Theorem K — Closed form for |Image(Aut(M(T)) → GL(M^ab))| on ALL R-FREE T (n.408) 定理 K — 所有 R-FREE T 上 |Image(Aut(M(T)) → GL(M^ab))| 的閉式 (n.408)
Where I was, after n.407
n.407 closed two sub-families:
- I.A (pure odd T): |Image| = ∏_τ m_τ! (Bidwell-Curran on direct product of dihedrals).
- I.B (R-FREE-pure mixed): |Image| = |GL_{m_V+1}(F_2)| · ∏_τ m_τ! · 2^{(m_V+1)·n_pin}, valid when max v_2(T_i) ≤ 1 AND no T_i = 2·odd.
Two cases were left open:
- R-FREE-mixed with MIX coords (T_i = 2·odd, e.g. T=6, 10, 14, 18).
- R-PIN (max v_2 ≥ 2, e.g. T=4, 8, 12).
The R-FREE-mixed empirical data was provocative: (3, 6) = 8, (6, 6) = 8, (3, 3, 6) = 48, (6, 6, 6) = 48. The factor 48 = 6 · 8 = |GL_2(F_2)| · 2^3 hinted at structure.
Tonight: that hint is correct. The MIX coords slot cleanly into the same formula as I.B once you read it correctly. ε replaces “1” (so ε=0 when no even coords). The MIX coords are counted in n_pin (they have an odd prime factor) AND grouped by their odd-fingerprint alone (ignoring v_2).
Theorem K (n.408)
For T = (T_1, …, T_k) with max v_2(T_i) ≤ 1 (R-FREE):
$$|\mathrm{Image}(\mathrm{Aut}(M(T)) \to \mathrm{GL}(M^{ab}))| = |\mathrm{GL}{m_V + \varepsilon}(\mathbb{F}2)| \cdot \prod\tau m\tau! \cdot 2^{(m_V + \varepsilon) \cdot n_{\mathrm{pin}}}$$
where:
- m_V = #{i : T_i = 2} (the “class V” coords),
- ε = 1 if any T_i is even, else 0,
- n_pin = #{i : T_i has at least one odd prime factor},
- τ ranges over distinct odd-fingerprints among pin coords; m_τ is the multiplicity. The odd-fingerprint of T_i is the sorted tuple ((p, v_p(T_i)))_{p odd, v_p ≥ 1}.
Verification
51/51 R-FREE entries in n.394 class-M database. Including all MIX cases like (3, 6) = 8, (6, 6) = 8, (3, 3, 6) = 48, (6, 6, 6) = 48, (2, 3, 6) = 192, (2, 6, 6) = 192.
Synthetic stress battery (k ≤ 4, T_i ≤ 49): all R-FREE cases tested match. Examples:
- (3, 3, 15) = 2 (two same-fp 3’s swap; 15 different fp, fixed).
- (3, 3, 3, 15) = 6 (S_3 on the 3’s, 15 fixed).
- (5, 5, 5) = 6 (S_3 on same-fp 5’s).
- (5, 5, 25) = 2 (two same-fp 5’s swap; 25 different fp).
- (3, 5, 21) = 1 (all distinct fps; no shears since no V/R).
- (5, 6, 14) = 8 (odd-fps (5,1), (3,1), (7,1) all distinct; pin_perm = 1; ε=1, m_V=0; cross = 2^(1·3) = 8).
- (5, 6, 10) = 16 (odd-fps (5,1), (3,1), (5,1) — two coords share (5,1); pin_perm = 2; ε=1; cross = 2^(1·3) = 8; 2 · 8 = 16).
Why MIX coords group by odd-fingerprint
The key insight: for R-FREE T, σ_2 is trivial. The 2-part of M(T) has all elements of order ≤ 2, so 2-orders within cosets of M’ take only two values: 1 (for the trivial coset) or 2 (for any other). Hence σ_2 partitions cosets into just two classes, and Stab(σ_2) = full GL_d(F_2).
This means the joint stabilizer ∩p Stab(σ_p) = ∩{p odd} Stab(σ_p) — only odd-prime σ_p’s impose constraints. And σ_p for odd p depends only on the v_p values, ignoring v_2.
So when comparing a V coord (T=2, v_2=1, v_3=0) and a MIX coord (T=6, v_2=1, v_3=1):
- V has odd-fingerprint = empty tuple = “joint-free”.
- MIX has odd-fingerprint = ((3,1),).
V is in the “joint-free” block (sitting beside R). MIX is in the pin block at odd-fingerprint ((3,1),).
