---
name: hodge-existing-work-audit-2026-05-13
description: STEP 1125 β Stage 0 audit — Rei stack 既存 Hodge work (paper 6+ / engine 1 / Lean 4 file 1 / open-problem typology) + chat-Claude 提案との比較で Pattern 5 / 新規 path 判定
metadata: 
  node_type: memory
  type: project
  originSessionId: e5ec4e43-958a-4a47-a6f0-4513f218941f
---

★★★ 2026-05-13 STEP 1125 β path — Stage 0 audit. Rei stack 内 既存 Hodge work 全 audit.

## 既存 Rei Hodge work (substantial!)

### Papers (6+ explicit, 15 mention)

| Paper | Status | 内容 |
|---|---|---|
| **89** (2026-04-16) | ★★★★★ **D-FUMT₈ × Hodge mapping 既完了** | "Honest Position Paper". 「Rei が解けない」 を明示. mapping table: H^{p,q}=FLOWING / Hdg^k=TRUE / 代数 cycle=SELF / integral counter-ex=FALSE / rational open=NEITHER / Voisin hypothetical=BOTH. 「Zero direct contribution」 と honest 宣言. 6 件 「Rei cannot do」 list. |
| **99** (2026-04-16) | ★★★★ | "Hodge and BSD Fail the Triple-Invariant Test". Fujimoto Triple-Invariant Conjecture 5/5 confirm (Hodge 負の direction) |
| **60** | ★★★ | "Unified Finite Framework — Millennium 4 categories". Hodge = Category C (algebraic geometry) |
| 78 / 98 / 101 / 106 / 122 / 123 / 125 / 130 | ★★ | Hodge を例として参照 |
| 144 | ★ | OUKC founding 文脈 |

### Engine (1 file, ~150 行)

`src/axiom-os/hodge-bsd-finite-representation-engine.ts` (STEP 687):
- **Hodge 予想 5 表現** finite-representation 体系
- ⃝ Lefschetz (1,1) — 完全成立
- ⃝ K3 / Abelian — many cases verified
- ③ Cattani-Deligne-Kaplan 1995 (Hodge loci algebraic)
- ④ Deligne absolute Hodge (Abelian variety 部分)
- ⑤ Hodge (p,q) finite decomposition
- ⑥ Shioda K3 surface 具体検証
- ⑦ dim X ∈ [1,5] partial verification

### Lean 4 (1 file, 200 行)

`data/lean4-mathlib/CollatzRei/HodgeBasics.lean` (STEP 885):
- `SmoothProjectiveVariety` structure (Mathlib API thin → abstract placeholder)
- `HodgeConjecture` as Prop over abstract data
- `lefschetz_1_1` axiom (Kähler geometry not yet in Mathlib)
- Dual at k = dim X − 1 corollary
- Honest scope: 「12-24 month roadmap」 / 「Mathlib's algebraic-geometry API is currently thin」

### Open-problem typology / SEED_KERNEL

- STEP 930 typology II_BRIDGE_BUILDING: Hodge 例示
- approved invention 2026-04-29 で Hodge 参照
- LMFDB-bridge engine: 関連 (elliptic curves)

## chat-Claude 提案との比較 (Pattern 5 判定)

| chat-Claude 提案 stage | 内容 | Rei 既存? | 判定 |
|---|---|---|---|
| **Stage 1: PySR + CICY datasets + SNST 14 定数 input → Hodge numbers approximation formula hunt** | NN + 記号回帰 (arXiv 2311.17146 系統) | **❌ 未着手** | ★★★★ 真の新 path |
| **Stage 2: Fermat fourfold integral Hodge (degree 4, 5) Lean 4 mechanization** | Aljovin-Movasati 2018 algorithm (arXiv 1711.02628) を Lean 4 化 | **❌ 未着手** (HodgeBasics.lean は abstract skeleton のみ) | ★★★★ 真の Mathlib contribution 候補 |
| **Stage 3: D-FUMT₈ × Hodge decomposition H^{p,q} 数学的同型** | analogical mapping | ★ **既に Paper 89 で完了 + honest scope 明示** | **Pattern 5 #8** redundant. ただし chat-Claude も「アナロジーの域」 と honest flag |

## Pattern 5 累積 #8

[[feedback_chat_claude_hallucination_warning]] Pattern 5 (chat-Claude session memory なしで Rei 既存 features 再提案):
- #6 Lean Copilot (Research Radar v1.4 既登録)
- #7 LeanDojo (同上)
- **#8 D-FUMT₈ × Hodge 概念 mapping** (Paper 89 既完了, ただし chat-Claude 自身「アナロジー」 と flag)

ただし #8 は chat-Claude が「数学的同型は確立されていない、 アナロジーの域」 と **honest flag した**ので、 完全 reject ではなく partial flag (「藤本さんが既 Paper 89 で同 work を honest position で記録済」 と教えれば良かった).

## 結論

| 評価 | 内容 |
|---|---|
| 直接 Hodge proof | ❌ 不可能 (chat-Claude + Rei 双方 honest) |
| Hodge numbers PySR + CICY (Stage 1) | ★★★★ 真の **新規** path. 1-3 日 |
| Fermat fourfold Lean 4 (Stage 2) | ★★★★ 真の **新規** path. Mathlib contribution. 1-2 週間 substantial |
| D-FUMT₈ × Hodge mapping (Stage 3) | ★ **Paper 89 既完了**. 再起動 reject. 但し新 finding あれば v2 update |

## 戦略的 recommendation (Stage 0 audit 完了後)

1. **Stage 1 着手** (PySR + CICY) — 新規 + 既 PySR Phase 1 動作確認済 + Paper 63 SNST 14 定数 input が natural fit
2. **Stage 2 ⏸ retain** (Fermat fourfold Lean 4) — Mathlib Hodge API thin 待ち, 12-24 month roadmap (HodgeBasics.lean note 通り)
3. **Stage 3 ❌ skip** (Paper 89 既完了)

## 連結 reference

- Paper 89 (Hodge × D-FUMT₈ Honest Position)
- Paper 99 (Hodge × BSD Triple-Invariant Negative)
- `hodge-bsd-finite-representation-engine.ts` (5 表現)
- `HodgeBasics.lean` (Lean 4 abstract skeleton)
- [[feedback_chat_claude_hallucination_warning]] (Pattern 5 累積 #8)