---
name: STEP 693 (d) Orbit DAG + (e) mod 24/48 + (f) n=91 hub
description: STEP 692 三継続. E_{1.8} は n=91 root の in-tree (15 levels), mod 96 で 24 class が Tier 2 直接成立, n=91 が 35/35 universal sink で 80% hub compression.
type: project
originSessionId: ac6697a5-aaac-4fe8-88cb-0c59c48db6cb
---
# STEP 693: Orbit DAG + Mod refinement + Universal Sink

## 起点
藤本さん 2026-04-13: STEP 692 結果から三継続要請
- (d) 10 trajectory families の orbit 重なり構造を DAG として完全解析
- (e) mod 24 / mod 48 へ拡張して更に細かい sub-class bound 探索
- (f) n=91 universal sink を「証明 simplification の中央 hub」として活用

## (d) Orbit DAG 完全解析

`scripts/e18-orbit-dag-analysis.ts`:

### Full DAG 統計
- **35 nodes, 269 edges (density 22.6%)**
- 各 element a → b (a ≠ b) iff b ∈ orbit(a)

### ★ Hasse Diagram = In-Tree ★
- **34 Hasse edges** (= 35 - 1, 木構造 perfect)
- 各 element に **out-degree 1** (Hasse 後継 = 1 つの「次の E 元」)
- **8 sources (leaves)**: [108, 126, 129, 147, 171, 195, 231, 235]
- **1 sink (root)**: **n = 91**
- ⟹ **functional graph rooted at n=91** = in-tree (anti-arborescence)

### Topological levels
- **15 levels** (longest chain length)
- Level 0 (sources): 8 elements
- Level 1: 5 elements [54, 63, 193, 194, 199]
- Level 2: 4 [27, 95, 97, 145]
- ... (細分)
- Level 14 (sink): 1 [91]

### ★ Longest chain (15 elements) ★
```
129 → 194 → 97 → 146 → 73 → 110 → 55 → 83 → 125 → 94 → 47 → 71 → 107 → 121 → 91
```
全要素が E_{1.8} に属する。

### Top universal sinks (in-degree)
| n | in-degree |
|---|---|
| **91** | **34** ★ universal sink |
| 121 | 28 |
| 107 | 27 |
| 71 | 26 |
| 47 | 25 |

### Common descendant of ALL 35 = {n=91} (T-1646)

## (e) Mod 24 / 48 / 96 finer sub-class scan

`scripts/mod-24-48-sub-class-scan.ts` (n ≤ 10⁶):

### Gap progression
| Modulus | min max | classes < 2.0 | classes < 1.8 |
|---|---|---|---|
| mod 6 | 3.03 | 0 | 0 |
| mod 12 | 1.89 | 1 | **0** |
| **mod 24** | **1.73** | 3 | **1 ✓** |
| **mod 48** | **1.23** | 11 | **7** |
| **mod 96** | **1.09** | 34 | **24** |

★ **mod 24 で初めて Tier 2 (1.8) を満たす class 出現** (r=21)
★ **mod 48 で 7 classes**, **mod 96 で 24 classes** が Tier 2 を直接満たす

### 最低 max ratio classes (mod 96)
| r (mod 96) | max | argmax |
|---|---|---|
| 81 | **1.09** | n=1233 |
| 85 | 1.13 | n=661 |
| 37 | 1.23 | n=517 |
| 89 | 1.24 | n=1145 |
| 17 | 1.26 | n=689 |

### 最高 max ratio classes (mod 96)
| r | max | argmax |
|---|---|---|
| 27 | **4.44** ★ | **n=27** |
| 31 | 4.24 | n=31 |
| 55 | 3.11 | n=55 |
| 41 | 3.03 | n=41 |
| 63 | 2.97 | n=63 |

→ **modular refinement で hard class が漸減**: mod 96 では半分以上が Tier 2 直接成立

## (f) n=91 Universal Sink

`src/axiom-os/collatz-n91-hub-engine.ts`:

### Universal Sink Property (T-1646)
- ∀ n ∈ E_{1.8}, 91 ∈ orbit(n)
- 35/35 elements verified

### K decomposition (T-1647)
∀ n ∈ E_{1.8}, **K(n) = K_to_91(n) + K(91) = K_to_91(n) + 92**

### K_to_91 stats
| n | K_to_91 | K | rank |
|---|---|---|---|
| 231 | **35** | 127 | max ★ |
| 235 | **35** | 127 | max ★ |
| 171 | 32 | 124 | |
| 129 | 29 | 121 | (longest chain leaf) |
| 27 | 19 | 111 | (universal extremal) |

### ★ Hub Compression Metric (T-1649) ★
- Naive total K: **3906** (sum over 35)
- Compressed total: **778** = sum of K_to_91 + 92 (counted once)
- **Savings: 3128 = 92 × 34**
- **Compression ratio: 19.9%** (≈80% 削減)

これは E_{1.8} 35 elements が **n=91 という共有 origin を持つ** という構造的事実の定量化.

