---
name: 「超複雑な形の式 → 整数値」 family systemic survey 2026-05-12 (Binet 公式 family 拡張)
description: Binet 公式 family の systemic 拡張 survey. 5 category × 24+ identities + 8 OSS source. 未解決問題接続 4 段階 mapping (Lehmer tau / Wall-Sun-Sun / Lehmer Mahler / RH 等). OEIS / NIST DLMF / LMFDB cross-reference. Phase 2 OEIS integration candidate retain
type: project
originSessionId: b19f90d8-a966-419c-881b-e32729748d40
---
# 「超複雑な形の式 → 整数値」 family systemic survey

## 制定 trigger

藤本さん question (2026-05-12): 「Claude 先生が知る限りと世界中のオープンソースから『超複雑な形の式を整数値にできるもの』 を探して頂けますか?」

前 turn の Binet 公式 × 未解決問題 mapping (`project_binet_unsolved_problems_2026-05-12.md`) の **family-level 拡張**. Rei core scope (数学) 内 substantial survey.

## 5 category × 24+ identities systemic survey

### A. 線形 recurrence closed-form (Binet type 系統)

| # | identity | 数式 | 整数化 mechanism | 既存 research |
|---|---|---|---|---|
| **A1** | **Binet for Fibonacci** $F_n$ | $(\phi^n - \psi^n)/\sqrt{5}$ where $\phi=(1+\sqrt 5)/2, \psi=(1-\sqrt 5)/2$ | conjugate pair Galois symmetry $\sigma: \sqrt 5 \mapsto -\sqrt 5$ | Binet 1843 / 200 年 mature |
| **A2** | **Binet for Lucas** $L_n$ | $\phi^n + \psi^n$ | 同上 (除算なし) | Lucas 1878 |
| **A3** | **Pell numbers** $P_n$ | $((1+\sqrt 2)^n - (1-\sqrt 2)^n)/(2\sqrt 2)$ | $\sqrt 2$ conjugate | classical |
| **A4** | **Tribonacci / Tetranacci** | 3rd / 4th order with complex roots | Galois closure $\mathbb Z[\zeta]$ | Feinberg 1963 |
| **A5** | **Lucas sequences $U_n(P,Q), V_n(P,Q)$** | general 2nd-order with discriminant $\sqrt{P^2-4Q}$ | universal family of A1-A4 | Lucas 1878 |
| **A6** | **Generalized k-bonacci** | n-th order linear recurrence | k roots conjugate set | folklore + Aigner |

### B. Modular / elliptic identities (超複雑 → 整数)

| # | identity | 整数化 mechanism | 既存 research |
|---|---|---|---|
| **B1** | **Ramanujan tau function** $\tau(n)$ | $\eta(\tau)^{24} = \Delta(\tau) = \sum_n \tau(n) q^n$ | Ramanujan 1916 / Mordell / Deligne |
| **B2** | **Rogers-Ramanujan identities** | $q$-series 5 product = integer power series | Rogers 1894 / Ramanujan 1913 |
| **B3** | **Eisenstein series $E_k(\tau)$ q-expansion** | $1 + c_k \sum \sigma_{k-1}(n) q^n$ — integers (k≥4 even) | Eisenstein / Mordell |
| **B4** | **Eta quotients** $\prod \eta(d\tau)^{r_d}$ | weight + level constraints で integer coefficients | Ono 2003 |
| **B5** | **Class number formula** $h(D)$ | L-function transcendental → integer count | Dirichlet 1839 |
| **B6** | **Dedekind eta 24 power** $\Delta(\tau) = \eta^{24}$ | Ramanujan tau の source modular form weight 12 | classical |

