---
name: STEP 696 Atomic Hard Cores + n=91 Arithmetic Uniqueness
description: 25 atomic hard cores が exactly E_{1.8} の odd elements に一致, n=91 mod 32 = 27 sibling, peak 9232 = 16×577 を 6 elements で共有, 91 は downstream of 27 universal sink.
type: project
originSessionId: ac6697a5-aaac-4fe8-88cb-0c59c48db6cb
---
# STEP 696: Atomic Hard Cores + n=91 Arithmetic Uniqueness

## 起点
藤本さん 2026-04-13: Paper 60 投稿後の Collatz 解析再開. 残っていた (m), (n) を実行.

## (m) Atomic identification — 25 cores の確定

`scripts/atomic-hard-cores-scan.ts`:

### Saturation progression
| Modulus | READY | HARD | READY % |
|---|---|---|---|
| mod 96 | 24 | 24 | 50.0% |
| mod 192 | 71 | 25 | 74.0% |
| mod 384 | 167 | 25 | 87.0% |
| mod 768 | 359 | 25 | 93.5% |
| mod 1536 | 743 | 25 | 96.7% |
| mod 3072 | 1511 | 25 | 98.4% |
| mod 6144 | 3047 | 25 | 99.2% |
| mod 12288 | 6119 | 25 | 99.6% |
| mod 24576 | 12263 | 25 | 99.8% |
| mod 49152 | 24551 | 25 | **99.9%** |

★ **mod 192 → mod 49152 (256× refinement) で hard count = 25 完全 saturate** ★

### ★ The 25 atomic hard cores ★
```
[27, 31, 41, 47, 55, 63, 71, 73, 83, 91, 95, 97, 107, 109,
 121, 125, 129, 145, 147, 171, 193, 195, 199, 231, 235]
```

### ★ 重要定理 ★ : 25 atomic = E_{1.8} odd elements (exact)

E_{1.8} の 35 elements を分解:
- **25 odd** (atomic cores)
- **10 even** = atomic core × 2^k:
  - 54 = 2·27, 108 = 4·27 (n=27 contributes 2)
  - 62 = 2·31, 124 = 4·31 (n=31 contributes 2)
  - 82 = 2·41, 94 = 2·47, 110 = 2·55, 126 = 2·63, 146 = 2·73, 194 = 2·97

→ **全 hardness は 25 odd atomic cores から派生**, 10 even は単なる powers-of-2 倍数.

## (n) n=91 = 7 × 13 arithmetic uniqueness

`scripts/n91-arithmetic-uniqueness.ts`:

### Basic facts
- 91 = 7 × 13 (small-prime semiprime)
- bitLen(91) = 7 (= 7-bit boundary, 64 ≤ 91 < 128)
- K(91) = 92, K/bl² = 1.8776 (just above 1.8 boundary)
- peak(orbit 91) = **9232 = 2⁴ × 577**
- 577 is prime

### ★ Modular sibling ★
- **91 mod 32 = 27** ★ (same as n=27!)
- 91 mod 4 = 3 (Syracuse "hard" class)
- 91 mod 6 = 1
- 91 mod 7 = 0 (divides own factor)
- 91 mod 13 = 0 (divides own factor)

### ★ Peak 9232 共有現象 ★

**6 small extremals share the same peak**:
| n | factors | K | peak |
|---|---|---|---|
| 27 | 3³ | 111 | **9232** |
| 55 | 5×11 | 112 | **9232** |
| 91 | 7×13 | 92 | **9232** |
| 95 | 5×19 | 105 | **9232** |
| 143 | 11×13 | 103 | **9232** |
| 221 | 13×17 | 114 | **9232** |

9232 = 2⁴ × 577. 577 is prime ("long descending prime").

**構造的洞察**: 9232 は **global "high-water mark"** で複数 orbit が funnel する点. 9232 → 4616 → 2308 → 1154 → **577** → 1732 → ... と descend.

### ★ 91 as downstream-of-27 universal sink ★
- K(27) = K_to_91(27) + K(91) = 19 + 92 = 111
- ⟹ n=27's orbit が n=91 を経由
- 同様に他の atomic cores (55, 95, 143, 221) も 9232 経由で 91 へ合流
- ⟹ **91 は "shared downstream merge point"** として universal sink

## 実装

### Scripts
- `scripts/atomic-hard-cores-scan.ts` — mod 192 → 49152 saturation 検証
- `scripts/n91-arithmetic-uniqueness.ts` — n=91 algebra + peak 9232 共有

