.
+"""
+import sqlite3
+from collections import defaultdict
+import numpy as np
+from scipy.stats import gaussian_kde
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+FIRM_A = '勤業眾信聯合'
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+ALIAS = {'勤業眾信聯合': 'A', '安侯建業聯合': 'B', '資誠聯合': 'C', '安永聯合': 'D'}
+SEED = 42
+POP = np.array([bin(i).count('1') for i in range(256)], dtype=np.uint8)
+
+
+def load():
+    conn = sqlite3.connect(f'file:{DB}?mode=ro', uri=True)
+    cur = conn.cursor()
+    cur.execute("""
+        SELECT s.assigned_accountant, a.firm, s.source_pdf,
+               s.feature_vector, s.dhash_vector
+        FROM signatures s JOIN accountants a ON s.assigned_accountant=a.name
+        WHERE s.assigned_accountant IS NOT NULL AND a.firm IS NOT NULL
+          AND s.feature_vector IS NOT NULL AND s.dhash_vector IS NOT NULL""")
+    rows = cur.fetchall()
+    conn.close()
+    return rows
+
+
+def wilson(k, n, z=1.96):
+    if n == 0:
+        return (None, None)
+    p = k/n; d = 1+z*z/n
+    c = (p+z*z/(2*n))/d
+    h = z*np.sqrt(p*(1-p)/n+z*z/(4*n*n))/d
+    return (max(0.0, c-h), min(1.0, c+h))
+
+
+def prep(rows, keep_fn):
+    keep = [r for r in rows if keep_fn(r[1])]
+    feats = np.stack([np.frombuffer(r[3], np.float32) for r in keep]).astype(np.float32)
+    feats /= np.clip(np.linalg.norm(feats, axis=1, keepdims=True), 1e-9, None)
+    dh = np.stack([np.frombuffer(r[4], np.uint8) for r in keep])  # (n,8)
+    cpas = np.array([r[0] for r in keep])
+    firms = np.array([ALIAS.get(r[1], 'X') for r in keep])
+    docs = np.array([r[2] for r in keep])
+    return feats, dh, cpas, firms, docs
+
+
+def crossover(feats, cpas, label):
+    by = defaultdict(list)
+    for i, c in enumerate(cpas):
+        by[c].append(i)
+    by = {c: np.array(v) for c, v in by.items() if len(v) >= 2}
+    accts = list(by.keys())
+    rng = np.random.default_rng(SEED)
+    N = 100_000
+    intra = np.empty(N, np.float32); inter = np.empty(N, np.float32)
+    ks = rng.integers(0, len(accts), N)
+    for t in range(N):
+        idx = by[accts[ks[t]]]
+        a, b = rng.choice(idx, 2, replace=False)
+        intra[t] = feats[a] @ feats[b]
+        i, j = rng.choice(len(accts), 2, replace=False)
+        inter[t] = feats[rng.choice(by[accts[i]])] @ feats[rng.choice(by[accts[j]])]
+    xs = np.linspace(0.3, 1.0, 10000)
+    diff = gaussian_kde(intra)(xs) - gaussian_kde(inter)(xs)
+    cross = [float(x) for x in xs[np.where(np.diff(np.sign(diff)))[0]] if 0.6 < x < 0.99]
+    print(f'  [{label}] crossover {[f"{x:.4f}" for x in cross]}  '
+          f'(intra {intra.mean():.4f} / inter {inter.mean():.4f})')
+
+
+def pool_sim(feats, dh, cpas, firms, docs, label):
+    n = len(cpas)
+    cpa_idx = defaultdict(list)
+    for i, c in enumerate(cpas):
+        cpa_idx[c].append(i)
+    cpa_idx = {c: np.array(v) for c, v in cpa_idx.items()}
+    pool_size = {c: len(v)-1 for c, v in cpa_idx.items()}
+    rng = np.random.default_rng(SEED)
+    sig_hc = np.zeros(n, bool)
+    doc_hcmc = defaultdict(bool)
+    for si in range(n):
+        c = cpas[si]; npool = pool_size[c]
+        if npool <= 0:
+            continue
+        same = cpa_idx[c]
+        draw = rng.integers(0, n, size=npool + same.size + 20)
+        cand = draw[~np.isin(draw, same)][:npool]
+        cosv = feats[cand] @ feats[si]
+        cg = cosv > 0.95
+        if cg.any():
+            dist = POP[dh[cand] ^ dh[si]].sum(axis=1)
+            sig_hc[si] = bool((cg & (dist <= 5)).any())
+            if (cg & (dist <= 15)).any():
+                doc_hcmc[docs[si]] = True
+        else:
+            doc_hcmc.setdefault(docs[si], doc_hcmc[docs[si]] if docs[si] in doc_hcmc else False)
+    # ensure every doc present
+    for d in docs:
+        doc_hcmc.setdefault(d, False)
+    k = int(sig_hc.sum())
+    lo, hi = wilson(k, n)
+    print(f'\n  [{label}] per-SIGNATURE any-pair HC (cos>0.95 & dh<=5): '
+          f'{k/n:.4f} ({k}/{n}) Wilson95% [{lo:.4f},{hi:.4f}]')
+    for f in sorted(set(firms)):
+        m = firms == f
+        print(f'      Firm {f}: {sig_hc[m].sum()/m.sum():.4f} ({int(sig_hc[m].sum())}/{int(m.sum())})')
+    # per-doc, with firm of first sig
+    dfirm = {}
+    for i, d in enumerate(docs):
+        dfirm.setdefault(d, firms[i])
+    dl = list(doc_hcmc.keys())
+    dv = np.array([doc_hcmc[d] for d in dl])
+    df = np.array([dfirm[d] for d in dl])
+    dk = int(dv.sum()); dm = len(dv)
+    dlo, dhi = wilson(dk, dm)
+    print(f'  [{label}] per-DOCUMENT HC+MC (cos>0.95 & dh<=15): '
+          f'{dk/dm:.4f} ({dk}/{dm}) Wilson95% [{dlo:.4f},{dhi:.4f}]')
+    for f in sorted(set(df)):
+        m = df == f
+        print(f'      Firm {f}: {dv[m].sum()/m.sum():.4f} ({int(dv[m].sum())}/{int(m.sum())})')
+
+
+rows = load()
+SCOPES = [('ABCD', lambda fm: fm in BIG4),
+          ('BCD-only', lambda fm: fm in BIG4 and fm != FIRM_A),
+          ('BCD+nonBig4', lambda fm: fm != FIRM_A)]
+
+print('=== KDE crossover ===')
+for name, fn in SCOPES[:2]:
+    f, _, c, _, _ = prep(rows, fn)
+    crossover(f, c, name)
+
+print('\n=== per-signature & per-document inter-CPA ICCR (full) ===')
+for name, fn in SCOPES:
+    f, dh, c, fm, dc = prep(rows, fn)
+    pool_sim(f, dh, c, fm, dc, name)

signature_analysis/49_firmA_outofsample.py

@@ -0,0 +1,104 @@
+#!/usr/bin/env python3
+"""Script 49: Firm A as out-of-sample target against a clean BCD baseline.
+(1) A signatures scored against a BCD-only candidate pool (true out-of-sample
+    inter-firm coincidence).