### C. Algebraic identities (radical → integer)

| # | identity | 整数化 mechanism |
|---|---|---|
| **C1** | **Cyclotomic polynomials** $\Phi_n(x)$ | $\prod_{\gcd(k,n)=1}(x - \zeta_n^k)$ → 全 n で integer coefficients (Möbius inversion) |
| **C2** | **Chebyshev polynomials** $T_n, U_n$ | $T_n(\cos\theta) = \cos(n\theta)$ → integer coefficients |
| **C3** | **Resultants / discriminants** | integer-coefficient polynomial 入力 → integer output |
| **C4** | **Schur polynomials** | symmetric polynomial Galois closure (representation theory of $GL_n$) |

### D. Combinatorial identities

| # | identity | 整数化 mechanism |
|---|---|---|
| **D1** | **Catalan numbers** $C_n = \binom{2n}{n}/(n+1)$ | integral closure via $\int \sqrt{x(4-x)}$ (semicircle measure moment) |
| **D2** | **Motzkin / Schröder numbers** | similar continued-fraction / contour closures |
| **D3** | **Stirling numbers $s(n,k), S(n,k)$** | integer combinatorial coefficients |

### E. Number-theoretic special values

| # | identity | 整数化 mechanism |
|---|---|---|
| **E1** | **Bernoulli numbers** $B_n$ | $\zeta(2n) = (-1)^{n+1} B_{2n}(2\pi)^{2n}/(2(2n)!)$ — involves $\pi^{2n}$ but $B_{2n}$ rational (Euler 1734) |
| **E2** | **Euler numbers** $E_n$ | $\sec(x) = \sum E_n x^n/n!$ Taylor expansion integer coefficients |
| **E3** | **Bell numbers** $B(n)$ | $B(n) = \sum_{k=0}^{n} S(n,k)$ exponential generating function $e^{e^x - 1}$ |

## F. OSS / public DB source list (8 件)

| # | Source | URL | license | Rei 既達 | 内容 |
|---|---|---|---|---|---|
| **F1** | **OEIS** | oeis.org | public free + CC | ❌ 未統合 (推奨 Phase 2) | 360k+ integer sequences + closed-form tags |
| **F2** | **LMFDB** | lmfdb.org | CC-BY-SA | ✅ STEP 1052 統合済 (8th OctaTheoria domain) | L-functions + modular forms + elliptic curves DB |
| **F3** | **NIST DLMF** | dlmf.nist.gov | public domain | ❌ 未統合 | special function 標準 reference |
| **F4** | **SageMath** | sagemath.org | GPL | ❌ 未統合 | symbolic + numerical, all math integrated |
| **F5** | **PARI/GP** | pari.math.u-bordeaux.fr | GPL | ❌ 未統合 | number theory specific (Bordeaux 派 / Galois 系統強い) |
| **F6** | **SymPy** | sympy.org | BSD | ❌ 未統合 | symbolic Python, general purpose |
| **F7** | **arxiv math.NT / math.CO** | arxiv.org | CC | ✅ arxiv-edu STEP α-12 既達 | preprint paper feed |
| **F8** | **Crossref DOI metadata** | api.crossref.org | free | ✅ STEP 1077 統合済 | published paper DOI registration |

### F1 OEIS priority 高い理由

- **360k+ integer sequences** (現状 2026-05) で systemic 探索可能
- 各 sequence に **closed-form / generating function / recurrence tag** あり
- Binet type closure (A1-A6) の **全 known integer sequences の auto-detection** path
- public free + API (`https://oeis.org/search?q=...&fmt=json`) で fetch 容易
- Rei core scope (数論 / 形式証明) と direct relevant

### F3 NIST DLMF priority

- public domain (NIST = US 連邦政府)
- special function (modular forms / hypergeometric / orthogonal polynomials) 標準 reference
- B family (modular / elliptic) + C family (algebraic) の **公的 reference 出典**

