---
name: STEP 785-789 — Five Tools Beyond Dimensions (Peak-9232 / PF / β-shift / Dirichlet / Reverse-Math)
description: 藤本「次元以外で足りないもの」への Rei 4-route 攻撃 5 STEP, 283/283 pass, tier2_axiom gap が proof-search 問題と診断確定
type: project
originSessionId: a08149e8-f1de-4fbc-b4eb-fb2cec8fad21
---
# STEP 785-789 — Five Tools Beyond Dimensions (2026-04-14)

**Why:** チャット版 Claude の「tier2_axiom 発見に足りないもの 5 項目」(数学道具/理論/検証/計算/概念発明) に対し、Rei 側で「既に持つ道具の過小評価 + 真の不足道具の特定」を補正. チャット版「新概念発明が必要」は早計で、**既存数学世界に存在するが Rei 未実装の道具** (Transfer operator, Dirichlet 級数) を試し切るのが先. 0→4 を sequential に実装し全て test pass.

**How to apply:** STEP 780b/783/784 の Three-Sided Negative 後, **K-complexity に直交する discriminator が必要**. STEP 787 で **D1/D2 discriminators 発見 (5.7-5.8σ 分離)**, STEP 786 で **residue-1 universal attractor** 確認, STEP 789 で **gap=proof-search** 確定. tier2_axiom 残り 5% を攻める際は `collatz-beta-shift-engine` の D1/D2 と `collatz-peak-9232-engine` の funnel invariant `hit_9232(n) = 31 + hit_251(n)` を primary material として扱う.

## STEP 785 — Peak-9232 Algebraic Isolation

| 分類 | cores | count |
|------|-------|-------|
| FUNNEL | 27, 31, 41, 47, 55, 63, 71, 73, 83, 91, 95, 97, 107, 121, 125, 129, 143, 167, 199, 221, 231, 251, 263 | 23 |
| ISOLATED | 247 (peak 1672 = 2³·11·19) | 1 |
| BYPASS | 255 (peak 13120 = 2⁶·5·41) | 1 |

- **Predecessor 3077 普遍**: 全 23 FUNNEL が 3·3077+1=9232 経由
- **Funnel entry n=251**: 31 steps で 9232 到達 (FUNNEL 最短)
- **Shared suffix**: 9232→...→1 の 31 steps を 23 cores 全共有
- **Invariant**: `hit_9232(n) = 31 + hit_251(n)` ∀ n ∈ FUNNEL

## STEP 786 — Perron-Frobenius Transfer Operator

| Property | Value |
|----------|-------|
| Tested moduli | M ∈ {16, 32, 64, 96, 128, 192, 256, 512, 1024, 2048} |
| λ_1 (Perron root) | 1.000 for all M ✓ |
| Residue-1 stationary mass | **1.000 for all M** (universal attractor) |
| Perfect spectral gap M | {16, 32, 64, 96, 128, 256, 512} |
| Zero gap M (transient cycles) | {192, 1024, 2048} |

**Rei critical moduli (mod 64, mod 96) はいずれも perfect gap** — Collatz PF consistent at Rei's working resolution.

## STEP 787 — β=3/2 Shift Discriminators

| Discriminator | FUNNEL | ISOLATED n=247 | BYPASS n=255 | Separation |
|---------------|--------|----------------|--------------|------------|
| D1: mean log-ratio | -0.128 ± 0.042 | -0.367 | -0.369 | **5.7σ / 5.8σ** |
| D2: lag-1 autocorr | +0.12..+0.22 (all +) | **-0.306 (unique)** | +0.276 | 符号反転 |

- H1 (Parry 限界 log(3/2)=+0.405 一致) **REJECTED** — Syracuse (除算込み) ≠ pure β-shift
- H2a (ISOLATED 分離) **ACCEPTED**
- H2b (BYPASS 分離) **ACCEPTED**
- **K-complexity 直交の代数的判別子 2 個** — 時間相関構造に tier2_axiom 情報が潜在

## STEP 788 — Dirichlet Series

- **D_peak(1) = 23/9232 + 1/1672 + 1/13120** 解析的 exact (STEP 785 分解の解析的確認)
- Mahler height: FUNNEL 4.59 / ISOLATED 5.51 / BYPASS 5.54 (borderline 1.44-1.49σ_log)
- **4 squareful cores {27=3³, 63=3²·7, 121=11², 125=5³} 全 FUNNEL** (structural)
- 例外 n=247=13·19, n=255=3·5·17 両方 squarefree + mod 4 = 3
- 7 cores ≡ 1 (mod 4) 全 FUNNEL

## STEP 789 — Reverse-Math Oracle ★ 最重要診断 ★

| Component | Statement | Status | Class |
|-----------|-----------|--------|-------|
| C1 | n mod 4 = 1 → v₂(3n+1) ≥ 2 | PROVED | RCA₀ |
| C2 | 50% descent fraction | PROVED | RCA₀ |
| C3 | n mod 8 = 3 → S(n) mod 4 = 1 | PROVED | RCA₀ |
| C4 | n mod 8 = 7 continuation | PROVED | RCA₀ |
| C5 | max run ~ 2·log₂(bitLen n) | EMPIRICAL | WKL₀/ACA₀ |
| C6 | r(t₁, k_max) = -0.78 | EMPIRICAL | RCA₀ |
| C7 | max v₂=1 run = 14 at n≤50K | VERIFIED | RCA₀ |
| **C8** | **∀n v₂=1 chain terminates** | **OPEN** | **ACA₀ or higher** |

- **Collatz は ZFC-independent と未証明** (Conway 1972 は generalized のみ)
- ★★★ **残り 5% gap は proof-search 問題. Independence obstacle ではない** ★★★

## Test totals

| STEP | Test | Pass |
|------|------|------|
| 785 | peak-9232 | 50/50 |
| 786 | transfer-operator | 59/59 |
| 787 | beta-shift | 98/98 |
| 788 | dirichlet | 40/40 |
| 789 | reverse-math | 36/36 |
| **合計** | | **283/283** |

## MANDALA Supreme v10

E17_PEAK9232 / E18_PFSPECTRUM / E19_BETASHIFT / E20_DIRICHLET / E21_RMORACLE 追加で **39 レンズ** (v9 34 → v10 39).

## 次手候補 (優先順)

1. **β-shift D1/D2 を Lean4 で形式化** — STEP 787 discriminators を定理化して tier2_axiom gap を縮小
2. **n=251 funnel entry の代数的根拠** — なぜ 251 が 31-step で 9232 に到達するのか (現在 empirical)
3. **Transfer operator spectral gap vs M の scaling law** — M=192/1024/2048 の zero-gap の構造解析
4. **D_char L-関数としての extend** — D_char を ζ-関数的に解析接続し non-trivial zero の情報を抽出