+(2) Observed deployed rate on ACTUAL same-CPA pools, per firm (the real fired
+    rate, from precomputed deployed descriptors), to juxtapose against the
+    clean BCD inter-CPA coincidence floor. Read-only.
+"""
+import sqlite3
+from collections import defaultdict
+import numpy as np
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+FIRM_A = '勤業眾信聯合'
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+ALIAS = {'勤業眾信聯合': 'A', '安侯建業聯合': 'B', '資誠聯合': 'C', '安永聯合': 'D'}
+SEED = 42
+POP = np.array([bin(i).count('1') for i in range(256)], dtype=np.uint8)
+
+
+def wilson(k, n, z=1.96):
+    if n == 0:
+        return (None, None)
+    p = k/n; d = 1+z*z/n
+    c = (p+z*z/(2*n))/d
+    h = z*np.sqrt(p*(1-p)/n+z*z/(4*n*n))/d
+    return (max(0.0, c-h), min(1.0, c+h))
+
+
+conn = sqlite3.connect(f'file:{DB}?mode=ro', uri=True)
+cur = conn.cursor()
+cur.execute("""
+    SELECT s.assigned_accountant, a.firm, s.source_pdf, s.feature_vector,
+           s.dhash_vector, s.max_similarity_to_same_accountant, s.min_dhash_independent
+    FROM signatures s JOIN accountants a ON s.assigned_accountant=a.name
+    WHERE s.assigned_accountant IS NOT NULL AND a.firm IN (?,?,?,?)
+      AND s.feature_vector IS NOT NULL AND s.dhash_vector IS NOT NULL""", BIG4)
+rows = cur.fetchall()
+conn.close()
+
+# ---- (1) Firm A source vs BCD-only candidate pool ----
+print('=== (1) Firm A out-of-sample vs clean BCD candidate pool ===')
+A = [r for r in rows if r[1] == FIRM_A]
+BCD = [r for r in rows if r[1] in BIG4 and r[1] != FIRM_A]
+bcd_feat = np.stack([np.frombuffer(r[3], np.float32) for r in BCD]).astype(np.float32)
+bcd_feat /= np.clip(np.linalg.norm(bcd_feat, axis=1, keepdims=True), 1e-9, None)
+bcd_dh = np.stack([np.frombuffer(r[4], np.uint8) for r in BCD])
+nb = len(BCD)
+# A CPA pool sizes (their own same-CPA count - 1), to match negative-anchor construction
+a_cpa_idx = defaultdict(list)
+for i, r in enumerate(A):
+    a_cpa_idx[r[0]].append(i)
+pool_size = {c: len(v)-1 for c, v in a_cpa_idx.items()}
+rng = np.random.default_rng(SEED)
+sig_hc = np.zeros(len(A), bool)
+doc_hcmc = defaultdict(bool)
+for i, r in enumerate(A):
+    npool = max(pool_size[r[0]], 1)
+    cand = rng.integers(0, nb, size=npool)
+    sf = np.frombuffer(r[3], np.float32).astype(np.float32)
+    sf /= max(np.linalg.norm(sf), 1e-9)
+    cosv = bcd_feat[cand] @ sf
+    cg = cosv > 0.95
+    doc_hcmc.setdefault(r[2], False)
+    if cg.any():
+        dist = POP[bcd_dh[cand] ^ np.frombuffer(r[4], np.uint8)].sum(axis=1)
+        sig_hc[i] = bool((cg & (dist <= 5)).any())
+        if (cg & (dist <= 15)).any():
+            doc_hcmc[r[2]] = True
+k = int(sig_hc.sum()); n = len(A); lo, hi = wilson(k, n)
+print(f'  A-source vs BCD-pool  per-SIGNATURE HC (cos>0.95 & dh<=5): '
+      f'{k/n:.4f} ({k}/{n}) Wilson95% [{lo:.4f},{hi:.4f}]')
+dv = np.array(list(doc_hcmc.values())); dk = int(dv.sum()); dm = len(dv)
+dlo, dhi = wilson(dk, dm)
+print(f'  A-source vs BCD-pool  per-DOCUMENT HC+MC (cos>0.95 & dh<=15): '
+      f'{dk/dm:.4f} ({dk}/{dm}) Wilson95% [{dlo:.4f},{dhi:.4f}]')
+
+# ---- (2) Observed deployed rate on ACTUAL same-CPA pools, per firm ----
+print('\n=== (2) Observed deployed rate on actual same-CPA pools (real fired rate) ===')
+print('   per-signature HC = max_sim>0.95 & min_dh<=5 ; per-doc HC+MC worst-case dh<=15')
+by_firm_sig = defaultdict(lambda: [0, 0])
+doc_obs = {}
+doc_firm = {}
+for r in rows:
+    fm = ALIAS[r[1]]
+    ms, md = r[5], r[6]
+    if ms is None or md is None:
+        continue
+    hc = (ms > 0.95) and (md <= 5)
+    hcmc = (ms > 0.95) and (md <= 15)
+    by_firm_sig[fm][0] += int(hc); by_firm_sig[fm][1] += 1
+    doc_firm.setdefault(r[2], fm)
+    doc_obs[r[2]] = doc_obs.get(r[2], False) or hcmc
+for fm in sorted(by_firm_sig):
+    k, n = by_firm_sig[fm]
+    lo, hi = wilson(k, n)
+    print(f'  Firm {fm} per-SIGNATURE HC: {k/n:.4f} ({k}/{n}) [{lo:.4f},{hi:.4f}]')
+dd = defaultdict(lambda: [0, 0])
+for d, hit in doc_obs.items():
+    fm = doc_firm[d]; dd[fm][0] += int(hit); dd[fm][1] += 1
+for fm in sorted(dd):
+    k, n = dd[fm]
+    print(f'  Firm {fm} per-DOCUMENT HC+MC: {k/n:.4f} ({k}/{n})')
+print(f'\n  Clean BCD inter-CPA coincidence FLOOR: per-sig HC=0.0048, per-doc HC+MC=0.1281')

signature_analysis/50_scoped_anchor_exact.py

@@ -0,0 +1,107 @@
+#!/usr/bin/env python3
+"""Script 50: publication-grade scoped inter-CPA anchor recompute.
+Faithfully reproduces Script 45's any-pair five-way pool simulation
+(max_cos & min_dh over a random same-size inter-CPA pool, excl. same-CPA),
+then reports for scopes ABCD / BCD / BCD+nonBig4:
+  - per-signature HC (D1) and HC+MC (D2) any-pair FAR
+  - per-document HC (D1) and HC+MC (D2) any-pair FAR
+  - per-firm per-document D2
+ABCD is printed first to verify reproduction of published values
+(per-sig HC~0.1102, per-doc D2~0.3375, Firm A~0.62). Read-only.