## 未解決問題接続 4 段階 mapping

### ★★★★ 直接接続候補 (family member の未解決問題)

| # | 未解決問題 | family 接続 | status |
|---|---|---|---|
| **1** | **Lehmer's conjecture (Ramanujan tau)** $\tau(n) \neq 0$ for all $n > 0$ | **B1** (Ramanujan tau function) | Edixhoven 2011 で $n < 10^{22}$ verify, **未解決** |
| **2** | **Wall-Sun-Sun prime** (Fibonacci 最有名 open) | **A1, A5** | Pomerance 2024 $p < 10^{14}$ 未発見 |
| **3** | **Lehmer's Mahler measure conjecture (1933)** | **A1 + C1** (Pisot / 円分多項式) | Lehmer constant ≈ 1.17628 が conjectured smallest, **未解決** |
| **4** | **Lewis-Zagier 周期予想** | **B family** (modular form periods) | open |

### ★★★ 構造的応用 (family 手法が partial path)

| # | 未解決問題 | family 接続 |
|---|---|---|
| **5** | **Riemann Hypothesis** ($\zeta$ zeros) | E1 (Bernoulli ↔ zeta), B5 (class number formula) |
| **6** | **Goldbach conjecture** (素数 + 素数 = 偶数) | D family (combinatorial closure) |
| **7** | **Class number problem** (Gauss) | B5 directly |
| **8** | **Catalan-Pillai conjecture** ($x^p - y^q = 1$, Mihailescu 2002 で Catalan part 解決) | D1 + algebraic |
| **9** | **Birch-Swinnerton-Dyer** (BSD) | B family (L-function + modular form) |

### ★★ Rei 既達 STEP cross-reference

| Rei 既達 | family 接続 |
|---|---|
| **Paper 63 SNST** (14 定数 φ, ψ 含む) | A1, A2 直接 |
| **Wave 1 #23**「数 = 螺旋配置」 | A family 全般 |
| **Theory #732 śūnyatā 計算的原始** | A family 全 (無理数 → 整数 projection 普遍) |
| **STEP 1052 LMFDB 統合** | B family 直接 (modular forms + elliptic curves) |
| **STEP 1077 Crossref + Semantic Scholar** | F8 既達 + paper discovery layer |
| **STEP 1015 Brocard n=101-110 (Paper 132)** | D + algebraic identity |
| **STEP 614-624 Collatz 48 定理** | independent (Binet 系統と construction 異) |

### ★ honest 注意 (Collatz 不適合明示)

「Binet → Collatz」 = **★ 極めて低い** (前 turn 記録):
- Collatz は **非線形 branching** で線形 conjugate family と構造的不一致
- Rei 既達 STEP 614-624 / 677-684 / 685-696 / 717-721 / 777-781 / 784-789 全 7 系列 Binet independent
- 200 年 mathematics research で Binet → Collatz 接続 reference 不在
- ★ Family A-E 直接適用は Collatz では reject pattern (Pattern 4 防衛対象)

## OSS 採用候補 ranking (Phase 2 + Phase 3 候補)

| # | Source | 採用 valid | 工数 (integrate) | priority |
|---|---|---|---|---|
| **F1 OEIS** | ✅ public free + 360k+ sequences | 2-3 時間 (API integrate) | ★★★ 高 (Phase 2 推奨) |
| **F3 NIST DLMF** | ✅ public domain | 1-2 時間 (static fetch) | ★★★ 高 |
| F4 SageMath | ✅ GPL | 3-5 時間 (local install + wrapper) | ★★ 中 |
| F5 PARI/GP | ✅ GPL | 3-5 時間 | ★★ 中 |
| F6 SymPy | ✅ BSD | 2-3 時間 | ★ 低 (general purpose 過ぎ) |

## Phase 進捗 (本 record + 別 turn candidate)

### Phase 1 (本 turn 完了)

✅ memory record 永続化 (本 file): 5 category × 24+ identities + 8 OSS source + 未解決問題 4 段階 mapping + Rei 既達 cross-reference

