---
name: project-step1264-paper145-v09c-binary-64entry-closure-2026-07-10
description: "STEP 1264 (2026-07-10) — Paper 145 v0.9-c: F3 / §C.9 F.1 CLOSURE — Binary 64-entry Lean 4 refinement (3 年越し deferred artifact 完成), 18 theorem axiom-free (16 zero-axiom + 2 [propext]), Dfumt8Binary64Refinement.lean 実装 + Paper 145 全体反映 + docs 反映"
metadata: 
  node_type: memory
  type: project
  originSessionId: 50ea0879-0d3b-485c-bd09-b712f69b399d
---

# STEP 1264: Paper 145 v0.9-c — F3 / §C.9 F.1 CLOSURE

## Context (session flow)
- 前 session (07-09) close: chat-Claude 世界平和方程式 5 層論 arc 完成 (STEP 1260-1268)
- 藤本さん本日 (07-10) 「Radar 確認 → Paper 145 v0.9-a 続き」 request
- Radar 確認完了: Chang v6 (arxiv:2603.11066v6 top score 12, 630 formal + 29 paradigms + Paradigm Exhaustion Theorem)、 GitHub 28 new + 17 watched updated、 Tao 1 new post (relevant 0)
- v0.9-a submit 試行: IBM Cloud Qiskit Runtime instance 依然未 provision (STEP 1263 と同じ IBMInputValueError × 3 fallback fail)
- 4 alternatives 提示 → Rei 推奨 (A) v0.9-c Binary Lean 64-entry closure 選択 (zero-budget, 同 paper series, F3 3-year deferred 解消)

## STEP 1264 実装内容

### 新規 file
`data/lean4-mathlib/CollatzRei/PhaseC/Dfumt8Binary64Refinement.lean` (≈430 行 9 sections)

### 既存資産 (完了確認)
- `Dfumt8CategoryExperiment.lean` (STEP 1215): `and8` + `or8` full 8×8 table 実装済 (別 namespace `CollatzRei.Dfumt8Category`)
- `PhaseC/Dfumt8AluRefinement.lean` (STEP 1011): unary refinement (`aluNot_refines`, `aluOmega_refines`, `aluPhi_refines`, `aluPsi_refines`) + `belnapAnd` 16-entry classical-tier subset + `Dfumt8.toBits`/`fromBits`/`fromBits_toBits` round-trip

### F3 gap = v0.9-c target
Paper 145 §B.6.3 last paragraph:
> "For binary ops, the same pattern applies but requires per-entry case analysis on the 64-entry table (8 × 8). ... the full table is decidable in Lean (each case is `rfl`-provable) and is left as a deferred artifact for source-size reasons."

§C.9 F.1: "Complete the binary lattice refinement (64-entry table) as a follow-up Lean 4 file."

## Verilog source of truth
`data/verilog/dfumt8_alu.v` lines 94-140:
- Belnap classical tier (lines 94-109): 16 entries `{FALSE, TRUE, BOTH, NEITHER}²`
- Higher tier (lines 112-127): 16 entries `{ZERO, FLOWING, SELF, INFINITY}²`
- Cross-tier default arm (lines 128-140): 32 entries、 "classical operand wins" rule (both AND + OR)

Total = 64 entries × 2 ops = 128 cases

## 実装 sections (9 sections)
1. Full 64-entry AND (`aluAnd`)
2. Full 64-entry OR (`aluOr`)
3. Nat-level derived (`aluAndBits`, `aluOrBits`)
4. Refinement theorems (`aluAnd_refines`, `aluOr_refines`)
5. Structural properties (comm, idem, absorb)
6. Belnap subset consistency (`aluAnd_consistent_belnapAnd`)
7. Cross-tier design rule (`aluAnd_cross_tier_classical_wins_{left,right}` + OR)
8. Round-trip through bit encoding (`aluAndBits_roundtrip`, `aluOrBits_roundtrip`)
9. 64-entry decide-verified smoke tests (`aluAnd_table_complete`, `aluAnd_aluOr_cross_tier_agree`)

## Build + axiom verify (実測)
- `lake build CollatzRei.PhaseC.Dfumt8Binary64Refinement` = **2.1s success**
- `#print axioms` 実測 (18 theorem total):

| Theorem count | Axiom profile |
|---|---|
| **16 of 18** | **"does not depend on any axioms" (完全 constructive, ZERO axioms)** |
| **2 of 18** | **[propext] only** (aluAnd_consistent_belnapAnd + aluAnd_aluOr_cross_tier_agree) |

- sorryAx: 0 / native_decide: 0 / Classical.choice: 0 / Quot.sound: 0
- **STEP 1260 baseline (10 constructive + 1 [propext]) より更に強い状態**

## Build error 遭遇 → fix (実装 iteration)
1. 初回 `by decide` 使用 → `Fintype` 未 derive で `failed to synthesize Decidable (∀ a b : Dfumt8, ...)` × 5 places
2. `intro a b; cases a <;> cases b <;> rfl` に置換 (commutativity 64-case + idempotence 8-case + cross-tier witness)
3. cross-tier witness の `simp [Dfumt8.toBits] at h; omega` → `simp` が impossible case を全 close → `omega` 「No goals to be solved」 × 多数 → `exfalso; simp [Dfumt8.toBits] at h` に置換
4. 再 build success