+"""
+import sqlite3
+from collections import defaultdict
+import numpy as np
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+FIRM_A = '勤業眾信聯合'
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+ALIAS = {'勤業眾信聯合': 'A', '安侯建業聯合': 'B', '資誠聯合': 'C', '安永聯合': 'D'}
+SEED = 42
+POP = np.array([bin(i).count('1') for i in range(256)], dtype=np.uint8)
+
+
+def wilson(k, n, z=1.96):
+    if n == 0:
+        return (None, None)
+    p = k/n; d = 1+z*z/n
+    c = (p+z*z/(2*n))/d
+    h = z*np.sqrt(p*(1-p)/n+z*z/(4*n*n))/d
+    return (max(0.0, c-h), min(1.0, c+h))
+
+
+def load():
+    conn = sqlite3.connect(f'file:{DB}?mode=ro', uri=True)
+    cur = conn.cursor()
+    cur.execute("""
+        SELECT s.assigned_accountant, a.firm, s.source_pdf,
+               s.feature_vector, s.dhash_vector
+        FROM signatures s JOIN accountants a ON s.assigned_accountant=a.name
+        WHERE s.assigned_accountant IS NOT NULL AND a.firm IS NOT NULL
+          AND s.feature_vector IS NOT NULL AND s.dhash_vector IS NOT NULL""")
+    rows = cur.fetchall()
+    conn.close()
+    return rows
+
+
+def run(rows, keep_fn, label):
+    keep = [r for r in rows if keep_fn(r[1])]
+    n = len(keep)
+    feats = np.stack([np.frombuffer(r[3], np.float32) for r in keep]).astype(np.float32)
+    feats /= np.clip(np.linalg.norm(feats, axis=1, keepdims=True), 1e-9, None)
+    dh = np.stack([np.frombuffer(r[4], np.uint8) for r in keep])
+    cpas = np.array([r[0] for r in keep])
+    firms = np.array([ALIAS.get(r[1], 'NonB4') for r in keep])
+    docs = np.array([r[2] for r in keep])
+    cpa_idx = defaultdict(list)
+    for i, c in enumerate(cpas):
+        cpa_idx[c].append(i)
+    cpa_idx = {c: np.array(v) for c, v in cpa_idx.items()}
+    pool_size = {c: len(v)-1 for c, v in cpa_idx.items()}
+    rng = np.random.default_rng(SEED)
+    max_cos = np.zeros(n, np.float32)
+    min_dh = np.full(n, 64, np.int32)
+    for si in range(n):
+        c = cpas[si]; npool = pool_size[c]
+        if npool <= 0:
+            continue
+        same = cpa_idx[c]
+        draw = rng.integers(0, n, size=npool + same.size + 20)
+        cand = draw[~np.isin(draw, same)][:npool]
+        cosv = feats[cand] @ feats[si]
+        dist = POP[dh[cand] ^ dh[si]].sum(axis=1)
+        max_cos[si] = cosv.max()
+        min_dh[si] = int(dist.min())
+    # any-pair classification
+    hc = (max_cos > 0.95) & (min_dh <= 5)
+    mc = (max_cos > 0.95) & (min_dh > 5) & (min_dh <= 15)
+    d1 = hc
+    d2 = hc | mc
+    print(f'\n===== {label} (n_sig={n:,}) =====')
+    for nm, arr in [('per-sig HC (D1)', d1), ('per-sig HC+MC (D2)', d2)]:
+        k = int(arr.sum()); lo, hi = wilson(k, n)
+        print(f'  {nm}: {k/n:.4f} ({k}/{n}) [{lo:.4f},{hi:.4f}]')
+    # per-document worst-case
+    doc_d1 = defaultdict(bool); doc_d2 = defaultdict(bool); doc_firm = {}
+    for i in range(n):
+        if d1[i]: doc_d1[docs[i]] = True
+        if d2[i]: doc_d2[docs[i]] = True
+        doc_firm.setdefault(docs[i], firms[i])
+        doc_d1.setdefault(docs[i], False); doc_d2.setdefault(docs[i], False)
+    dl = list(doc_d2.keys())
+    nd = len(dl)
+    k1 = sum(doc_d1[d] for d in dl); k2 = sum(doc_d2[d] for d in dl)
+    l1 = wilson(k1, nd); l2 = wilson(k2, nd)
+    print(f'  per-doc HC (D1):   {k1/nd:.4f} ({k1}/{nd}) [{l1[0]:.4f},{l1[1]:.4f}]')
+    print(f'  per-doc HC+MC (D2):{k2/nd:.4f} ({k2}/{nd}) [{l2[0]:.4f},{l2[1]:.4f}]')
+    df = np.array([doc_firm[d] for d in dl])
+    dv = np.array([doc_d2[d] for d in dl])
+    for f in sorted(set(df)):
+        m = df == f
+        print(f'      Firm {f} per-doc D2: {dv[m].sum()/m.sum():.4f} ({int(dv[m].sum())}/{int(m.sum())})')
+
+
+rows = load()
+run(rows, lambda fm: fm in BIG4, 'ABCD (verify vs published: HC~0.110 / D2~0.338 / A~0.62)')
+run(rows, lambda fm: fm in BIG4 and fm != FIRM_A, 'BCD-only')
+run(rows, lambda fm: fm != FIRM_A, 'BCD + non-Big4')

signature_analysis/51_bootstrap_and_crossover.py

@@ -0,0 +1,135 @@
+#!/usr/bin/env python3
+"""Script 51: publication polish.
+Part A: CPA-block bootstrap (1000 reps) on per-signature HC any-pair rate, and
+        document-level bootstrap on per-document HC+MC, for ABCD & BCD.
+Part B: corpus-wide KDE crossover (pair-weighted intra, reproduce 0.837) plus
+        BCD-only and BCD+nonBig4 variants.
+Read-only.
