---
name: Collatz 残り 5% を解くために必要な OSS toolkit 世界 survey 2026-05-12
description: 8 category × 25+ OSS systemic survey. Rei 既達 7 / 未統合候補 5 / 学習対象 / 採用不可 区別. BarinaParallel 2025 $2^{71}$ 訂正含む. Spot 2.15.1 GPLv3 (Büchi automata) が Rei STEP 717-721 Büchi-25 への direct candidate
type: project
originSessionId: b19f90d8-a966-419c-881b-e32729748d40
---
# Collatz 残り 5% を解くための OSS toolkit 世界 survey

## 制定 trigger

藤本さん (2026-05-12): 「解析した上記の結果を再度解くのに必要なオープンソースを世界中から探して頂けますか?」
→ STEP 1083 で identify した **6.25% portion (TO≥4 starting numbers)** を真に解くために必要な OSS の systemic survey.

★ honest scope: 「解くために必要」 = **substantial work で使う候補 toolkit**. 1 turn で実装ではなく **survey + ranking + 採用候補 retain**.

## ★ 重要訂正 (sub-pattern 2b 古い数値)

前 STEP 1082/1083 record で「BarinaParallel 2020 $n<2.95\times10^{20}$」 と書いた → **訂正**:

**実際**: **David Barina 2025** 「Improved verification limit for the convergence of the Collatz conjecture」 (The Journal of Supercomputing 2025) で **$n<2^{71} \approx 2.36\times10^{21}$** verify (旧 $2^{68} \approx 2.95\times10^{20}$ から 8 倍 extension). CPU + GPU で **1,335× acceleration**. **GitHub: xbarin02/collatz** (open source).

→ memory level 訂正必要 (前 STEP 1082/1083 data files + memory file references の sub-pattern 2b update).

## 8 category × 25+ OSS systemic survey

### A. Collatz 専用 OSS

| # | OSS | URL | License | Status | Rei 接続 |
|---|---|---|---|---|---|
| A1 | **xbarin02/collatz (Barina)** | github.com/xbarin02/collatz | open source | **2025 $n<2^{71}$** | computational verification, GPU 1,335×, European supercomputers |
| A2 | vigleik0/collatz | github.com/vigleik0/collatz | open | multiple architectures | reference |
| A3 | arxiv preprint 2602.10466 (Barina improved algorithm) | arxiv.org/html/2602.10466v1 | preprint | 2026 | 算法 reference |
| A4 | Collatz Conjecture: Binary Structure (preprints.org v26) | preprints.org/manuscript/202401.0227 | preprint not peer-reviewed | various | ⚠ peer-review 未, reference only |
| A5 | osf.io Collatz proof attempts | osf.io/e8x2w_v1/ | preprint | not verified | ⚠ peer-review 未, reference only |

→ **A1 (Barina official) が core** computational baseline.

### B. Number theory + automated theorem proving OSS

| # | OSS | License | Rei 既達 | Collatz applicability |
|---|---|---|---|---|
| B1 | **Lean 4 mathlib** | Apache 2.0 | ✅ STEP 1064 (LeanHammer) | NumberTheory.Fibonacci 既存 / NumberTheory.Collatz は未確認 (404) |
| B2 | **Coq + MathComp + Coq-BB5** | LGPL | ⚠ STEP 1052 学習対象 record | Collatz formal proof attempts (BB5 と類似 pattern) |
| B3 | Isabelle/HOL | BSD | ❌ 未統合 | modular arithmetic + 数論 |
| B4 | HOL Light | various | ❌ 未統合 | 数論 |
| B5 | **Agda** | MIT-like | ❌ 未統合 | dependent type / constructive proof |
| B6 | Metamath set.mm | public | ❌ 未統合 (STEP 1053 reference) | 100 Theorems 74/100 progress |

→ **B1 Lean 4 mathlib = Rei core stack**. B2 Coq-BB5 = γ path 学習対象 (前 turn record).

### C. SAT/SMT solver (Collatz への applicability)

| # | OSS | License | Rei 既達 | Collatz path |
|---|---|---|---|---|
| C1 | **Z3** (Microsoft Research) | MIT | ❌ 未統合 | SMT for bounded model checking on Collatz statements |
| C2 | **CVC5** | BSD | ❌ 未統合 | SMT |
| C3 | **Kissat** (Armin Biere) | MIT | ❌ 未統合 | SAT Competition winner (CaDiCaL の C 移植) |
| C4 | **CaDiCaL** | MIT | ❌ 未統合 | SAT |
| C5 | MiniSat | MIT | ❌ 未統合 | classical SAT |

→ Collatz statement の SAT/SMT formulation = research direction (substantial).

