---
name: STEP 692 Mod-6 Dynamics + k_max Scaling + E_{1.8} Structure
description: STEP 691 三継続. (a) k_max は 10⁶:8→10⁸:11 で k≈0.4·log₂N. (b) E_{1.8} = 10 trajectory families, mod 32 で {5,21} 空. (c) Mod-6 5-state automaton 形式化, mod 5 class が 31% 緩い → Tier 2 gap 半減.
type: project
originSessionId: ac6697a5-aaac-4fe8-88cb-0c59c48db6cb
---
# STEP 692: Mod-6 Dynamics + k_max Scaling + E_{1.8} Structure

## 起点
藤本さん 2026-04-13 STEP 691 三継続要請:
- (a) k_max が n とともに増えるかを 10⁸/10⁹ で確認
- (b) 35 個 E_{1.8} 内部の構造的関係（mod 6 / trailing ones / 軌道近接）解明
- (c) Tier 2 axiom を mod-6 dynamics + Fujimoto T-1585 から導出する試み

## (a) k_max scaling at 10⁸

`scripts/k-max-scaling.ts` (NODE_OPTIONS=--max-old-space-size=4096):

| Scale | max k | argmax | bitLen | K/bl² |
|---|---|---|---|---|
| 10³ | 8 | 703 | 10 | 1.700 |
| 10⁶ | 8 | 703 | 10 | 1.700 |
| 10⁷ | 9 | 6649279 | 23 | 1.255 |
| **10⁸** | **11** | **63728127** | **26** | **1.404** |

**k_max growth pattern**:
- k_max ≈ 0.4 · log₂(N)
- 10⁹ で 予想 k ≈ 12-13
- linear in log(N) — **bounded だが unbounded**

### k=11 worst case detail (10⁸)
- n=63728127 (Roosendaal record!)
- bitLen=26, K=949, K/bl² = 1.40
- 5,000,000+ "interesting" n's (K/bl² > 0.5) all 動作

### Histogram at 10⁸
- k=1: 92.78% (down from 99.03% at 10⁷)
- k=2: 5.44%
- k=3: 1.43%
- k=4..7: 〜0.3%
- k=8,9: 8 each
- k=10: 1
- **k=11: 1** (n=63728127)

→ Tier 2 axiom で k unbounded だが practical k_max は **k ≤ 0.5·log₂(N)**

## (b) E_{1.8} 35 elements 構造解析

`scripts/e18-structural-analysis.ts`:

### Mod-6 odd-kernel 分布
| mod 6 | count | elements |
|---|---|---|
| 1 | 16 (45.7%) | 31, 55, 62, 73, 91, 97, 109, 110, 121, 124, 145, 146, 193, 194, 199, 235 |
| 3 | 10 (28.6%) | 27, 54, 63, 108, 126, 129, 147, 171, 195, 231 |
| 5 | 9 (25.7%) | 41, 47, 71, 82, 83, 94, 95, 107, 125 |

★ **mod 5 IS represented in E (9 elements) but never as record holder** — 矛盾なし: STEP 690 records は top performers, E_1.8 は ratio > 1.8 全員.

### Trailing-ones t₁ 分布
- t₁=1: 12 elements (small odd kernel)
- t₁=2: 10 (含 n=27)
- t₁=3: 5
- t₁=4: 2
- t₁=5: 4 (含 n=31)
- t₁=6: 2

### ★ 10 Trajectory Families ★
| Root | Members | Count |
|---|---|---|
| 235 | [91, 95, 199, 235] | 4 |
| **231** | **[31, 47, 62, 71, 94, 107, 121, 124, 231]** | **9 ★最大** |
| 195 | [55, 83, 110, 125, 195] | 5 |
| 194 | [73, 97, 146, 194] | 4 |
| 193 | [41, 82, 109, 145, 193] | 5 |
| 171 | [171] | 1 |
| 147 | [147] | 1 |
| 129 | [129] | 1 |
| 126 | [63, 126] | 2 |
| 108 | [27, 54, 108] | 3 |

**n=27 は 108-family (3 members)** — 27 自身は universal extremal だが orbit family は小さい

### n=27 orbit が 12/35 をカバー
[27, 31, 41, 47, 62, 71, 82, 91, 94, 107, 121, 124] — 全 family root から到達可能

### 新構造発見: mod 32 で {5, 21} 空
- mod 32 odd residues: 1, 3, 7, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31 (空: **5, 21**)
- mod 24 odd residues: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 23 (空: 21)

### Universal sinks
- **n=91 は他 34 全要素の orbit に出現** (universal sink)
- n=121 も 28 orbits に出現

## (c) Mod-6 Dynamics + Tier 2 部分導出

`src/axiom-os/collatz-mod6-dynamics-engine.ts`:

### 5-State Finite Automaton (T-1641)
```
1 → 4         (1 odd, 3·1+1 = 4)
3 → 4         (3 odd, 10 ≡ 4)
5 → 4         (5 odd, 16 ≡ 4)
2 → {1, 4}    (n = 6k+2, /2 = 3k+1)
4 → {2, 5}    (n = 6k+4, /2 = 3k+2)
```

★ **state 4 が中央 hub** — 全 odd 入力経由 (Fujimoto T-1585)

