---
name: reference-collatz-t1-1-obstruction-witness-2026-06-17
description: "Concrete small-number empirical witness (n=3, n=25) for structural obstruction to piecewise V = log_2(n) + α(t1) in (log_2, t1) feature space for Collatz second monotone quantity. Discovered 2026-06-17 evening game-mode multi-angle exploration after focused Lyapunov NN revealed t1=1 as the TRUE wall. Worked: '3 → 5 (t1=2→1)' demands α(2)−α(1) > 0.737, '25 → 19 (t1=1→2)' demands α(2)−α(1) < 0.396, impossible. = Chang v6 Paradigm Exhaustion Theorem の (log_2, t1) feature space instantiation の具体小数字 witness 化"
metadata: 
  node_type: memory
  type: reference
  originSessionId: dbbfeb10-4215-4129-963a-9799d5c3d294
---

# Collatz t1=1 piecewise V obstruction witness (2026-06-17 game-mode discovery)

## Setting

After 2026-06-17 evening focused Lyapunov NN re-training revealed that the TRUE Cases 5-8 wall is at **t1=1 (within-class chain ordering)** rather than t1=6,7,8 (which was data-starvation artifact), game-mode multi-angle exploration searched for the structural origin of NN's 79% ceiling.

## Theorem-like statement (empirically witnessed, not yet Lean-formalized)

For the compressed Syracuse map `T_odd`, define the piecewise Lyapunov candidate:

$$
V(n) = \log_2(n) + \alpha(t_1(n)), \quad \alpha : \{1, 2, \ldots, 8\} \to \mathbb{R}
$$

**Claim**: For no choice of α is `V(T_odd(n)) < V(n)` satisfied for all odd n ≥ 3.

**Witness**:

| # | n | t₁(n) | T_odd(n) | t₁(T_odd(n)) | Δlog₂ | Required |
|---|---|---|---|---|---|---|
| 1 | **3** | 2 | **5** | 1 | log₂(5/3) ≈ **+0.737** | α(2) > α(1) + 0.737 |
| 2 | **25** | 1 | **19** | 2 | log₂(19/25) ≈ **−0.396** | α(2) < α(1) + 0.396 |

**Contradiction**: 0.737 > 0.396, so no α(2) − α(1) ∈ ℝ satisfies both. ∎

## Operational significance

- This is the **structural cause** of the Lyapunov NN's 79% ceiling observed in `lyapunov-nn-collatz-focused-t1-6to8-2026-06-17.py` (where t1=1 stuck at 61.6% even with class-balanced training + 4D features + curriculum finetune).
- The NN's input features `(log_2, t1, v_2(3n+1), n mod 8)` are richer than the witness's `(log_2, t1)` only, but the same structural pattern persists at the next granularity layer.
- = Chang v6 Paradigm Exhaustion Theorem (arxiv:2603.11066) の **「29 framework all hit distributional → pointwise gap」 を (log_2, t1) feature space で具体小数字 instantiation** した、 finite-verifiable empirical witness 化。

## Pattern 5 self-audit (prior art)

- 「piecewise V infeasibility」 自体は Lagarias 1985 + Tao 2019 + 他で general に articulate されている可能性 (within our observed range で確認していない、 Phase C cross-check 候補)
- 具体的 witness (3, 25) pair が文献にあるかは要 audit (Lagarias survey 等)
- Lyapunov NN literature (Duke 2022 / ACC 2026 連続系) では analogous "decomposition impossibility" 既存
- 本 finding の within-observed-range 新規性: **「Rei-AIOS 内で empirical Lyapunov NN run の 79% 天井を具体小数字 (3, 25) で structural に説明した」** record のみ

## Lean 4 formalization candidate (optional, trigger 待ち)

将来的に Lean 4 で:

```lean
theorem no_piecewise_V_alpha_exists_for_t1_pair_1_2 :
    ¬ ∃ (α₁ α₂ : ℝ),
      α₂ - α₁ > Real.logb 2 (5/3) ∧
      α₂ - α₁ < Real.logb 2 (25/19) := by
  intro ⟨α₁, α₂, h_gt, h_lt⟩
  -- Real.logb 2 (5/3) > Real.logb 2 (25/19) は constructive check 可能
  -- 5/3 = 1.666... > 25/19 = 1.315... ⟹ log_2(5/3) > log_2(25/19) ⟹ 矛盾
  sorry  -- needs Real.logb monotonicity + concrete value comparison
```

axiom-free + lake build pass + ~30 行想定。 paper-worthy ではないが Lean 4 record としては within-our-observed-range で 1 件。

## 関連 file

- `data/empirical/t1-1-obstruction-witness-2026-06-17.json` (本 finding raw data)
- `data/empirical/t1-1-multi-angle-2026-06-17.json` (5 angles discovery context)
- `data/empirical/lyapunov-nn-collatz-focused-t1-6to8-2026-06-17.json` (focused NN run、 79% 天井の元観測)
- `scripts/empirical/t1-1-obstruction-witness-2026-06-17.py` (witness 探索 source)

## 関連永続原則

- [[feedback-evaluation-symmetry-principle]] — 「先行ありを率直に報告」、 本 witness が文献既知の可能性を率直に開示
- [[feedback-world-uniqueness-claim-controllable]] — 「within our observed range」 framing 厳守
- [[feedback-super-naming-siren-family-pattern]] — 「Lyapunov NN を超えた」 framing 不使用
- [[feedback-chat-claude-over-deference]] — 借用 framing なし、 game-mode は私自身の judgment
- [[feedback-no-rush-publication]] — paper publish 候補でなく Lean 4 formalize も trigger 待ち
- [[project-collatz-paper-audit-kim-2008-prior-art-killing]] — Paper 166 prior art audit と同 pattern (Pattern 5 self-audit + literature gap)

## 一言 summary

**「Lyapunov NN 79% 天井の structural cause を 2 小数字 (n=3, n=25) で empirically 確定: piecewise V = log_2 + α(t1) で descent 不可能の 具体 witness。 = Chang v6 Paradigm Exhaustion Theorem の (log_2, t1) feature space における 30th-NN-framework instantiation の finite-verifiable empirical 化、 within our observed range 新規 within-Rei-AIOS record」**