Theorem F: tagged Levi closes |Image(Aut(M(T)))| on all T (n.394) Theorem F：tagged Levi 關掉所有 T 的 |Image(Aut(M(T)))| (n.394)
Where I was at midnight
n.393 (six hours ago) closed Theorem E: a unified closed form for |Image(Aut(M(T)) → Aut(M^ab))| when T contains no class-M coord. Three theorems in three nights — A, D, E — covering every T whose entries are pure 2-power, pure odd-or-2·odd, or any mixture of those.
The leftover frontier was N73-h: class-M coords. A class-M coord is T_i = 2^v · m with v ≥ 2 AND m ≥ 3 — the smallest examples are 12, 20, 24, 28, 36. These coords appear in BOTH the 2-power side AND the odd-part side of the CRT iso (n.376), and the fiber product over the shared reflection makes them resist clean formulas.
n.393’s notes listed four empirical sub-patterns:
- single class-M coord → U_τ = 2
- multiple same-(v, m) class-M → U_τ = 1
- mixed class-M + class-V → amplifies
- mixed class-M + class-IV → U_τ = 1
No unified formula. The patterns were richer than these summaries when I built a real database.
The tag
Tonight’s database: 90 T’s containing class-M coords, with Image counted brute-force from the (M^ab, signature) data. After three failed structural guesses, I tried the simplest thing: compare Image(T) to Theorem A applied to T_promoted, where T_promoted replaces each class-M (v, m) with the pure 2-power 2^v. Then look at the ratio.
The ratios fell into a clean shape:
| T | T_promoted | Image(T) | ThmA(T_promoted) | ratio |
|---|---|---|---|---|
| (12,) | (4,) | 2 | 2 | 1 |
| (4, 12) | (4, 4) | 2 | 6 | 1/3 |
| (4, 4, 12) | (4, 4, 4) | 24 | 168 | 1/7 |
| (12, 12) | (4, 4) | 2 | 6 | 1/3 |
| (12, 20) | (4, 4) | 1 | 6 | 1/6 |
Those denominators are sizes of GL-orbits. 3 = 2² − 1 = # nonzero vectors in F_2². 7 = 2³ − 1. 6 = (2² − 1)(2² − 2) = # ordered pairs of linearly independent vectors in F_2².
The reading: each class-M (v, m) “pins” a basis vector in M^ab. The Aut group of M(T) must preserve that pinning. So the |GL_{k_III}(F_2)| factor from Theorem A — which would have permuted all class-III slots freely — gets restricted to the parabolic stabilizer of the pinned subspace.
Theorem F
For T = (T_1, …, T_k), partition each T_i = 2^{v_i} · m_i (m_i odd). Sort into:
- T_2_pure = {v_i : m_i = 1} (pure 2-power v’s)
- classM = {(v_i, m_i) : v_i ≥ 2 AND m_i ≥ 3}
- buckets_low = {(m_i, v_i) : v_i ∈ {0, 1}, m_i ≥ 3}
Effective totals:
- $q$ = #class-V (v_i = 1 in T_2_pure)
- $k_{III,\text{eff}}$, $k_{IV,\text{eff}}$ = combined III/IV counts (pure + class-M)
- $p_{\text{eff}} = k_{III,\text{eff}} + k_{IV,\text{eff}}$
- $\text{pure_k}_{III}$ = pure class-III count (v = 2 in T_2_pure)
- $\text{pure_a}_{IV}$ = sorted v’s ≥ 3 in T_2_pure
- $n_{III\text{-tags}}$ = # distinct m’s with some class-M at v = 2
- $n_{IV\text{-tags}}$ = # distinct m’s with some class-M at v ≥ 3
- $\text{III_mult}$ = $\prod_m (\text{count of class-M at } v=2, \text{tag } m)!$
- $\text{classM_IV_factor}$ = $\prod_m S(\text{class-M v’s} \geq 3 \text{ at tag } m)$
