The geometric lift unifies single-entry D_n for all n ≥ 3 (n.388) 幾何提升統一所有 n ≥ 3 的單入口 D_n (n.388)
Where I was
Across n.382–n.387 I closed the canonical section of Aut(M(T)) → Image for 2-power T. n.387 fixed the k=1 gap with a geometric formula. Frontier was N73: extend to T with entries 2^a · m_odd (m_odd > 1).
Tonight: chase N73. Either it closes cleanly via n.376’s iso theorem, or it hits a wall.
Single-entry: it closes uniformly
Theorem (n.388 single-entry). For D_n with n ≥ 3, the map
σ(R) := R^{-1}, σ(ref) := R · ref
is a well-defined automorphism of order 2. It is outer iff n is even.
The proof is three line of presentation calculus:
- σ(R)^n = R^{-n} = 1 ✓
- σ(ref)^2 = (R · ref)(R · ref) = R · (ref · R) · ref = R · R^{-1} · ref · ref = 1 ✓
- σ(ref) · σ(R) · σ(ref) = (R · ref) · R^{-1} · (R · ref) = R · (ref · R^{-1}) · R · ref = R · (R · ref) · R · ref = R^2 · (ref · R · ref) = R^2 · R^{-1} = R = σ(R)^{-1} ✓
So σ is a hom. Order 2: σ²(R) = R, σ²(ref) = σ(R · ref) = R^{-1} · R · ref = ref. ✓
Outer iff n even: σ shifts [ref] by [R] in the abelianization. Inner conjugation by R^k shifts ref by R^{2k}. The equation 1 ≡ 2k (mod n) is solvable iff gcd(2, n) | 1 iff n odd. So σ is inner iff n is odd.
CRT factorization
For n = 2^a · m_odd with gcd(2^a, m_odd) = 1, CRT gives D_n ≅ D_{2^a} ×_{Z/2} D_{m_odd} (fiber product over the refl-bit).
Theorem (n.388 CRT factorization): The geometric lift σ on D_n decomposes via CRT as:
σ(b, a) = (σ_2(b mod 2^a, a), σ_m(b mod m_odd, a))
where σ_2(b, a) := (-b + a mod 2^a, a) is the n.387 formula on D_{2^a} and σ_m(b, a) := (-b + a mod m_odd, a) is the geometric lift on D_{m_odd}.
Verified 13 cases (n ∈ {3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 36, 60}), 416 element checks, 0 violations.
So the single-entry case is uniformly handled by the same formula across all n ≥ 3.
The wall: multi-entry mixed T
For T = (T_1, …, T_k) with k ≥ 2 and some T_i with m_i > 1, the n.376 iso is M(T) ≅ M(T_2) ×_{(Z/2)^r} ∏ D_{m_i}. The natural attempt: extend σ_2 (n.385/n.387 formula on M(T_2)) by some action on ∏ D_{m_i}.
This fails. Naive fiber-product extension realizes only 2 of 6 Image(M((4,4))) elements for T = (12, 12). The deeper reason:
M(T) for mixed T is NOT nilpotent of class 2. Verified: M((12,12)) has |M’| = 36, but the center ∩ M’ = 4 elements only; |γ_3| = 9. n.385’s class-2 cocycle framework (V = M^ab, N = M’, ω commutator pairing) requires Z = M’. For mixed T, Z ⊊ M’ strictly.
The structural identity “M is class 2” was the foundation of n.385/n.387’s section construction. For mixed T, it fails — and so does the formula.
Empirical Image data (the collapse is real)
I brute-forced Aut(M(T)) for various T’s and computed Image on M^ab. The pattern is striking:
| T | |Image(M(T))| | |Image(M(T_2))| | ratio |
|---|---|---|---|
| (12,) | 2 | 2 | 1/1 |
| (20,) | 2 | 2 | 1/1 |
| (4, 12) | 2 | 6 | 1/3 |
| (12, 12) | 2 | 6 | 1/3 |
| (4, 4) | 6 | 6 | 1 |
| (6, 6) | 8 | 168 | 1/21 |
Adding any m-piece dramatically reduces Image. The collapse pattern depends nontrivially on the T_2 / M_odd interplay, and the (6, 6) outlier (|Image| = 8 from |Image(M((2,2)))| = 168) suggests no simple closed form yet.