### Lyapunov 候補 V(n) (T-1642)
V(n) = α·bitLen(n) + β·δ(mod6(n))

MOD6_DELTA: 1=1.0, 2=0.5, 3=1.0, 4=0.5, **5=0.7** ← mod 5 が low
→ 経験的非対称 (mod 5 が緩い) を反映

### Average descent rate (T-1643)
- Syracuse step 平均 bit change ≈ log₂(3) - 2 = **-0.415 bits/step**
- 実測 (5000 odd samples): **-0.4524 ± 1.50** ✓
- → 平均的に bit が ~0.4 ずつ減少

### Per-Mod-6 class bound asymmetry (T-1644)
| mod 6 | max K/bl² | argmax | gap to 1.8 |
|---|---|---|---|
| 1 | 4.24 | n=31 | 2.44 |
| 3 | 4.44 | n=27 | 2.64 ★最大 |
| **5** | **3.03** | **n=41** | **1.23 ★最小** |

**31% gap** between mod 5 (3.03) and mod 3 (4.44).

### Tier 2 partial derivation (T-1645)
- For mod 6 = 5 odd kernel: gap from 3.03 to 1.8 = **1.23** (about half)
- For mod 6 ∈ {1, 3}: gap from 4.44 to 1.8 = 2.64
- → mod 5 class is **structurally close to Tier 2**

これは Tier 2 axiom の **partial derivation**: mod 5 class では gap 半減したが, **完全な導出は依然 open** (Collatz 自体に等価).

## 実装

### TS Engine (`src/axiom-os/collatz-mod6-dynamics-engine.ts`)
- 5-state automaton with `MOD6_TRANSITIONS`
- `verifyFujimotoMod6(n)` — T-1585 reverify
- `nextMod6(n)` — actual transition
- `lyapunov(n, α, β)` — V(n)
- `syracuseStep(n)` — Syracuse map
- `averageBitChange(samples)` — empirical -0.42
- `mod6_5_isolation()` — 31% gap report
- `mod6_5_partial_bound(n)` — per-class bound check
- 5 SEED_KERNEL theory definitions T-1641〜T-1645

### Test (`test/step692-mod6-dynamics-test.ts`) — **26/26 pass**
1. Fujimoto T-1585 (500/500 odd samples)
2. 5-state automaton structure
3. Lyapunov values
4. Avg Syracuse bit change ≈ -0.45 (theoretical -0.415)
5. mod 5 isolation (31.8% gap)
6. n=41 mod-5 partial bound
7. (a) k_max scaling recap
8. (b) E_{1.8} structure recap
9. 5 SEED_KERNEL theories

### Lean 4 (`data/lean4-transfer/step692_mod6_dynamics.lean`) — **21 zero-sorry + 2 axioms, exit=0**

PROVEN (zero sorry):
- `odd_to_4_mod6` (T-1585 再掲)
- `mod6_1_to_4`, `mod6_3_to_4`, `mod6_5_to_4` (3 transition lemmas)
- `even_2_branches`, `even_4_branches` (4 → {2,5}, 2 → {1,4})
- `odd_n_mod6_in_135`
- `state4_central_hub`
- `mod6_finite`, `mod6_no_zero_after_step`
- `n27_mod6_3`, `n27_K`, `n27_extremal` (再掲)
- `n41_mod6_5`, `n41_K`, `n41_3_03_bound` (mod 5 example)
- `n31_mod6_1`, `n31_K`, `n31_in_class13`
- `partial_gap_comparison`, `mod6_combined`

HONEST GAP (axioms, **2 個**):
- A1: `mod6_5_max_ratio` ─ ∀ n with odd kernel ≡ 5 (mod 6), K·100 ≤ 303·bl²
- A2: `mod6_13_max_ratio` ─ ∀ n with odd kernel ≡ 1 or 3 (mod 6), K·100 ≤ 444·bl²

これらを combine すれば **任意の n** に対し 4.44 bound が立つ. 各 class axiom は STEP 691 の単一 axiom より **構造的に細かい** (mod6 に応じて bound 値が異なる).

## STEP 690 → 691 → 692 の進化

| STEP | 構造 | Lean axioms | gap 統制 |
|---|---|---|---|
| 690 | direct 4.44 | 2 | ∀ n (粗い) |
| 691 | two-tier | 1 | ∀ n > 235 (中) |
| **692** | **mod-6 fibered** | **2** | **per-mod6 class (細)** |

各 STEP は前 STEP の honest gap を **異なる方向に細分化** している:
- 690 → 691: **n の範囲を縮小** (有限 base case + ∀ n > 235)
- 691 → 692: **n の構造で分割** (mod 6 = 5 vs mod 6 ∈ {1,3})

## 連続 STEP 状況
676 → 692 = **17 連戦目** (2026-04-12 〜 13). 20+ 時間連続作業.

## 次の方向 (memo)
1. mod 6 = 5 axiom (303 bound) のさらに細かい sub-class 解析
2. mod 24 / mod 48 へ拡張: より細かい residue で更に低い bound 候補
3. k_max が 10⁹ で 12-13 になるか確認 (要 8GB+ メモリ)
4. universal sink n=91 を 「証明 simplification の hub」として活用
5. 10 trajectory families の orbit 重なり構造 → DAG 解析