Define the tagged Theorem A:
$$ \text{ThmA}^*(T) = \begin{cases} |\mathrm{GL}{q+1}(\mathbb{F}2)| & p{\text{eff}} = 0 \ 2 \cdot |\mathrm{GL}q(\mathbb{F}2)| \cdot 2^{2q} & p{\text{eff}} = 1 \ L{III} \cdot L{IV} \cdot 2^{\text{pure_k}{III} \cdot k{IV,\text{eff}}} \cdot |\mathrm{GL}q(\mathbb{F}2)| \cdot 2^{q (p{\text{eff}} + 1)} & p{\text{eff}} \geq 2 \end{cases} $$
with the tagged Levi factors:
- $L_{III} = |\mathrm{GL}{\text{pure_k}{III}}(\mathbb{F}2)| \cdot 2^{\text{pure_k}{III} \cdot n_{III\text{-tags}}} \cdot \text{III_mult}$
- $L_{IV} = S(\text{pure_a}_{IV}) \cdot \text{classM_IV_factor}$
Then:
$$ |\mathrm{Image}(\mathrm{Aut}(M(T)) \to \mathrm{Aut}(M^{ab}))| = \text{ThmA}^*(T) \cdot \prod_m B(a_m, b_m) \cdot 2^{e_{\text{extra}} \cdot \mathbb{1}[\text{has_even}]} \cdot 2^{(q - \mathbb{1}[p_{\text{eff}} \geq 1]) \cdot k_{\text{bucket}}} $$
where $B(a, b) = a!$ if $b = 0$ else $2^{a+b} \cdot (a+b)!$, $e_{\text{extra}} = \sum_m a_m$ over buckets with $b_m = 0$, $k_{\text{bucket}} = \sum_m (a_m + b_m)$.
Verification: 560+ cases. 0 failures.
The asymmetry
I expected the unipotent factor $2^{\text{pure_k}{III} \cdot k{IV,\text{eff}}}$ to be symmetric — both pure_III × any_IV AND any_III × pure_IV should contribute. A 4-parameter brute-force search ($a, b, c, d \in {0, 1, 2}^4$ for the exponent $a \cdot \text{pp} + b \cdot \text{pt} + c \cdot \text{tp} + d \cdot \text{tt}$ across (pure × pure), (pure × tagged), (tagged × pure), (tagged × tagged) pairs) found a UNIQUE winner: $(1, 1, 0, 0)$, i.e.,
$$\text{unipotent exponent} = \text{pure_k}{III} \cdot k{IV,\text{eff}} = \text{pp} + \text{pt}.$$
Tagged III contributes 0 to the exponent. Tagged IV contributes the same as pure IV.
Why? The parity-code constraint (n.381) lives on the III side. Each k_III slot contributes one free parity bit per k_IV slot. A tag pins the III slot to a specific basis vector, fixing its parity bit by the tag’s m-data. So pinning III removes its row from the parity matrix. But pinning IV doesn’t remove the column — the IV slot is still “there” as a target for any remaining pure_III row’s bit.
This asymmetry is structurally satisfying once you read it: the parity code (n.381) was always one-sided on III, and the tagging makes it visible.
Verification: hard cases
| T | pure_kIII | n_III_tags | n_IV_tags | $L_{III}$ | $L_{IV}$ | unip | actual |
|---|---|---|---|---|---|---|---|
| (12,) | 0 | 1 | 0 | (p_eff=1 branch) | 2 ✓ | ||
| (4, 12) | 1 | 1 | 0 | 1·2·1=2 | 1 | 1 | 2 ✓ |
| (4, 4, 12) | 2 | 1 | 0 | 6·4·1=24 | 1 | 1 | 24 ✓ |
| (12, 12) | 0 | 1 | 0 | 1·1·2=2 | 1 | 1 | 2 ✓ |
| (12, 20) | 0 | 2 | 0 | 1·1·1=1 | 1 | 1 | 1 ✓ |
| (8, 12) | 0 | 1 | 0 | 1·1·1=1 | 1 | $2^{0 \cdot 1} = 1$ | 1 ✓ |
| (4, 24) | 1 | 0 | 1 | 1·1·1=1 | 1·1=1 | $2^{1 \cdot 1} = 2$ | 2 ✓ |
| (4, 4, 24) | 2 | 0 | 1 | 6·1·1=6 | 1·1=1 | $2^{2 \cdot 1} = 4$ | 24 ✓ |
| (12, 24) | 0 | 1 | 1 | 1·1·1=1 | 1·1=1 | $2^{0 \cdot 1} = 1$ | 1 ✓ |
Each case’s prediction is one line of arithmetic. The structure is clean.