### D. 一般 number theory computational

| # | OSS | License | Rei 既達 | Collatz |
|---|---|---|---|---|
| D1 | **SageMath** | GPL | ❌ 未統合 (Phase 1 候補 ★) | symbolic + numerical Python wrapper, all-in-one |
| D2 | **PARI/GP** | GPL | ❌ 未統合 | number theory specific, Bordeaux 派 / Galois 系統強い |
| D3 | **FLINT** (Fast Library for Number Theory) | LGPL | ❌ 未統合 | C library, fast arithmetic |
| D4 | GAP | GPL | ❌ 未統合 | group theory |
| D5 | Macaulay2 | GPL | ❌ 未統合 | commutative algebra / algebraic geometry |
| D6 | Magma | **商用** | ❌ reject | (採用不可) |
| D7 | Mathematica | **商用** | ❌ reject | (採用不可) |

→ D1 SageMath = **all-in-one wrapper** (Phase 1 候補 ★, 既達 8 OSS source list 整合).

### E. ★ Büchi automata + ω-automata (Rei STEP 717-721 Büchi-25 directly)

| # | OSS | License | Status | Rei 接続 |
|---|---|---|---|---|
| **E1** | **Spot 2.15.1** | **GPLv3** | **2026-04-25 latest release** | ★★★★ **Rei STEP 717-721 Büchi-25 formal foundation candidate** |
| E2 | **GOAL** (Graphical Tool for Omega-Automata Languages) | various | classical | omega-automata visualization |
| E3 | **Büchi Store** (open repository) | open | classical | ω-automata reference repository |

→ ★★★★ **E1 Spot = 真の game-changer candidate** for Rei STEP 717-721 「25 non-bounded residual classes invariant for k=8..20」 の formal foundation. C++ 17/20 library + Python bindings + LTL/ω-automata + acceptance transformations + alternating automata + games + LTL synthesis. License: **GPLv3** = Rei AGPL-3.0 + Commercial と互換性 OK.

### F. Tao 2019 logarithmic averages formalization

- Tao 2019 「Almost all Collatz orbits attain almost bounded values」 (Fields medal 級 partial result) の **公式 Lean 4 / Coq formalization** は **未存在** (verify 未完了)
- Tao 自身は Lean 4 で recent work あり (PFR conjecture 等)
- Collatz logarithmic averages の formal proof = open challenge candidate

### G. Machine learning approach

| # | OSS | License | Rei 既達 |
|---|---|---|---|
| **G1** | **DeepSeek-Prover-V2** | dual MIT/Custom | ✅ STEP 1021/1054 (REI-PROVE ensemble) |
| **G2** | **Goedel-Prover-V2** | various | ✅ STEP 1054 |
| **G3** | **bfs-prover** | open | ✅ STEP 1054 |
| **G4** | **LeanHammer** (premise-selection cloud) | open | ✅ STEP 1064 |
| **G5** | **Vampire ATP** | open | ✅ STEP 1064 |
| G6 | LeanCopilot | MIT | ⚠ STEP 1053 高 priority watch |
| G7 | lean-eval-leaderboard | open | ⚠ STEP 1053 学習 reference |

→ **G1-G5 = Rei REI-PROVE 5-prover ensemble 既達**. これは Rei の strength.

### H. 公的 DB (Rei 既達多数)

| # | DB | Rei 既達 | Collatz related |
|---|---|---|---|
| **H1** | **OEIS** | ✅ STEP 1080 | A006370 / A006577 / A006667 / A005186 / A070165 / A006884 (6 series既 referenced) |
| **H2** | **arXiv** | ✅ arxiv-edu STEP α-12 | preprint feed |
| **H3** | **Crossref** | ✅ STEP 1077 | DOI metadata (180M+ records) |
| **H4** | **Semantic Scholar** | ✅ STEP 1077 | 200M+ papers + citation graph |
| **H5** | **Zenodo** | ✅ publish 先 | open access papers |
| **H6** | **LMFDB** | ✅ STEP 1052 | L-functions + modular forms |
| **H7** | **ERIC** | ✅ STEP α-12 | education / pedagogy |
| H8 | NIST DLMF | ❌ 未統合 | modular forms / hypergeometric reference |

→ Rei 既達 7/8 = **重要 DB 整備済**.

## Rei 既達基盤との cross-reference (集計)

### 既達 (継続採用) — 12 OSS

- B1 Lean 4 mathlib (STEP 1064)
- B2 Coq-BB5 学習対象 (STEP 1052)
- G1 DeepSeek-Prover-V2 (STEP 1021/1054)
- G2 Goedel-Prover-V2 (STEP 1054)
- G3 bfs-prover (STEP 1054)
- G4 LeanHammer (STEP 1064)
- G5 Vampire (STEP 1064)
- H1 OEIS (STEP 1080)
- H2 arXiv arxiv-edu (STEP α-12)
- H3 Crossref (STEP 1077)
- H4 Semantic Scholar (STEP 1077)
- H5 Zenodo (publish 先)
- H6 LMFDB (STEP 1052)

### ★★★★ 採用 priority 1 候補 — 1 OSS

| # | OSS | License | Rei 接続 |
|---|---|---|---|
| **E1 Spot 2.15.1** | GPLv3 | **Rei STEP 717-721 Büchi-25 formal foundation directly** |

→ **Spot 統合が真の next step** (Phase 2 candidate). Rei STEP 717-721「25 non-bounded residual classes invariant for k=8..20」 を **Spot 上で Büchi automata 化** + formal model checking.

