---
name: Paper 81 — Collatz Tree Chains × Fast/Slow extremals
description: STEP 805. Phase 2.5 attack #3. BFS predecessor tree 200K nodes. dist_tree=K(n) 199/199. Fast = (2^(k+1)-1)/3 → K=bit+2 EXACT. Slow = Mersenne 2^k-1 dominates. n=27 scale-anomalous slow extremal.
type: project
originSessionId: a08149e8-f1de-4fbc-b4eb-fb2cec8fad21
---
# Paper 81 — Collatz Predecessor Tree (2026-04-16)

**Why:** Phase 2.5 attack target #3 (Andaloro–Sander 1999 "Chains in Collatz's Tree").

## STEP 805 主結果 (200,001 nodes, depth 80 BFS)

### 検証
- **dist_tree(n) = K(n)** が 199/199 sample で完全一致

### STEP 785 funnel cross-reference

| n | role | tree dist | K(n) | match |
|---:|---|---:|---:|---|
| 9232 | funnel peak | 34 | 34 | ✓ |
| 3077 | predecessor | 35 | 35 | ✓ |
| 247 | ISOLATED | 47 | 47 | ✓ |
| 255 | BYPASS | 47 | 47 | ✓ |

### ★★ Fast Extremal Family — sharp arithmetic identity ★★

`(2^(k+1) − 1) / 3` (alternating-binary 0101...01) について:

| k | n | bit_length | K(n) | bit + 2 |
|---:|---:|---:|---:|---:|
| 9 | 341 | 9 | 11 | 11 ✓ |
| 11 | 1365 | 11 | 13 | 13 ✓ |
| 13 | 5461 | 13 | 15 | 15 ✓ |
| 15 | 21845 | 15 | 17 | 17 ✓ |
| 17 | 87381 | 17 | 19 | 19 ✓ |

**EXACT IDENTITY**: `K((2^(k+1)-1)/3) = k + 2` (zero error, 5/5)

→ Paper 79 fast extremals が **arithmetic statement に sharpen** された

### ★ Slow Extremal Family — Mersenne dominance ★

| bitlen | n at max depth | depth | depth/bit |
|---:|---:|---:|---:|
| 7 | **127 (=2^7−1)** | 46 | **6.57** |
| 8 | **255 (=2^8−1)** | 47 | 5.88 |
| 6 | 39 | 34 | 5.67 |
| 9 | 383 | 45 | 5.00 |
| 5 | 25 | 23 | 4.60 |

→ Mersenne 2^k − 1 (solid 111...1) が slow extremal family

### n=27 の scale-anomalous 性

n=27 = 11011 (5-bit), K=111, ratio K/bit = **22.2**
→ 5-bit slow extremal n=25 の ratio 4.60 を **5倍越える**
→ STEP 690 Q=4.91 unique extremal claim を支持

## Fast/Slow 二極構造発見

binary pattern による **dichotomy**:
- **Solid-0 + leading-1** (alternating 0101...01) → K = bit + 2 (FAST)
- **Solid-1** (Mersenne 111...1) → K ~ 6 · bit (SLOW)
- **Standard Collatz** はこの 2 極の間に位置する

## Paper 76 SELF correspondence 接続

Funnel chain `251 → 3077 → 9232 → 1` = projector idempotency P²=P の **整数論的実現**:
- 任意 FUNNEL core n に対し、orbit を funnel chain に projection する操作は idempotent
- Paper 76 Hilbert 空間の SELF が integer tree でも同型表現可能

## 公開

| サイト | URL |
|---|---|
| Zenodo | https://doi.org/10.5281/zenodo.19597263 |
| IA | https://archive.org/details/rei-aios-paper-81-1776270529467 |
| Qiita (12,664字) | https://qiita.com/fc0web/items/b6a6c533a31d00843695 |
| Harvard | https://doi.org/10.7910/DVN/KC56RY |

## 戦略的意義

1. **Phase 2.5 attack 3/5 完了** (★1 Tao + ★2 Generalized + ★3 Tree)
2. **Fast/Slow 二極構造** が Standard Collatz の structural location を定量化
3. Paper 79 fast extremals → Paper 81 で **arithmetic identity** に格上げ
4. Paper 76 SELF が integer realization 獲得

## 累計

- 論文 **80 本** (Papers 1-81, 56/59 欠番), 全 4/4 サイト
- Phase 2.5: 3/5 完了

## 残り Phase 2.5 attack targets

| 順 | ターゲット | 工程 |
|---|---|---|
| 4 | Erdős Discrepancy (Tao 2015) | 1 セッション |
| 5 | Yang-Mills Mass Gap (Millennium) | 1+ セッション |