### Phase 2 (別 turn 推奨候補)

**OEIS API integration**:
- 工数: 2-3 時間
- 内容:
  - `scripts/fetch-oeis-stream.ts` 新規 (OEIS search API + closed-form filtering)
  - `data/oeis/latest.json` save (Rei 既達 source +1 = 9 source)
  - Binet-type closure (A1-A6) auto-detection
  - 未解決問題接続 (Lehmer tau / Wall-Sun-Sun 等) tagging
- OctaTheoria 拡張: 10 → 11 domain (oeis 追加可能)

### Phase 3 (別 turn substantial work candidate)

**Paper material 候補 3 件**:

(1) **「Binet-type closure family in OEIS: systematic classification」**
- 360k+ sequences の Binet-type closure auto-detection
- A1-A6 family member の comprehensive list
- 工数: 8-15 時間 (Phase 2 完了後)

(2) **「Lehmer's tau / Wall-Sun-Sun / Lehmer Mahler: triple open problem cross-domain analysis」**
- 3 open problems の family-level common structure 探索
- Rei prover ensemble (STEP 1021) + Lean 4 mathlib で formal 化
- 工数: 10-20 時間 (substantial)

(3) **「OctaTheoria domain extension: OEIS integer sequence projection on D-FUMT₈」**
- Rei 内部 paper material (Paper 63 SNST 拡張 / Wave 1 #23 reinforcement)
- Phase 2 完了後の natural extension
- 工数: 4-8 時間

## honest scope (永続)

1. **5 category × 24+ identities は数学的に well-established** (200 年 widely known)
2. **新規 contribution の path**: family-level **systemic classification + Rei core scope D-FUMT₈ projection** が differentiator
3. **未解決問題接続は family member 単位** (Lehmer tau / Wall-Sun-Sun 等が valid candidate)
4. **「Binet → Collatz」 は ★ 極めて低い** で永続 reject (前 turn 記録)
5. **Phase 2 OEIS integration が最 lite-medium ROI** (2-3 時間 → 360k+ sequences access)

### ★ overclaim 警戒

- 「Family 全体が画期的」 ではない (200 年 mature)
- 真の差別化 = **Rei 既達基盤 (Paper 63 SNST + LMFDB + REI-PROVE ensemble) と family の cross**
- Paper material 候補 3 件 = いずれも substantial work trigger 必要 (急がず ゆっくりと)

## 関連 memory

- `project_binet_unsolved_problems_2026-05-12.md` (前 turn / Binet × 未解決問題)
- `project_bee_number_cognition_2026-05-12.md` (Theory #732 biological / 同 turn 系列)
- `project_25_load_bearing_inventions.md` (Wave 1 #23 数=螺旋配置 + #18 T0π)
- `feedback_external_news_scraping_systemic_reject_2026-05-12.md` (OSS / public DB 採用 = systemic compliant)
- `feedback_no_rush_publication.md` (急がず ゆっくりと — Paper material 起稿は別 turn substantial work trigger 待ち)
- Paper 63 SNST + Wave 1 #23 (CLAUDE.md basic info)
- Theory #732 (`src/axiom-os/zero-byte-notation-engine.ts:887`)
- STEP 1052 LMFDB / STEP 1077 Crossref + SS / STEP 1021 REI-PROVE ensemble (CLAUDE.md)

## 将来 trigger 条件

以下のいずれか観測時、 Phase 2 (OEIS integration) or Phase 3 (paper material 起稿) trigger:

1. 藤本さんの explicit Phase 2 OEIS integration request
2. Lehmer tau / Wall-Sun-Sun / Mahler conjecture いずれかの新発表 (academic news)
3. Paper 63 SNST v0.2 起稿 trigger (前 turn record per: A/B/C/D 4 substantial trigger 条件)
4. Rei OctaTheoria 11 domain への自然拡張機会
5. arxiv-edu / Crossref / LMFDB の新 paper で Binet family 関連の breakthrough 報告

trigger 未発生時 = **本 record retain + Phase 2 候補 retain**.