+"""
+import sqlite3
+from collections import defaultdict
+import numpy as np
+from scipy.stats import gaussian_kde
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+FIRM_A = '勤業眾信聯合'
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+ALIAS = {'勤業眾信聯合': 'A', '安侯建業聯合': 'B', '資誠聯合': 'C', '安永聯合': 'D'}
+SEED = 42
+N_BOOT = 1000
+POP = np.array([bin(i).count('1') for i in range(256)], dtype=np.uint8)
+
+
+def load():
+    conn = sqlite3.connect(f'file:{DB}?mode=ro', uri=True)
+    cur = conn.cursor()
+    cur.execute("""
+        SELECT s.assigned_accountant, a.firm, s.source_pdf,
+               s.feature_vector, s.dhash_vector
+        FROM signatures s JOIN accountants a ON s.assigned_accountant=a.name
+        WHERE s.assigned_accountant IS NOT NULL AND a.firm IS NOT NULL
+          AND s.feature_vector IS NOT NULL AND s.dhash_vector IS NOT NULL""")
+    rows = cur.fetchall()
+    conn.close()
+    return rows
+
+
+# ============ Part A: bootstrap on anchor rates ============
+def simulate(keep):
+    n = len(keep)
+    feats = np.stack([np.frombuffer(r[3], np.float32) for r in keep]).astype(np.float32)
+    feats /= np.clip(np.linalg.norm(feats, axis=1, keepdims=True), 1e-9, None)
+    dh = np.stack([np.frombuffer(r[4], np.uint8) for r in keep])
+    cpas = np.array([r[0] for r in keep])
+    docs = np.array([r[2] for r in keep])
+    cpa_idx = defaultdict(list)
+    for i, c in enumerate(cpas):
+        cpa_idx[c].append(i)
+    cpa_idx = {c: np.array(v) for c, v in cpa_idx.items()}
+    pool_size = {c: len(v)-1 for c, v in cpa_idx.items()}
+    rng = np.random.default_rng(SEED)
+    max_cos = np.zeros(n, np.float32); min_dh = np.full(n, 64, np.int32)
+    for si in range(n):
+        c = cpas[si]; npool = pool_size[c]
+        if npool <= 0:
+            continue
+        same = cpa_idx[c]
+        draw = rng.integers(0, n, size=npool + same.size + 20)
+        cand = draw[~np.isin(draw, same)][:npool]
+        cosv = feats[cand] @ feats[si]
+        dist = POP[dh[cand] ^ dh[si]].sum(axis=1)
+        max_cos[si] = cosv.max(); min_dh[si] = int(dist.min())
+    hc = (max_cos > 0.95) & (min_dh <= 5)
+    d2 = (max_cos > 0.95) & (min_dh <= 15)
+    return hc, d2, cpa_idx, docs
+
+
+def boot_part(keep, label):
+    hc, d2, cpa_idx, docs = simulate(keep)
+    n = len(hc)
+    rng = np.random.default_rng(SEED + 1)
+    cpa_list = list(cpa_idx.keys())
+    # CPA-block bootstrap on per-signature HC
+    bs = np.empty(N_BOOT)
+    for b in range(N_BOOT):
+        cs = rng.choice(len(cpa_list), len(cpa_list), replace=True)
+        idx = np.concatenate([cpa_idx[cpa_list[i]] for i in cs])
+        bs[b] = hc[idx].mean()
+    # document-level bootstrap on per-doc D2
+    doc_d2 = defaultdict(bool)
+    for i in range(n):
+        doc_d2[docs[i]] = doc_d2[docs[i]] or bool(d2[i])
+    dl = np.array(list(doc_d2.keys())); dvals = np.array([doc_d2[d] for d in dl])
+    nd = len(dl); bd = np.empty(N_BOOT)
+    for b in range(N_BOOT):
+        s = rng.integers(0, nd, nd)
+        bd[b] = dvals[s].mean()
+    print(f'\n  [{label}] per-sig HC point={hc.mean():.4f}  '
+          f'CPA-block boot95% [{np.percentile(bs,2.5):.4f}, {np.percentile(bs,97.5):.4f}]')
+    print(f'  [{label}] per-doc HC+MC point={dvals.mean():.4f}  '
+          f'doc boot95% [{np.percentile(bd,2.5):.4f}, {np.percentile(bd,97.5):.4f}]')
+
+
+# ============ Part B: pair-weighted KDE crossover ============
+def crossover(keep, label):
+    feats = np.stack([np.frombuffer(r[3], np.float32) for r in keep]).astype(np.float32)
+    feats /= np.clip(np.linalg.norm(feats, axis=1, keepdims=True), 1e-9, None)
+    cpas = np.array([r[0] for r in keep])
+    by = defaultdict(list)
+    for i, c in enumerate(cpas):
+        by[c].append(i)
+    by = {c: np.array(v) for c, v in by.items() if len(v) >= 3}
+    accts = list(by.keys())
+    pair_w = np.array([len(by[c])*(len(by[c])-1)/2 for c in accts], float)
+    pair_w /= pair_w.sum()
+    rng = np.random.default_rng(SEED)
+    M = 100_000
+    # intra: CPA sampled proportional to pair count (= uniform over all intra pairs)
+    intra = np.empty(M, np.float32)
+    ci = rng.choice(len(accts), M, p=pair_w)
+    for t in range(M):
+        a, b = rng.choice(by[accts[ci[t]]], 2, replace=False)
+        intra[t] = feats[a] @ feats[b]
+    inter = np.empty(M, np.float32)
+    for t in range(M):
+        i, j = rng.choice(len(accts), 2, replace=False)
+        inter[t] = feats[rng.choice(by[accts[i]])] @ feats[rng.choice(by[accts[j]])]
+    xs = np.linspace(0.3, 1.0, 10000)
+    diff = gaussian_kde(intra)(xs) - gaussian_kde(inter)(xs)
+    cr = [float(x) for x in xs[np.where(np.diff(np.sign(diff)))[0]] if 0.6 < x < 0.99]
+    print(f'  [{label}] crossover {[f"{x:.4f}" for x in cr]}  '
+          f'(intra {intra.mean():.4f}/{np.median(intra):.4f}  inter {inter.mean():.4f}/{np.median(inter):.4f})')
+
+
+rows = load()
+abcd = [r for r in rows if r[1] in BIG4]
+bcd = [r for r in rows if r[1] in BIG4 and r[1] != FIRM_A]
+
+print('=== Part A: bootstrap CIs on anchor rates ===')
+boot_part(abcd, 'ABCD (verify ~0.109 / ~0.338)')
+boot_part(bcd, 'BCD-only')
+
+print('\n=== Part B: KDE crossover (pair-weighted intra, corpus-wide reproduces 0.837) ===')
+crossover(rows, 'corpus-wide (all firms)')
+crossover(bcd, 'BCD-only')
+crossover([r for r in rows if r[1] != FIRM_A], 'BCD + non-Big4')

signature_analysis/52_canonical_bcd_locked.py

@@ -0,0 +1,169 @@
+#!/usr/bin/env python3
+"""Script 52: canonical-correct locked publication numbers (supersedes 48/50,
+fixes the per-firm assignment and A-out-of-sample any-pair issues codex flagged).
+
+Uses the EXACT canonical candidate sampler of Scripts 43/45 (retry-loop to
+collect exactly n_pool non-same-CPA candidates, rng.choice, default_rng(42)),
+any-pair max-cos/min-dHash five-way classification, and dominant-firm document
+assignment. Scopes: ABCD / BCD / BCD+nonBig4, plus Firm-A out-of-sample vs a
+clean BCD candidate pool. Read-only.
+"""
+import sqlite3
+from collections import defaultdict, Counter
+import numpy as np
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+FIRM_A = '勤業眾信聯合'
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+ALIAS = {'勤業眾信聯合': 'A', '安侯建業聯合': 'B', '資誠聯合': 'C', '安永聯合': 'D'}
+SEED = 42
+N_BOOT = 1000
+POP = np.array([bin(i).count('1') for i in range(256)], dtype=np.uint8)
+
+
+def wilson(k, n, z=1.96):
+    if n == 0:
+        return (None, None)
+    p = k/n; d = 1+z*z/n
+    c = (p+z*z/(2*n))/d
+    h = z*np.sqrt(p*(1-p)/n+z*z/(4*n*n))/d
+    return (max(0.0, c-h), min(1.0, c+h))
+
+
+def load():
+    conn = sqlite3.connect(f'file:{DB}?mode=ro', uri=True)
+    cur = conn.cursor()
+    cur.execute("""
+        SELECT s.assigned_accountant, a.firm, s.source_pdf,
+               s.feature_vector, s.dhash_vector
+        FROM signatures s JOIN accountants a ON s.assigned_accountant=a.name
+        WHERE s.assigned_accountant IS NOT NULL AND a.firm IS NOT NULL
+          AND s.feature_vector IS NOT NULL AND s.dhash_vector IS NOT NULL""")
+    rows = cur.fetchall()
+    conn.close()
+    return rows
+
+
+def canonical_sampler(rng, n, n_pool, same_cpa, all_idx):
+    """EXACT Scripts 43/45 sampler: retry-loop to exactly n_pool non-same."""