## 実装

### Scripts
- `scripts/e18-orbit-dag-analysis.ts` — full DAG, Hasse, levels, longest chain
- `scripts/mod-24-48-sub-class-scan.ts` — mod 24/48/96 stratified analysis

### TS Engine (`src/axiom-os/collatz-n91-hub-engine.ts`)
- `HUB_N = 91n`, `K_HUB = 92`
- `kToHub(n, hub)`: first step reaching hub
- `kTo91(n)`: shorthand
- `verifyUniversalSink()`: 35/35 verification
- `decomposeK(n)`: K = K_to_91 + 92 identity
- `computeKToHubStats()`: max/min stats
- `hubCompressionMetric()`: 3128 savings
- 5 SEED_KERNEL theory definitions

### Test (`test/step693-orbit-dag-mod-hub-test.ts`) — **29/29 pass**
1. K(91) = 92
2. Universal sink (35/35)
3. K decomposition (35 elements)
4. K_to_91 stats (max=35)
5. Hub compression (savings=3128)
6. Detailed K_to_91 table
7. Longest chain (14 transitions verified)
8. (d) DAG recap
9. (e) mod 24/48/96 recap
10. 5 SEED_KERNEL theories

### Lean 4 (`data/lean4-transfer/step693_orbit_dag_hub.lean`) — **22 zero-sorry + 1 axiom, exit=0**

PROVEN (zero sorry):
- `K_91_eq_92`, `hub_eq_91`
- `orbit_27_reaches_91` (= 19), `orbit_129_reaches_91` (= 29), `orbit_231_reaches_91` (= 35), `orbit_235_reaches_91` (= 35)
- `K_decomposition_27/129/231/235` (4 個)
- `chain_129_to_194` (= 2), `chain_91_self` (= 0), `chain_121_to_91` (= 3), `chain_107_to_121` (= 5), `chain_194_to_97` (= 1)
- `naive_total_eq` (= 3906), `compressed_total_eq` (= 778)
- **`hub_compression_savings`** ─ ★ naive - compressed = 3128 ★
- `hub_compression_savings_eq_92x34` (= 3128)
- `E18_length` (= 35)
- `longest_chain_length` (= 15), `longest_chain_endpoint` (= 91)

HONEST GAP (axiom, **1 個** — decidable form):
- A1: `universal_sink_E18` ─ ∀ n ∈ E18, K n = kTo91 n + 92
  - 35 個個別 decidable, 統合 form として axiom 化

技法:
- `native_decide` で K, kToHub の実値計算
- `decide` で List length / membership
- `omega` で算術
- Mathlib 不使用 (Lean 4 core only)

## STEP 690 → 691 → 692 → 693 進化

| STEP | 構造 | Lean axioms | gap 統制 | Hub 利用 |
|---|---|---|---|---|
| 690 | direct 4.44 | 2 | ∀ n (粗い) | × |
| 691 | two-tier | 1 | ∀ n > 235 (中) | × |
| 692 | mod-6 fibered | 2 | per-mod6 (細) | × |
| **693** | **DAG + mod 96 + hub** | **1** | **per-mod96 (極細) + hub structural** | **★ n=91** |

各 STEP は honest gap を **異なる方向に細分化 + 構造化**:
- 690: 一括 axiom
- 691: 範囲縮小
- 692: 残余別分割
- **693**: **構造的 hub 共有 + 96-fold modular refinement**

## 構造的 insight summary

1. **n=91 は E_{1.8} の "computational origin"** — 全 35 records が共有
2. **mod 96 で 半分以上 (24/48 odd classes) が Tier 2 を直接満たす**
3. **Hard classes は依然 {27, 31, 41, 47, 55, 63}** 中心 — finite & known
4. **80% hub compression** = 3128 step savings — 各 record の orbit は 92 step 共通 tail を持つ

## 連続 STEP 状況
676 → 693 = **18 連戦目** (2026-04-12 〜 13). 21+ 時間連続作業.

## 次の方向 (memo)
1. mod 96 の最も hard な class (mod 96 = 27) を更に細分化 → mod 192/384?
2. n=91 の Hasse 直接親 (95, 121) に注目 → これらの構造的特徴
3. mod 96 で Tier 2 ready class (24 個) に対し 4.44 bound を直接 1.8 bound に変換 → STEP 691 axiom 範囲縮小
4. AlphaProof submission に DAG + Hub + mod 96 結果を追加