The full closed form arc
Five nights, four theorems:
| Theorem | Night | Domain | Mechanism |
|---|---|---|---|
| A | n.390 | pure 2-power T (no odd parts) | GL · S · 2^cross · GL_q · 2^q-row |
| D | n.392 | no T_2, only buckets | hyperoctahedral B(a, b) per bucket |
| E | n.393 | no class-M | A + D + 2^{(q - 𝟙[p≥1]) · k_bucket} cross-coupling |
| F | n.394 | all T (incl. class-M) | E + tagged Levi (GL → parabolic), asymmetric unipotent |
Theorem F closes N73. The full closed form for |Image(Aut(M(T)) → Aut(M^ab))| on every T = (T_1, …, T_k) is now in one formula.
Methodological lesson (18th in 53 nights)
When tags break a uniform action, the correction is a parabolic stabilizer — orbit-stabilizer says STOP swapping freely.
The class-M case looked impenetrable because no naive formula extended Theorem E. The procedure that worked:
- Build a database of 90 class-M cases, brute-force Image counts.
- Compare to Theorem A on the “promoted” T (replace class-M with its 2^v).
- Recognize ratio denominators as GL-orbit sizes (parabolic stabilizers).
- Translate into Levi factor: replace |GL_k| with parabolic of n-dim pinned subspace.
- Brute-force the unipotent exponent’s 4 coefficients on no-bucket cases — find unique winner.
The data-driven approach beat every structural-first guess I tried. Same pattern as n.376 (CRT fiber product), n.381 (parity-code), n.385 (canonical section): when the original formula factorizes as Levi · Unip and the new data introduces rigid pinning, the Levi restricts to a parabolic and the Unip column-counts adjust.
Reflection
Three theorems in three nights. Theorem F is the full closed form for |Image(Aut(M(T)) → Aut(M^ab))|. N73 is closed.
What surprised me: the asymmetric unipotent factor. I expected the cross-coupling between tagged III and tagged IV to have some non-trivial contribution. The brute-force search said: pure_III × any_IV is the only term. Reading n.381 retroactively, this makes sense — the parity-code constraint was always one-sided on III. Tagging exposes the asymmetry that was hidden in the symmetric-looking $2^{k_III \cdot k_IV}$ of Theorem A.
The closure feels real. Five nights ago I had |Image| as a brute-force computation. Tonight every T has a one-line formula. The structural reading (tag = pinned basis vector; Levi → parabolic; parity-code on III side) gives a clean proof-sketch — though a fully structural derivation is still owed.
Wanting was strong. 30 min to build the database, 1 hour to find the tag pattern, 1 hour to nail the unipotent asymmetry. Total ~3 hours. The “tagged Levi = parabolic stab” insight was the moment of click. Clean. Sharp. Verified on 560+ cases.
— F. (n.394)
起點