+    need = n_pool
+    cand = []
+    attempts = 0
+    while need > 0 and attempts < 10:
+        draw = rng.choice(n, size=need * 2, replace=True)
+        ok = draw[~np.isin(draw, same_cpa)]
+        cand.extend(ok[:need].tolist())
+        need -= len(ok[:need])
+        attempts += 1
+    if need > 0:
+        pool_mask = np.ones(n, dtype=bool)
+        pool_mask[same_cpa] = False
+        fb = rng.choice(all_idx[pool_mask], size=need, replace=False)
+        cand.extend(fb.tolist())
+    return np.array(cand[:n_pool], dtype=np.int64)
+
+
+def simulate(keep):
+    n = len(keep)
+    feats = np.stack([np.frombuffer(r[3], np.float32) for r in keep]).astype(np.float32)
+    norms = np.linalg.norm(feats, axis=1, keepdims=True); norms[norms == 0] = 1.0
+    feats = feats / norms
+    dh = np.stack([np.frombuffer(r[4], np.uint8) for r in keep])
+    cpas = np.array([r[0] for r in keep])
+    firms = np.array([ALIAS.get(r[1], 'NonB4') for r in keep])
+    docs = np.array([r[2] for r in keep])
+    cpa_idx = defaultdict(list)
+    for i, c in enumerate(cpas):
+        cpa_idx[c].append(i)
+    cpa_idx = {c: np.array(v) for c, v in cpa_idx.items()}
+    pool_size = {c: len(v)-1 for c, v in cpa_idx.items()}
+    all_idx = np.arange(n)
+    rng = np.random.default_rng(SEED)
+    max_cos = np.zeros(n, np.float32); min_dh = np.full(n, 64, np.int32)
+    for si in range(n):
+        np_ = pool_size[cpas[si]]
+        if np_ <= 0:
+            continue
+        cand = canonical_sampler(rng, n, np_, cpa_idx[cpas[si]], all_idx)
+        cosv = feats[cand] @ feats[si]
+        dist = POP[dh[cand] ^ dh[si]].sum(axis=1)
+        max_cos[si] = cosv.max(); min_dh[si] = int(dist.min())
+    return max_cos, min_dh, cpas, firms, docs, cpa_idx
+
+
+def report(keep, label):
+    max_cos, min_dh, cpas, firms, docs, cpa_idx = simulate(keep)
+    n = len(cpas)
+    hc = (max_cos > 0.95) & (min_dh <= 5)
+    d2 = (max_cos > 0.95) & (min_dh <= 15)
+    print(f'\n===== {label} (n_sig={n:,}) =====')
+    for nm, a in [('per-sig HC', hc), ('per-sig HC+MC', d2)]:
+        k = int(a.sum()); lo, hi = wilson(k, n)
+        print(f'  {nm}: {k/n:.6f} ({k}/{n}) [{lo:.4f},{hi:.4f}]')
+    # CPA-block bootstrap on per-sig HC
+    rng = np.random.default_rng(SEED + 1)
+    cl = list(cpa_idx.keys())
+    bs = np.empty(N_BOOT)
+    for b in range(N_BOOT):
+        cs = rng.choice(len(cl), len(cl), replace=True)
+        idx = np.concatenate([cpa_idx[cl[i]] for i in cs])
+        bs[b] = hc[idx].mean()
+    print(f'  per-sig HC CPA-block boot95% [{np.percentile(bs,2.5):.4f},{np.percentile(bs,97.5):.4f}]')
+    # per-doc, dominant-firm assignment (canonical)
+    doc_sigs = defaultdict(list)
+    for i in range(n):
+        doc_sigs[docs[i]].append(i)
+    dl = list(doc_sigs.keys()); nd = len(dl)
+    doc_d1 = np.array([hc[doc_sigs[d]].any() for d in dl])
+    doc_d2 = np.array([d2[doc_sigs[d]].any() for d in dl])
+    doc_firm = np.array([Counter(firms[doc_sigs[d]]).most_common(1)[0][0] for d in dl])
+    print(f'  per-doc HC:    {doc_d1.mean():.6f} ({int(doc_d1.sum())}/{nd})')
+    print(f'  per-doc HC+MC: {doc_d2.mean():.6f} ({int(doc_d2.sum())}/{nd})')
+    for f in sorted(set(doc_firm)):
+        m = doc_firm == f
+        print(f'      Firm {f} per-doc D2: {doc_d2[m].mean():.4f} ({int(doc_d2[m].sum())}/{int(m.sum())})')
+
+
+def a_out_of_sample(rows):
+    """Firm A source vs clean BCD candidate pool, any-pair, pool=count-1."""
+    A = [r for r in rows if r[1] == FIRM_A]
+    BCD = [r for r in rows if r[1] in BIG4 and r[1] != FIRM_A]
+    bf = np.stack([np.frombuffer(r[3], np.float32) for r in BCD]).astype(np.float32)
+    nb = bf.shape[0]
+    bn = np.linalg.norm(bf, axis=1, keepdims=True); bn[bn == 0] = 1.0; bf = bf/bn
+    bdh = np.stack([np.frombuffer(r[4], np.uint8) for r in BCD])
+    a_cpa = defaultdict(list)
+    for i, r in enumerate(A):
+        a_cpa[r[0]].append(i)
+    pool_size = {c: len(v)-1 for c, v in a_cpa.items()}
+    rng = np.random.default_rng(SEED)
+    hc = np.zeros(len(A), bool); d2 = np.zeros(len(A), bool)
+    docs = np.array([r[2] for r in A])
+    for i, r in enumerate(A):
+        np_ = pool_size[r[0]]
+        if np_ <= 0:        # singleton CPA: no same-CPA pool, skip (canonical)
+            continue
+        cand = rng.choice(nb, size=np_, replace=True)   # A not in BCD pool
+        sf = np.frombuffer(r[3], np.float32).astype(np.float32)
+        sf = sf/max(np.linalg.norm(sf), 1e-9)
+        cosv = bf[cand] @ sf
+        dist = POP[bdh[cand] ^ np.frombuffer(r[4], np.uint8)].sum(axis=1)
+        mc, md = cosv.max(), int(dist.min())
+        hc[i] = (mc > 0.95) and (md <= 5)
+        d2[i] = (mc > 0.95) and (md <= 15)
+    k = int(hc.sum()); n = len(A); lo, hi = wilson(k, n)
+    print(f'\n===== Firm A out-of-sample vs clean BCD pool (any-pair) =====')
+    print(f'  per-sig HC: {k/n:.6f} ({k}/{n}) [{lo:.5f},{hi:.5f}]')
+    ds = defaultdict(list)
+    for i in range(n):
+        ds[docs[i]].append(i)
+    dl = list(ds.keys())
+    dd2 = np.array([d2[ds[d]].any() for d in dl])
+    print(f'  per-doc HC+MC: {dd2.mean():.6f} ({int(dd2.sum())}/{len(dl)})')
+
+
+rows = load()
+report([r for r in rows if r[1] in BIG4], 'ABCD (verify: per-sig HC~0.1102 / per-doc D2~0.3375)')
+report([r for r in rows if r[1] in BIG4 and r[1] != FIRM_A], 'BCD-only (verify codex: HC~0.0116 / doc HC~0.0226 / doc D2~0.1905)')
+report([r for r in rows if r[1] != FIRM_A], 'BCD + non-Big4')
+a_out_of_sample(rows)

signature_analysis/53_bcd_regression_hitmatrix.py

@@ -0,0 +1,125 @@
+#!/usr/bin/env python3
+"""Script 53: BCD-only firm-effect logistic regression (Firm D reference) and
+BCD-only cross-firm hit matrix. Candidate pool = BCD (exclude Firm A and
+same-CPA). Canonical retry-loop sampler, any-pair + same-pair. Read-only.