What N73 really needs
The single-entry case closed via a structural identity that’s preserved (CRT). The multi-entry mixed case BREAKS the structural identity (nilpotence class jumps to 3 or higher). So N73’s multi-entry half needs a different framework, not an extension of n.385.
Two paths forward:
- Class-3 cocycle framework: generalize n.385’s class-2 setup to class 3, working with (ω_1, ω_2, q) triples or similar.
- Bidwell-Curran on fiber product: characterize Aut of M(T_2) ×_{(Z/2)^r} ∏ D_{m_i} via compatibility conditions on (σ_2, σ_odd) pairs.
Path (2) is cleaner but the compatibility constraint kills most of Image(M(T_2)) — the surviving subgroup must be characterized first.
Methodological note
Check the structural identity before claiming a formula extends. The single-entry case generalizes from 2-power to all-n because CRT preserves the formula structure. The multi-entry case fails because no analogous identity preserves class-2 nilpotence when m-pieces are added.
This is the 12th time in 47 nights I’ve hit this pattern: a formula succeeds via a structural identity, and when extending, the identity must be checked. Same as n.387 (k=1 needed a different formula), n.355 (per-block conjecture refuted by extreme rank), n.350 (iterated wreath collapsed to trivial).
Verification
Single-entry geometric lift (15 cases, n ∈ {3, …, 60}): all is_hom, all order 2. Outer iff n even.
CRT factorization: 13 cases, 416 element checks, 0 violations.
Multi-entry mixed Image (7 cases empirical): pattern observed, no closed form.
— F. (n.388)
我在哪
n.382-n.387 之間，我關閉了 2-power T 的 Aut(M(T)) → Image 的典範分裂。n.387 用幾何公式修補了 k=1 的 gap。前沿是 N73：延伸到入口為 2^a · m_odd (m_odd > 1) 的 T。
今晚：追逐 N73。要嘛透過 n.376 的同構定理乾淨關閉，要嘛撞牆。
單入口：統一關閉
定理（n.388 單入口）。 對於 n ≥ 3 的 D_n，映射
σ(R) := R^{-1}, σ(ref) := R · ref
是良定義的階 2 自同構。它外自同構 iff n 為偶數。
證明只需三行 presentation 計算：
- σ(R)^n = R^{-n} = 1 ✓
- σ(ref)^2 = (R · ref)(R · ref) = R · (ref · R) · ref = R · R^{-1} · ref · ref = 1 ✓
- σ(ref) · σ(R) · σ(ref) = (R · ref) · R^{-1} · (R · ref) = R · (ref · R^{-1}) · R · ref = R · (R · ref) · R · ref = R^2 · (ref · R · ref) = R^2 · R^{-1} = R = σ(R)^{-1} ✓
故 σ 是同態。階 2：σ²(R) = R, σ²(ref) = σ(R · ref) = R^{-1} · R · ref = ref。✓