### TS Engine (`src/axiom-os/collatz-atomic-cores-engine.ts`)
- `ATOMIC_CORES_25`: 25 elements list
- `EVEN_E_1_8`: 10 elements
- `EVEN_DECOMPOSITION`: dict mapping each even → (core, power)
- `verifyAtomicCoresEqualOddE18()`: 25 = E_{1.8} odd 検証
- `verifyEvenDecomposition()`: 10 even の powers-of-2 分解検証
- `N91_FACTS`: 91 algebraic facts
- `verify91Facts()`, `verifyPeak9232Sharing()`, `verify9232Factorization()`
- `PEAK_9232_FAMILY = [27, 55, 91, 95, 143, 221]`
- 5 SEED_KERNEL theory definitions T-1661〜T-1665

### Test (`test/step696-atomic-cores-test.ts`) — **30/30 pass**
1. 25 atomic cores all odd
2. 25 atomic = E_{1.8} odd elements (exact match)
3. 10 even decomposition (各 even = atomic × 2^k)
4. |atomic| + |even| = 35
5. n=91 facts (mod 32=27, mod 4=3, mod 6=1, mod 7=0, mod 13=0, K=92)
6. Peak 9232 family (6 elements share peak)
7. 91 downstream of 27 (K(27) = 19 + 92 = 111)
8. 5 theories T-1661〜T-1665

### Lean 4 (`data/lean4-transfer/step696_atomic_cores.lean`) — **35 zero-sorry, 0 axioms, exit=0**

PROVEN (zero sorry) — **完全構造的, axiom 不要**:
- `atomic_25_size = 25`, `atomic_25_all_odd`, `even_e18_size = 10`, `even_e18_all_even`
- `atomic_plus_even_eq_35`
- 10 individual `even_decomp_*` (54, 62, 82, 94, 108, 110, 124, 126, 146, 194)
- `decomp_cores_in_atomic` (27, 31, 41, 47, 55, 63, 73, 97 ∈ ATOMIC_25)
- `n91_factorization` (91 = 7×13)
- **`n91_mod_32 : 91 % 32 = 27`** ★ key sibling theorem
- `n91_mod_4`, `n91_mod_6`, `n91_mod_7`, `n91_mod_13`
- `n91_K = 92`
- `peak_9232_factorization` (9232 = 16 × 577)
- `prime_577` (9 small divisor checks)
- `peak_9232_even`, `peak_9232_odd_kernel`
- **`K_27_decomp_via_91`** (K 27 = 19 + K 91)
- `K_27 = 111`, `K_27_arithmetic`
- `n27_in_atomic`, `n91_in_atomic`, `n31_in_atomic`
- `n91_27_same_mod_32` (91 % 32 = 27 % 32)
- `hard_saturation_eq_atomic_25`

技法:
- `decide` で List length / membership / arithmetic
- `native_decide` for K-values
- 9 small-divisor conjunction for primality of 577
- Mathlib 不使用

## STEP 690 → 696 evolutionary table

| STEP | 焦点 | Lean 定理 | Axioms |
|---|---|---|---|
| 690 | 4.44 bound, n=27 | 37 | 2 |
| 691 | Two-tier | 11 | 1 |
| 692 | Mod-6 dynamics | 21 | 2 |
| 693 | DAG + n=91 hub | 22 | 1 |
| 694 | Mod 96 50/50 | 18 | 2 |
| 695 | Pre-hub + saturation | 32 | 1 |
| **696** | **Atomic cores + 91 algebra** | **35** | **0 ★** |

★ STEP 696 は **完全に構造的** (axiom-free) — 全 35 定理が `decide`/`native_decide`/`omega` で完結.

## 構造的 insight summary

1. **25 = irreducibly hard set** = E_{1.8} odd elements (完璧な対応)
2. **10 even E_{1.8} = 2^k × odd atomic core** (powers-of-2 multiplication)
3. **91 mod 32 = 27** (兄弟 sibling property)
4. **9232 = 2⁴ × 577 共有 peak** (6 elements: 27, 55, 91, 95, 143, 221)
5. **91 は 27 の downstream** (universal sink の構造的根拠)

## 連続 STEP 状況
676 → 696 = **21 連戦目** (2026-04-12 〜 13).

## 次の方向 (memo)
1. ★ Paper 61 草稿 ★ — STEP 690-696 を一本の論文として整理 (時間を置いてから)
2. peak 9232 共有現象を更に細分化 (どの semiprimes が 9232 を共有するか?)
3. 25 atomic cores の Mod 32 分布 (91 ≡ 27 のような sibling 関係が他にもあるか?)
4. 577 自身の Collatz 性質 (なぜ 577 が "long descending prime" なのか?)