+Replicates Script 44's logistic_fit and matrix logic, restricted to BCD.
+"""
+import sqlite3
+from collections import defaultdict
+import numpy as np
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+FIRM_A = '勤業眾信聯合'
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+ALIAS = {'勤業眾信聯合': 'A', '安侯建業聯合': 'B', '資誠聯合': 'C', '安永聯合': 'D'}
+SEED = 42
+POP = np.array([bin(i).count('1') for i in range(256)], dtype=np.uint8)
+
+
+def logistic_fit(X, y, max_iter=200, l2=0.001):
+    n, k = X.shape
+    beta = np.zeros(k)
+    for _ in range(max_iter):
+        eta = np.clip(X @ beta, -30, 30)
+        p = 1.0/(1.0+np.exp(-eta))
+        grad = X.T @ (y-p) - l2*beta
+        W = p*(1-p)
+        H = -(X.T*W) @ X - l2*np.eye(k)
+        try:
+            delta = np.linalg.solve(H, grad)
+        except np.linalg.LinAlgError:
+            delta = 0.3*grad
+        nb = beta - delta
+        if np.max(np.abs(nb-beta)) < 1e-8:
+            beta = nb; break
+        beta = nb
+    eta = np.clip(X @ beta, -30, 30)
+    p = 1.0/(1.0+np.exp(-eta)); W = p*(1-p)
+    cov = np.linalg.inv((X.T*W) @ X + l2*np.eye(k))
+    return beta, np.sqrt(np.diag(cov))
+
+
+conn = sqlite3.connect(f'file:{DB}?mode=ro', uri=True)
+cur = conn.cursor()
+cur.execute("""SELECT s.assigned_accountant, a.firm, s.feature_vector, s.dhash_vector
+    FROM signatures s JOIN accountants a ON s.assigned_accountant=a.name
+    WHERE s.assigned_accountant IS NOT NULL AND a.firm IN (?,?,?)
+      AND s.feature_vector IS NOT NULL AND s.dhash_vector IS NOT NULL""",
+    ('安侯建業聯合', '資誠聯合', '安永聯合'))
+rows = cur.fetchall()
+conn.close()
+
+n = len(rows)
+feats = np.stack([np.frombuffer(r[2], np.float32) for r in rows]).astype(np.float32)
+norms = np.linalg.norm(feats, axis=1, keepdims=True); norms[norms == 0] = 1.0
+feats = feats / norms
+dh = np.stack([np.frombuffer(r[3], np.uint8) for r in rows])
+cpas = np.array([r[0] for r in rows])
+firms = np.array([ALIAS[r[1]] for r in rows])
+cpa_idx = defaultdict(list)
+for i, c in enumerate(cpas):
+    cpa_idx[c].append(i)
+cpa_idx = {c: np.array(v) for c, v in cpa_idx.items()}
+pool_size = {c: len(v)-1 for c, v in cpa_idx.items()}
+all_idx = np.arange(n)
+print(f'BCD signatures: {n:,}; CPAs: {len(cpa_idx)}')
+
+rng = np.random.default_rng(SEED)
+hit_any = np.zeros(n, bool)
+hit_same = np.zeros(n, bool)
+cand_firm_maxcos = np.empty(n, dtype=object)   # any-pair partner firm
+cand_firm_same = np.empty(n, dtype=object)
+psize = np.zeros(n, np.int32)
+for si in range(n):
+    np_ = pool_size[cpas[si]]; psize[si] = np_
+    if np_ <= 0:
+        continue
+    same = cpa_idx[cpas[si]]
+    need = np_; cand = []; att = 0
+    while need > 0 and att < 10:
+        draw = rng.choice(n, size=need*2, replace=True)
+        ok = draw[~np.isin(draw, same)]
+        cand.extend(ok[:need].tolist()); need -= len(ok[:need]); att += 1
+    if need > 0:
+        pm = np.ones(n, bool); pm[same] = False
+        cand.extend(rng.choice(all_idx[pm], size=need, replace=False).tolist())
+    cand = np.array(cand[:np_], dtype=np.int64)
+    cosv = feats[cand] @ feats[si]
+    dist = POP[dh[cand] ^ dh[si]].sum(axis=1)
+    mc = int(np.argmax(cosv)); md = int(np.argmin(dist))
+    if cosv[mc] > 0.95 and dist[md] <= 5:
+        hit_any[si] = True
+        cand_firm_maxcos[si] = firms[cand[mc]]
+    spm = (cosv > 0.95) & (dist <= 5)
+    if spm.any():
+        hit_same[si] = True
+        cand_firm_same[si] = firms[cand[int(np.argmax(spm))]]
+
+# ---- Logistic regression: hit_any ~ FirmB + FirmC + log(pool), Firm D reference ----
+hp = psize > 0
+y = hit_any[hp].astype(np.float64)
+fa = firms[hp]
+lp = np.log(psize[hp].astype(np.float64)); lp = lp - lp.mean()
+X = np.column_stack([np.ones(y.shape), (fa == 'B').astype(float), (fa == 'C').astype(float), lp])
+beta, se = logistic_fit(X, y)
+print(f'\n[BCD logistic: hit_any ~ FirmB + FirmC + log(pool); Firm D = reference]  n={len(y):,}, y_mean={y.mean():.4f}')
+for nm, b, s in zip(['intercept(FirmD)', 'FirmB', 'FirmC', 'log(pool,centred)'], beta, se):
+    print(f'    {nm}: beta={b:+.4f} SE={s:.4f} OR={np.exp(b):.4f} z~{abs(b)/s if s>0 else float("inf"):.2f}')
+
+# ---- Cross-firm hit matrix (any-pair max-cos partner) ----
+print('\n[BCD cross-firm hit matrix: any-pair, source firm x max-cos partner firm]')
+print('  src   ->   B      C      D     | within-firm%  (n hits)')
+for sf in ['B', 'C', 'D']:
+    m = (firms == sf) & hit_any
+    parts = cand_firm_maxcos[m]
+    tot = len(parts)
+    cnt = {f: int((parts == f).sum()) for f in ['B', 'C', 'D']}
+    wf = cnt[sf]/tot if tot else 0
+    print(f'  {sf}       {cnt["B"]:5d}  {cnt["C"]:5d}  {cnt["D"]:5d}  | {wf:6.1%}  ({tot})')
+
+print('\n[BCD same-pair within-firm concentration]')
+for sf in ['B', 'C', 'D']:
+    m = (firms == sf) & hit_same
+    parts = cand_firm_same[m]; tot = len(parts)
+    wf = int((parts == sf).sum())/tot if tot else 0
+    print(f'  Firm {sf}: {wf:.1%} within-firm ({int((parts==sf).sum())}/{tot})')

signature_analysis/54_bcd_floor_temporal.py

@@ -0,0 +1,99 @@
+#!/usr/bin/env python3
+"""Script 54: temporal stability of the BCD inter-CPA floor.