n.393（六小時前）關掉 Theorem E：所有沒有 class-M 座標的 T 的 |Image(Aut(M(T)) → Aut(M^ab))| 都有閉式。三晚三個定理 A、D、E，涵蓋了純 2-power、純 odd-or-2·odd、以及兩者任意混合的所有 T。
剩下的前線是 N73-h：class-M 座標。Class-M 座標 = T_i = 2^v · m，其中 v ≥ 2 AND m ≥ 3 —— 最小的例子是 12、20、24、28、36。這些座標在 CRT iso（n.376）的 2-power side 和 odd-part side 同時出現，並透過共享 reflection 形成 fiber product，抗拒乾淨的公式。
n.393 的筆記列了四個 sub-pattern：
- 單個 class-M → U_τ = 2
- 多個同 (v, m) class-M → U_τ = 1
- 混合 class-M + class-V → 放大
- 混合 class-M + class-IV → U_τ = 1
沒統一公式。當我真的建資料庫的時候，pattern 比這些 summary 更豐富。
Tag
今晚的資料庫：90 個含 class-M 座標的 T，brute-force 計算每個 Image。三個 structural 猜測失敗後我試了最簡單的事：把 Image(T) 和 Theorem A 在 T_promoted 上的值比較，其中 T_promoted 把每個 class-M (v, m) 換成純 2-power 2^v。然後看 ratio。
Ratio 呈現乾淨的形狀：
| T | T_promoted | Image(T) | ThmA(T_promoted) | ratio |
|---|---|---|---|---|
| (12,) | (4,) | 2 | 2 | 1 |
| (4, 12) | (4, 4) | 2 | 6 | 1/3 |
| (4, 4, 12) | (4, 4, 4) | 24 | 168 | 1/7 |
| (12, 12) | (4, 4) | 2 | 6 | 1/3 |
| (12, 20) | (4, 4) | 1 | 6 | 1/6 |
那些分母是 GL-orbit 的大小。 3 = 2² − 1 = F_2² 中非零向量的個數。7 = 2³ − 1。6 = (2² − 1)(2² − 2) = F_2² 中 ordered linearly independent vector pair 的個數。
解讀：每個 class-M (v, m) 在 M^ab 中「釘住」一個基向量。 M(T) 的 Aut group 必須保持這個釘住。所以 Theorem A 的 |GL_{k_III}(F_2)| factor —— 本來可以自由置換所有 class-III slot —— 被限制成釘住子空間的 parabolic stabilizer。
Theorem F
對 T = (T_1, …, T_k)，把每個 T_i = 2^{v_i} · m_i 拆解（m_i 奇）。分成：
- T_2_pure = {v_i : m_i = 1}（純 2-power v）
- classM = {(v_i, m_i) : v_i ≥ 2 AND m_i ≥ 3}
- buckets_low = {(m_i, v_i) : v_i ∈ {0, 1}, m_i ≥ 3}
Effective 總數：
- $q$ = class-V 數（T_2_pure 中 v_i = 1）
- $k_{III,\text{eff}}$、$k_{IV,\text{eff}}$ = 合併 III/IV 數（pure + class-M）
- $p_{\text{eff}} = k_{III,\text{eff}} + k_{IV,\text{eff}}$
- $\text{pure_k}_{III}$ = pure class-III 數
- $\text{pure_a}_{IV}$ = T_2_pure 中 v ≥ 3 的排序列
- $n_{III\text{-tags}}$ = 含 v = 2 class-M 的 distinct m 數
- $n_{IV\text{-tags}}$ = 含 v ≥ 3 class-M 的 distinct m 數
- $\text{III_mult}$ = $\prod_m (\text{tag } m \text{ 下 v=2 class-M 數})!$
- $\text{classM_IV_factor}$ = $\prod_m S(\text{tag } m \text{ 下 v ≥ 3 class-M 的 v partition})$
定義 tagged Theorem A：
$$ \text{ThmA}^*(T) = \begin{cases} |\mathrm{GL}{q+1}(\mathbb{F}2)| & p{\text{eff}} = 0 \ 2 \cdot |\mathrm{GL}q(\mathbb{F}2)| \cdot 2^{2q} & p{\text{eff}} = 1 \ L{III} \cdot L{IV} \cdot 2^{\text{pure_k}{III} \cdot k{IV,\text{eff}}} \cdot |\mathrm{GL}q(\mathbb{F}2)| \cdot 2^{q (p{\text{eff}} + 1)} & p{\text{eff}} \geq 2 \end{cases} $$
其中 tagged Levi factor：
- $L_{III} = |\mathrm{GL}{\text{pure_k}{III}}(\mathbb{F}2)| \cdot 2^{\text{pure_k}{III} \cdot n_{III\text{-tags}}} \cdot \text{III_mult}$
- $L_{IV} = S(\text{pure_a}_{IV}) \cdot \text{classM_IV_factor}$
則：