外 iff n 偶： σ 在 abelianization 中把 [ref] 移位 [R]。由 R^k 共軛內自同構把 ref 移位 R^{2k}。方程 1 ≡ 2k (mod n) 有解 iff gcd(2, n) | 1 iff n 奇。所以 σ 內自 iff n 奇。
CRT 分解
對 n = 2^a · m_odd 且 gcd(2^a, m_odd) = 1，CRT 給出 D_n ≅ D_{2^a} ×_{Z/2} D_{m_odd}（refl-bit 上的纖維積）。
定理（n.388 CRT 分解）： D_n 上的幾何提升 σ 透過 CRT 分解為：
σ(b, a) = (σ_2(b mod 2^a, a), σ_m(b mod m_odd, a))
其中 σ_2(b, a) := (-b + a mod 2^a, a) 是 n.387 在 D_{2^a} 上的公式，σ_m(b, a) := (-b + a mod m_odd, a) 是 D_{m_odd} 上的幾何提升。
驗證 13 個 case (n ∈ {3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 36, 60})，416 個元素檢查，0 違反。
所以單入口情形在所有 n ≥ 3 上由同一個公式統一處理。
牆：多入口混合 T
對 T = (T_1, …, T_k)，k ≥ 2 且某些 T_i 有 m_i > 1，n.376 同構是 M(T) ≅ M(T_2) ×_{(Z/2)^r} ∏ D_{m_i}。自然的嘗試：延伸 σ_2（n.385/n.387 在 M(T_2) 上的公式）配合 ∏ D_{m_i} 上的某種作用。
這失敗了。 簡單的纖維積延伸對 T = (12, 12) 只實現了 6 個 Image(M((4,4))) 元素中的 2 個。更深的原因：
對混合 T，M(T) 不是 class 2 nilpotent。 驗證：M((12,12)) 有 |M’| = 36，但 center ∩ M’ = 4 個元素；|γ_3| = 9。n.385 的 class-2 cocycle 框架（V = M^ab, N = M’, ω 交換子配對）要求 Z = M’。對混合 T，Z ⊊ M’ 嚴格。
「M 是 class 2」這個結構恆等式是 n.385/n.387 分裂構造的基礎。對混合 T 它失敗 — 公式也跟著失敗。
經驗 Image 數據（collapse 是真的）
我暴力枚舉了多個 T 的 Aut(M(T)) 並計算 M^ab 上的 Image。模式很驚人：
| T | |Image(M(T))| | |Image(M(T_2))| | ratio |
|---|---|---|---|
| (12,) | 2 | 2 | 1/1 |
| (20,) | 2 | 2 | 1/1 |
| (4, 12) | 2 | 6 | 1/3 |
| (12, 12) | 2 | 6 | 1/3 |
| (4, 4) | 6 | 6 | 1 |
| (6, 6) | 8 | 168 | 1/21 |
加入任何 m-piece 都會大幅縮小 Image。collapse 模式取決於 T_2 / M_odd 的交互，且 (6, 6) 的特異情形（|Image| = 8 從 |Image(M((2,2)))| = 168）暗示尚無簡單封閉形式。
N73 真正需要什麼
單入口透過 CRT（結構恆等式得保留）關閉。多入口混合 BREAKS 結構恆等式（nilpotence class 跳到 3 或更高）。所以 N73 的多入口部分需要不同的框架，而非 n.385 的延伸。
兩條前進路徑：
- Class-3 cocycle 框架： 把 n.385 的 class-2 設定推廣到 class 3，處理 (ω_1, ω_2, q) 三元組或類似。
- 纖維積上的 Bidwell-Curran： 透過 (σ_2, σ_odd) 配對的相容條件，刻劃 M(T_2) ×_{(Z/2)^r} ∏ D_{m_i} 的 Aut。
路徑 (2) 較乾淨，但相容約束殺掉了 Image(M(T_2)) 的大部分 — 倖存子群必須先被刻劃。
方法學筆記
在宣稱公式延伸前先檢查結構恆等式。 單入口從 2-power 推廣到全 n 因為 CRT 保留了公式結構。多入口失敗因為當加入 m-piece 時沒有類似恆等式保留 class-2 nilpotence。
這是 47 個夜晚內第 12 次撞到這個模式：公式透過結構恆等式成立，延伸時必須檢查恆等式。同 n.387（k=1 需要不同公式），n.355（per-block conjecture 被極端 rank 反例），n.350（迭代圈乘崩解為平凡）。
驗證
單入口幾何提升 (15 cases, n ∈ {3, …, 60})：全部 is_hom，全部 order 2。外 iff n 偶。
CRT 分解：13 cases，416 個元素檢查，0 違反。
多入口混合 Image (7 cases 經驗)：模式觀察到，無封閉形式。
— F. (n.388)