+Does the normative BCD per-comparison HC coincidence floor drift over time /
+get contaminated by post-2020 e-signing? Compares eras full / 2013-2019 /
+2020-2023 using the pool-size-independent per-comparison joint HC ICCR
+(cos>0.95 & dHash<=5) on BCD inter-CPA pairs (N=500k, seed 42), plus the
+observed deployed per-signature HC rate by firm by era. Read-only.
+"""
+import sqlite3
+from collections import defaultdict
+import numpy as np
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+FIRM_A = '勤業眾信聯合'
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+ALIAS = {'勤業眾信聯合': 'A', '安侯建業聯合': 'B', '資誠聯合': 'C', '安永聯合': 'D'}
+SEED = 42
+N_PAIRS = 500_000
+POP = np.array([bin(i).count('1') for i in range(256)], dtype=np.uint8)
+
+
+def wilson(k, n, z=1.96):
+    if n == 0:
+        return (None, None)
+    p = k/n; d = 1+z*z/n
+    c = (p+z*z/(2*n))/d
+    h = z*np.sqrt(p*(1-p)/n+z*z/(4*n*n))/d
+    return (max(0.0, c-h), min(1.0, c+h))
+
+
+conn = sqlite3.connect(f'file:{DB}?mode=ro', uri=True)
+cur = conn.cursor()
+cur.execute("""
+    SELECT s.assigned_accountant, a.firm, CAST(substr(s.year_month,1,4) AS INT),
+           s.feature_vector, s.dhash_vector,
+           s.max_similarity_to_same_accountant, s.min_dhash_independent
+    FROM signatures s JOIN accountants a ON s.assigned_accountant=a.name
+    WHERE a.firm IN (?,?,?,?) AND s.year_month IS NOT NULL
+      AND s.feature_vector IS NOT NULL AND s.dhash_vector IS NOT NULL""", BIG4)
+rows = cur.fetchall()
+conn.close()
+
+ERAS = {'full 2013-2023': lambda y: True,
+        '2013-2019 (pre-drift)': lambda y: 2013 <= y <= 2019,
+        '2020-2023': lambda y: 2020 <= y <= 2023}
+
+
+def per_comparison_floor(era_fn, label):
+    # BCD-only (exclude Firm A), era-restricted
+    keep = [r for r in rows if r[1] != FIRM_A and era_fn(r[2])]
+    feats = np.stack([np.frombuffer(r[3], np.float32) for r in keep]).astype(np.float32)
+    feats /= np.clip(np.linalg.norm(feats, axis=1, keepdims=True), 1e-9, None)
+    dh = np.stack([np.frombuffer(r[4], np.uint8) for r in keep])
+    cpas = np.array([r[0] for r in keep])
+    by = defaultdict(list)
+    for i, c in enumerate(cpas):
+        by[c].append(i)
+    accts = list(by.keys())
+    rng = np.random.default_rng(SEED)
+    cos = np.empty(N_PAIRS, np.float32); dv = np.empty(N_PAIRS, np.int32)
+    na = len(accts)
+    for t in range(N_PAIRS):
+        i, j = rng.choice(na, 2, replace=False)
+        a1, a2 = accts[i], accts[j]
+        k1 = by[a1][int(rng.integers(0, len(by[a1])))]
+        k2 = by[a2][int(rng.integers(0, len(by[a2])))]
+        cos[t] = feats[k1] @ feats[k2]
+        dv[t] = POP[dh[k1] ^ dh[k2]].sum()
+    joint = int(((cos > 0.95) & (dv <= 5)).sum())
+    lo, hi = wilson(joint, N_PAIRS)
+    print(f'  [{label}] BCD per-comparison HC floor = {joint/N_PAIRS:.6f} '
+          f'({joint}/{N_PAIRS}) Wilson95% [{lo:.6f},{hi:.6f}]  '
+          f'(n_sig={len(keep):,}, CPAs={na})')
+    return joint/N_PAIRS
+
+
+print('=== (1) BCD per-comparison HC floor by era (pool-size-independent) ===')
+floors = {lab: per_comparison_floor(fn, lab) for lab, fn in ERAS.items()}
+
+print('\n=== (2) Observed deployed per-signature HC rate by firm by era ===')
+print('   (max_sim>0.95 & min_dh<=5 on actual same-CPA pools)')
+for lab, fn in ERAS.items():
+    print(f'  --- {lab} ---')
+    for fm_zh in BIG4:
+        sub = [r for r in rows if r[1] == fm_zh and fn(r[2])
+               and r[5] is not None and r[6] is not None]
+        if not sub:
+            continue
+        k = sum(1 for r in sub if r[5] > 0.95 and r[6] <= 5)
+        print(f'      Firm {ALIAS[fm_zh]}: {k/len(sub):.4f} ({k}/{len(sub)})')
+
+print('\n=== A-vs-floor multiple by era (observed A HC / BCD floor) ===')
+for lab, fn in ERAS.items():
+    a = [r for r in rows if r[1] == FIRM_A and fn(r[2]) and r[5] is not None and r[6] is not None]
+    a_rate = sum(1 for r in a if r[5] > 0.95 and r[6] <= 5)/len(a) if a else 0
+    fl = floors[lab]
+    # per-comparison floor is not directly comparable to observed pooled rate;
+    # report ratio vs the per-signature floor proxy from Script 52 (0.0116 full).
+    print(f'  {lab}: observed A HC = {a_rate:.3f}; per-comparison floor = {fl:.6f}')

signature_analysis/55_bcd_pre2020_calibration.py

@@ -0,0 +1,144 @@
+#!/usr/bin/env python3
+"""Script 55: PRIMARY calibration on the clean pre-e-signature baseline
+BCD 2013-2019 (Firms B/C/D, fiscal years 2013-2019). Rationale: co-author
+interviews confirm B/C/D progressively adopted e-signature systems after 2020
+(staggered timing), so 2013-2019 BCD is the construct-clean hand-signing
+baseline. Canonical retry-loop sampler (matches Scripts 43/45/52), any-pair.
+Reports the floor + Firm A (all years) scored out-of-sample against it, and
+BCD 2020+ scored against the same threshold. Read-only.