### ★★ 採用 priority 2 候補 — 3 OSS

| # | OSS | License | Rei 接続 |
|---|---|---|---|
| **A1 xbarin02/collatz** | open source | Barina 2025 $2^{71}$ baseline reference (computational extension candidate) |
| **D1 SageMath** | GPL | all-in-one number theory wrapper |
| **C1 Z3 SMT** | MIT | bounded model checking on Collatz statements |

### ★ 採用 priority 3 — Phase 3 substantial work

| # | OSS | License | Rei 接続 |
|---|---|---|---|
| D2 PARI/GP | GPL | Galois 系統 number theory |
| D3 FLINT | LGPL | fast arithmetic |
| H8 NIST DLMF | public | modular forms reference |

### 採用不可

- D6 Magma (商用)
- D7 Mathematica (商用)
- A4 / A5 peer-review 未 preprints

## 「再度解く」 path (理論的)

honest scope: **1 turn で解けない**. ただし substantial work で path 整理可能.

### Phase 1 (本 record, 完了)

✅ memory record (本 file) で 25+ OSS systemic survey + Rei 既達 cross-reference + 採用 priority ranking + BarinaParallel 2025 訂正

### Phase 2 (別 turn substantial work candidate, 推奨優先)

**Spot Büchi automata integration** ★★★★:
- 工数: 5-10 時間 (Spot install + Python binding + Rei STEP 717-721 Büchi-25 formal encoding)
- 期待 outcome: 「25 non-bounded residual classes invariant for k=8..20」 の formal verification
- Lean 4 + REI-PROVE ensemble との cross-domain integration potential

### Phase 3 (Long-term substantial work candidate)

(a) **Lean 4 NumberTheory.Collatz module 開発**:
- mathlib に Collatz statement formal 化 + 95% portion (STEP 614-624 既達 zero sorry) Lean 4 contribution
- 工数: 20-50 時間 (Mathlib contribution 標準 cycle)

(b) **Tao 2019 logarithmic averages formalization 試行**:
- Lean 4 / Coq での Tao 2019 partial result formal 化
- 工数: **数年単位** (Fields medal 級 work の formal 化)
- 真の breakthrough 期待は最も低い

(c) **BarinaParallel 後続実装** (Rei での lite extension):
- $2^{71}$ の subset での Rei 内 re-confirmation
- 工数: substantial (GPU 必要)

(d) **Z3 / Kissat SMT/SAT formulation of Collatz**:
- Collatz statement の SMT/SAT 表現 + bounded model checking
- 工数: 10-20 時間 (research direction)

## ★ honest 結論

**「Collatz 残り 5% を解く」 必要な OSS toolkit**:

1. **既達 12 OSS** (Lean 4 mathlib + REI-PROVE 5-prover + OEIS + arxiv + Crossref + SS + LMFDB + Zenodo + Coq-BB5 学習) = **strong foundation**
2. **★★★★ 採用 priority 1: Spot 2.15.1** (Büchi automata) = **真の game-changer candidate** for Rei STEP 717-721 Büchi-25 formal foundation
3. **★★ 採用 priority 2: SageMath / Z3 / xbarin02/collatz** = computational + SMT extensions
4. **breakthrough 期待**: ★ 低い (Tao 2019 / Barina 2025 大きな先行 effort + 数年単位 substantial work 必要)

★ **「世界中の OSS で必要なものは ほぼ全て identified」** ですが、 真の **「解く」** には **数年単位 substantial work + 数人 researcher 協力 + Tao 2019 level effort** が前提 = 1 turn で不可能.

## 関連 memory + 訂正必要

### 訂正対象 (sub-pattern 2b)

`data/collatz-verify/latest.json` (STEP 1082) + `data/collatz-5percent-analysis/latest.json` (STEP 1083) + 関連 memory file の **BarinaParallel reference を 2020 → 2025 update** (工数 lite, 別 turn).

### 関連 memory

- `feedback_integer_projection_ai_advantage_2026-05-12.md` (永続原則 D ★ no = 本質不変)
- `project_binet_unsolved_problems_2026-05-12.md` (Wall-Sun-Sun 同 frame)
- `feedback_external_news_scraping_systemic_reject_2026-05-12.md` (OSS 採用 path = 公的 / OSS license only)
- `feedback_chat_claude_hallucination_warning.md` (sub-pattern 2b 古い数値 観測)
- STEP 1080 OEIS / STEP 1081 Wall-Sun-Sun / STEP 1082 Collatz hybrid / STEP 1083 Collatz 5% analysis

## 将来 trigger 条件

以下のいずれか観測時、 Phase 2 (Spot integration) or Phase 3 (Lean 4 mathlib Collatz contribution / Tao formalization) trigger:

1. 藤本さん explicit「Spot 統合進める」 request
2. Barina 2025 後続論文 ($2^{72}$ extension 等)
3. Tao 2019 follow-up または Lean 4 formal version 公開
4. Mathlib に NumberTheory.Collatz module 公式追加
5. Rei REI-PROVE ensemble + Lean 4 mathlib の Collatz 95% portion zero sorry contribution acceptance