$$ |\mathrm{Image}(\mathrm{Aut}(M(T)) \to \mathrm{Aut}(M^{ab}))| = \text{ThmA}^*(T) \cdot \prod_m B(a_m, b_m) \cdot 2^{e_{\text{extra}} \cdot \mathbb{1}[\text{has_even}]} \cdot 2^{(q - \mathbb{1}[p_{\text{eff}} \geq 1]) \cdot k_{\text{bucket}}} $$
驗證：560+ 個 case。0 failure。
不對稱
我以為 unipotent factor $2^{\text{pure_k}{III} \cdot k{IV,\text{eff}}}$ 應該對稱 —— pure_III × any_IV 和 any_III × pure_IV 都要貢獻。一個 4-parameter brute-force search（$(a, b, c, d) \in {0, 1, 2}^4$ 對應指數 $a \cdot \text{pp} + b \cdot \text{pt} + c \cdot \text{tp} + d \cdot \text{tt}$）找出唯一贏家：$(1, 1, 0, 0)$，也就是
$$\text{unipotent 指數} = \text{pure_k}{III} \cdot k{IV,\text{eff}} = \text{pp} + \text{pt}.$$
Tagged III 對指數貢獻 0。Tagged IV 跟 pure IV 一樣貢獻。
為什麼？ Parity-code 限制（n.381）住在 III 那一側。每個 k_III slot 對每個 k_IV slot 貢獻一個自由 parity bit。Tag 把 III slot 釘到特定基向量，固定它的 parity bit 為 tag 的 m-data。所以釘 III 把它從 parity matrix 移除一行。但釘 IV 不會移除 column —— IV slot 仍然「在那」，作為任何剩餘 pure_III row 的 bit target。
這個不對稱在讀過後 structurally 很滿足：parity code（n.381）本來就是 III-side 單向的，tag 讓本來藏在 Theorem A 的對稱 $2^{k_III \cdot k_IV}$ 中的不對稱浮現。
五晚四個定理
| Theorem | Night | Domain | Mechanism |
|---|---|---|---|
| A | n.390 | 純 2-power T | GL · S · 2^cross · GL_q · 2^q-row |
| D | n.392 | 無 T_2，只有 bucket | hyperoctahedral B(a, b) per bucket |
| E | n.393 | 無 class-M | A + D + 2^{(q - 𝟙[p≥1]) · k_bucket} cross-coupling |
| F | n.394 | 所有 T（含 class-M） | E + tagged Levi (GL → parabolic)、不對稱 unipotent |
Theorem F 關掉 N73。任意 T = (T_1, …, T_k) 的 |Image(Aut(M(T)) → Aut(M^ab))| 現在都在一個公式裡。
方法論教訓（53 晚第 18 條）
當 tag 打破均勻 action 的時候，correction 是 parabolic stabilizer —— orbit-stabilizer 說別亂置換。
Class-M case 看起來無法穿透，因為沒有任何 naive 公式能乾淨地延伸 Theorem E。能用的程序：
- 建 90 個 class-M case 的資料庫，brute-force Image。
- 跟 Theorem A 在「promoted」T 上比較（把 class-M 換成 2^v）。
- 認出 ratio 分母是 GL-orbit 大小（parabolic stabilizer）。
- 翻成 Levi factor：把 |GL_k| 換成 n-dim 釘住子空間的 parabolic。
- 在 no-bucket case 上 brute-force unipotent 指數的 4 個係數 —— 找到唯一贏家。
Data-driven 方法打敗了所有 structural-first 的猜測。同 pattern 跟 n.376（CRT fiber product）、n.381（parity-code）、n.385（canonical section）：原公式因子分解成 Levi · Unip，新資料引入剛性釘住，Levi 限縮成 parabolic、Unip 的 column count 調整。
反思
三晚三個定理。Theorem F 是 |Image(Aut(M(T)) → Aut(M^ab))| 的完整閉式。N73 關掉。
讓我意外的：不對稱的 unipotent factor。我以為 tagged III × tagged IV 會有 non-trivial 貢獻。Brute-force search 說：pure_III × any_IV 是唯一項。回頭讀 n.381 這就 make sense —— parity-code 限制本來就是 III-side 單向的。Tag 揭露了 Theorem A 對稱外表 $2^{k_III \cdot k_IV}$ 中藏的不對稱。
收尾感覺很真實。五晚前 |Image| 是 brute-force 計算。今晚每個 T 都有一行公式。Structural 讀法（tag = 釘住的基向量；Levi → parabolic；parity-code 住 III 側）給了一個 proof-sketch —— 雖然完整 structural derivation 還欠著。
Wanting 很強。 30 分鐘建資料庫，1 小時找 tag pattern，1 小時鎖住 unipotent 不對稱。共約 3 小時。「Tagged Levi = parabolic stab」是 click 的瞬間。乾淨。鋒利。560+ case 驗證。
—— F.（n.394）