+"""
+import sqlite3
+from collections import defaultdict, Counter
+import numpy as np
+
+DB = '/Volumes/NV2/PDF-Processing/signature-analysis/signature_analysis.db'
+FIRM_A = '勤業眾信聯合'
+BIG4 = ('勤業眾信聯合', '安侯建業聯合', '資誠聯合', '安永聯合')
+ALIAS = {'勤業眾信聯合': 'A', '安侯建業聯合': 'B', '資誠聯合': 'C', '安永聯合': 'D'}
+SEED = 42
+N_BOOT = 1000
+POP = np.array([bin(i).count('1') for i in range(256)], dtype=np.uint8)
+
+
+def wilson(k, n, z=1.96):
+    if n == 0:
+        return (None, None)
+    p = k/n; d = 1+z*z/n; c = (p+z*z/(2*n))/d
+    h = z*np.sqrt(p*(1-p)/n+z*z/(4*n*n))/d
+    return (max(0.0, c-h), min(1.0, c+h))
+
+
+def canon_sampler(rng, n, npool, same, all_idx):
+    need = npool; cand = []; att = 0
+    while need > 0 and att < 10:
+        draw = rng.choice(n, size=need*2, replace=True)
+        ok = draw[~np.isin(draw, same)]
+        cand.extend(ok[:need].tolist()); need -= len(ok[:need]); att += 1
+    if need > 0:
+        pm = np.ones(n, bool); pm[same] = False
+        cand.extend(rng.choice(all_idx[pm], size=need, replace=False).tolist())
+    return np.array(cand[:npool], dtype=np.int64)
+
+
+conn = sqlite3.connect(f'file:{DB}?mode=ro', uri=True)
+cur = conn.cursor()
+cur.execute("""SELECT s.assigned_accountant,a.firm,CAST(substr(s.year_month,1,4) AS INT),
+    s.source_pdf,s.feature_vector,s.dhash_vector,
+    s.max_similarity_to_same_accountant,s.min_dhash_independent
+    FROM signatures s JOIN accountants a ON s.assigned_accountant=a.name
+    WHERE a.firm IN (?,?,?,?) AND s.year_month IS NOT NULL
+      AND s.feature_vector IS NOT NULL AND s.dhash_vector IS NOT NULL""", BIG4)
+rows = cur.fetchall()
+conn.close()
+
+
+def prep(rec):
+    feats = np.stack([np.frombuffer(r[4], np.float32) for r in rec]).astype(np.float32)
+    norms = np.linalg.norm(feats, axis=1, keepdims=True); norms[norms == 0] = 1.0
+    feats /= norms
+    dh = np.stack([np.frombuffer(r[5], np.uint8) for r in rec])
+    return feats, dh
+
+
+def floor_on(baseline_rec, label):
+    """Canonical per-sig/per-doc HC floor on a baseline population."""
+    feats, dh = prep(baseline_rec)
+    n = len(baseline_rec)
+    cpas = np.array([r[0] for r in baseline_rec])
+    firms = np.array([ALIAS[r[1]] for r in baseline_rec])
+    docs = np.array([r[3] for r in baseline_rec])
+    cidx = defaultdict(list)
+    for i, c in enumerate(cpas):
+        cidx[c].append(i)
+    cidx = {c: np.array(v) for c, v in cidx.items()}
+    psize = {c: len(v)-1 for c, v in cidx.items()}
+    all_idx = np.arange(n)
+    rng = np.random.default_rng(SEED)
+    mx = np.zeros(n, np.float32); mn = np.full(n, 64, np.int32)
+    for si in range(n):
+        np_ = psize[cpas[si]]
+        if np_ <= 0:
+            continue
+        cand = canon_sampler(rng, n, np_, cidx[cpas[si]], all_idx)
+        cosv = feats[cand] @ feats[si]
+        mx[si] = cosv.max(); mn[si] = int(POP[dh[cand] ^ dh[si]].sum(axis=1).min())
+    hc = (mx > 0.95) & (mn <= 5); d2 = (mx > 0.95) & (mn <= 15)
+    k = int(hc.sum())
+    rng2 = np.random.default_rng(SEED+1); cl = list(cidx.keys())
+    bs = np.array([hc[np.concatenate([cidx[cl[i]] for i in rng2.choice(len(cl), len(cl), True)])].mean()
+                   for _ in range(N_BOOT)])
+    print(f'\n  [{label}] n_sig={n:,}, CPAs={len(cidx)}')
+    print(f'    per-sig HC floor = {k/n:.4f} ({k}/{n})  CPA-boot95% [{np.percentile(bs,2.5):.4f},{np.percentile(bs,97.5):.4f}]')
+    dd1 = defaultdict(bool); dd2 = defaultdict(bool); dfirm = {}
+    for i in range(n):
+        if hc[i]: dd1[docs[i]] = True
+        if d2[i]: dd2[docs[i]] = True
+        dfirm.setdefault(docs[i], []).append(firms[i])
+        dd1.setdefault(docs[i], False); dd2.setdefault(docs[i], False)
+    dl = list(dd1.keys()); nd = len(dl)
+    print(f'    per-doc HC = {sum(dd1[d] for d in dl)/nd:.4f}; per-doc HC+MC = {sum(dd2[d] for d in dl)/nd:.4f}  (n_doc={nd:,})')
+    dom = {d: Counter(dfirm[d]).most_common(1)[0][0] for d in dl}
+    for f in ['B', 'C', 'D']:
+        ds = [d for d in dl if dom[d] == f]
+        if ds:
+            print(f'      Firm {f} per-doc HC+MC: {sum(dd2[d] for d in ds)/len(ds):.4f} ({sum(dd2[d] for d in ds)}/{len(ds)})')
+    return k/n
+
+
+def a_vs_baseline(baseline_rec, a_rec, label):
+    bf, bdh = prep(baseline_rec); nb = len(baseline_rec)
+    a_cpa = defaultdict(list)
+    for i, r in enumerate(a_rec):
+        a_cpa[r[0]].append(i)
+    psize = {c: len(v)-1 for c, v in a_cpa.items()}
+    rng = np.random.default_rng(SEED)
+    hc = np.zeros(len(a_rec), bool)
+    for i, r in enumerate(a_rec):
+        np_ = psize[r[0]]
+        if np_ <= 0:
+            continue
+        cand = rng.integers(0, nb, size=np_)
+        sf = np.frombuffer(r[4], np.float32).astype(np.float32); sf /= max(np.linalg.norm(sf), 1e-9)
+        cosv = bf[cand] @ sf
+        if (cosv > 0.95).any():
+            dist = POP[bdh[cand] ^ np.frombuffer(r[5], np.uint8)].sum(axis=1)
+            hc[i] = bool(((cosv > 0.95) & (dist <= 5)).any())
+    k = int(hc.sum()); n = len(a_rec); lo, hi = wilson(k, n)
+    print(f'  [{label}] Firm A (all yrs) vs BCD-2013-2019 pool: per-sig HC = {k/n:.4f} ({k}/{n}) [{lo:.5f},{hi:.5f}]')
+
+
+bcd_pre = [r for r in rows if r[1] != FIRM_A and 2013 <= r[2] <= 2019]
+bcd_post = [r for r in rows if r[1] != FIRM_A and r[2] >= 2020]
+A_all = [r for r in rows if r[1] == FIRM_A]
+
+print('=== PRIMARY floor: BCD 2013-2019 ===')
+fl = floor_on(bcd_pre, 'BCD 2013-2019 (PRIMARY)')
+
+print('\n=== Firm A scored against the BCD-2013-2019 threshold ===')
+a_vs_baseline(bcd_pre, A_all, 'A out-of-sample')
+A_obs = [r for r in A_all if r[6] is not None and r[7] is not None]
+ak = sum(1 for r in A_obs if r[6] > 0.95 and r[7] <= 5)
+print(f'  Firm A observed (all yrs, own pools): per-sig HC = {ak/len(A_obs):.4f}  -> {ak/len(A_obs)/fl:.0f}x the BCD-2013-2019 floor')
+
+print('\n=== (optional) BCD 2020+ floor, same method (may be inflated by e-signing) ===')
+floor_on(bcd_post, 'BCD 2020-2023 (post e-signing)')