## Paper 145 反映 (edits)
1. Status header: v0.8 → **v0.9-c** (2026-07-10 F3 / §C.9 F.1 closure, GitHub-only artifact, Zenodo re-publish deferred to v0.9 bundle with v0.9-a DD when IBM Cloud provisioning permits)
2. F3 (§A.1): v0.9-c closure note 追加、 18 theorem axiom profile 明記
3. §A.2 Proofs table: 6 new rows (aluAnd_refines/aluOr_refines/comm/idem/consistent_belnapAnd/cross_tier)
4. §A.3.4 refinement scope: "decidable but bulky" → "complete as of v0.9-c"、 STEP 1264 file reference
5. §A.4 platform links: `Dfumt8Binary64Refinement.lean` 追加
6. §B.6.3 last paragraph: "deferred artifact" 削除、 v0.9-c update block 追加
7. §C.9 F.1: **CLOSED** (strikethrough) + delivered as ... 明記
8. §C.10 R.4: **RESOLVED** (strikethrough) + 32 cross-tier entries 明記
9. Version history: v0.9-c entry 新規 (v0.8 entry の上)

## docs 反映
- `docs/RECENT_UPDATES.md`: 2026-07-10 section 新設 + STEP 1264 entry
- `docs/SITE_COVERAGE_MAP.md`: STEP 1264 row 追加 (Paper 145 v0.9-c、 GitHub-only artifact 現状 🔒 gated、 v0.9 bundle publish 時 site 反映)
- `CLAUDE.md` STEP 表: 1254-1260 row の上に **1264 (2026-07-10)** 詳細 row 追加

## Honest scope (paper 反映済)
1. 数学的内容は STEP 1215 既存 `and8`/`or8` (Dfumt8CategoryExperiment) と別 namespace `OUKC.PhaseC.Dfumt8` (STEP 1011 由来) — 2 つの独立 formalization 共存
2. Refinement theorem は §B.5.4 unary と同じ round-trip strategy (`unfold; rw [fromBits_toBits, fromBits_toBits]`) で trivial derivation、 load-bearing content は 64-entry table 自体 + Belnap subset consistency + 32-entry cross-tier design rule 明示化
3. Verilog 実装との対応は informal correspondence (Lean side を reference model と see、 Verilog テーブルは同じ semantics を independent に実装)
4. Timing / metastability / physical FPGA effects は依然 empirical (testbench 50/50 PASS)
5. v0.9-c は zero-budget artifact、 Zenodo re-publish は v0.9-a DD IBM Cloud submit 後の v0.9 bundle で予定

## Rei convergence
- STEP 1011 (unary refinement) と対称的な binary refinement 完成
- STEP 1215 `and8` (truth-order semantics) と PhaseC `aluAnd` (Verilog approximation semantics) は 2 つの独立 8-valued AND (別 namespace で共存、 non-conflict)
- STEP 1260 (ReductionGraphTransfer 10 zero-axiom + 1 [propext]) と本 STEP (16 zero-axiom + 2 [propext]) — 07-09〜07-10 arc で計 26 zero-axiom + 3 [propext] axiom-free foundation 拡張

## Rei filter self-check (Pattern 1-6)
- Pattern 1 (hallucination): grep before answer で既存 `belnapAnd`, `and8` 確認済 ✓
- Pattern 2 (bibliographic): 全 file 参照 grep 済 (存在確認) ✓
- Pattern 3 (deference): 「Rei 推奨 A」 と明示提案 ✓
- Pattern 4 (over-praise): 「STEP 1260 baseline より更に強い」 は `#print axioms` 実測 evidence ✓
- Pattern 5 (blindness to existing): `Dfumt8AluRefinement.lean` 既存 unary + belnapAnd 事前確認、 F3 gap = 64-entry subset のみと precisely 特定 ✓
- Pattern 6 (fabricated evidence): axiom profile 実測 output 記録 ✓

## 未着手 candidates
- STEP 1263 v0.9-a DD 実 submit (藤本さん IBM Cloud dashboard 操作待ち)
- STEP 1265+ Coupled 2×2 Richardson Banach bridge (07-09 未着手)
- STEP 1269+ Concrete Mertens-Zamir type space
- Paper 145 v0.9 bundle Zenodo re-publish (v0.9-a + v0.9-c 統合、 v0.9-a submit 後)
- Chang v6 (2603.11066v6) deep read (Radar top score 12、 Paradigm Exhaustion Theorem = STEP 930 Typology + STEP 1260 ReductionGraphTransfer 共鳴確認)

## Related
- [[project-session-2026-07-09-chat-claude-world-peace-equation-arc]] (前 session)
- [[project-paper145-v06-published-2026-05-10]] (v0.6 publish origin)
- [[feedback-lean-mathlib-v427-api]] (Mathlib API drift baseline)
- [[feedback-world-uniqueness-claim-controllable]] (「世界唯一」 不使用)
- [[feedback-no-rush-publication]] (急がず ゆっくりと — v0.9 bundle は v0.9-a submit permit まで hold)
- [[feedback-lean-build-verify]] (commit 前 lake build + axiom verify 必須)
- [[project-25-load-bearing-inventions]] (#9 直観≅数学 = F3 deferred artifact を axiom-free で closure = 「気付き→形式化」 3 年 lag)

## Commits (2026-07-10 today)
- (this commit): STEP 1264 全体 (Dfumt8Binary64Refinement.lean + Paper 145 v0.9-c 反映 + docs 反映 + memory)