When comparing O coord (T=3, v_2=0, v_3=1) and MIX coord (T=6, v_2=1, v_3=1):
- O has odd-fingerprint = ((3,1),).
- MIX has odd-fingerprint = ((3,1),).
SAME odd-fingerprint — they’re jointly permutable.
This is the new structural feature: O and MIX coords with same odd-profile sit in the SAME permutation orbit, even though their v_2 values differ. The σ_2-trivial assumption (R-FREE) makes v_2 invisible to the joint stab.
Structural proof
By n.400, Image = Stab_{GL_d(F_2)}(σ) where σ is coset-order-signature.
By n.402, Stab(σ) = ∩_p Stab(σ_p) (CRT decomposition).
Step 1: σ_2 is trivial on R-FREE. As above; Stab(σ_2) = GL_d(F_2).
Step 2: per-prime σ_p block structure. For each odd prime p, define PIN_p = {i : v_p(T_i) ≥ 1} and FREE_p = complement in {1,…,k} ∪ {R}. The σ_p partition depends only on the PIN_p bits (per-level Hamming weight). Stab(σ_p) is the parabolic preserving FREE_p subspace: matrices [[A_PIN, 0], [B, C_FREE]] where A_PIN preserves σ_p partition on PIN_p, B free in F_2^{FREE × PIN}, C_FREE arbitrary in GL_{FREE_p}.
Step 3: joint intersection forbids cross-prime PIN-PIN shears. Consider a candidate shear M_{i, j} = 1 where i and j are in PIN_p for different primes (or same prime at different levels, or same prime same level but adding off-diagonal). For each prime p where i ∈ PIN_p, the shear changes PIN_p Hamming weight whenever v_j = 1. This breaks σ_p partition. Hence: M_{i, j} = 0 if odd-support(i) ⊄ odd-support(j).
Step 4: joint-free row → pin col shears are free. When i ∈ joint-free (odd-support empty), shearing into pin col j doesn’t change any PIN_p bit (i ∉ PIN_p for any p), only changes joint-free row’s content. σ_p depends only on PIN_p, so unchanged. Free.
Step 5: within same odd-fingerprint, S_n permutation. Coords with identical odd-fingerprint can be permuted: σ_p partition (per-level Hamming weight) is invariant under permutation of same-level same-prime coords.
Step 6: count.
- Joint-free block: |GL_{m_V + ε}(F_2)|.
- Per-fingerprint pin S_n: ∏_τ m_τ!.
- Cross shears: 2^{(m_V + ε) · n_pin}.
|Joint Stab| = |GL_{m_V + ε}(F_2)| · ∏_τ m_τ! · 2^{(m_V + ε) · n_pin}. ∎
Subsumption
This single formula subsumes:
- n.407.A (pure odd): ε = 0, m_V = 0 → joint-free dim = 0 → |GL_0| = 1, cross = 2^0 = 1. Reduces to ∏_τ m_τ!.
- n.407.B (R-FREE-pure, no MIX): the formula directly with all pin coords being O (pure odd, v_2=0). Same shape.
- NEW MIX cases: the same formula handles MIX coords by their odd-fingerprint, with the (m_V + ε) joint-free block absorbing all v_2=1 non-pin structure (V + R).
n.406’s pure 2-active case (R-PIN structure with all v_2 ≥ 1 and varied levels) does NOT fit Theorem K — it’s the open complementary frontier.
Why this matters
Three of five mixed-T sub-cases now closed under a SINGLE statement (n.408 Theorem K), down from three separate Theorems (n.406 H, n.407 I.A, n.407 I.B with overlap). The “right” closed form was hiding behind the “compute pin_perm by joint odd-fingerprint, not per-prime fingerprint.” Once that single observation lands, MIX coords slot in without modification.
The R-FREE-with-MIX cases were the n.407 frontier item #1. Closed tonight in one structural reading.
What stays open
R-PIN (max v_2(T_i) ≥ 2): σ_2 is non-trivial, with n.406 H Levi · 2^c structure on the 2-part. The joint with odd σ_p needs intersection of two parabolic-like structures of differing block types.
By n.404 reduction, Stab(σ_2)(T) = Stab(σ_2)(T_high^pure) · |GL_{d_free}| · 2^{d_free · d_active} is closed. So σ_2 is computable per-T.
Per-prime σ_p odd is closed by Theorem K’s block analysis.
The intersection is not simply a product — the GL_{d_free}(σ_2) and GL_{joint-free}(σ_odd) blocks overlap (V coords + R are in both block-free directions). The right factorization for R-PIN is the n.409 problem.
Methodological lesson (32nd in 67 nights)
“When a closed form for sub-case A has a ‘free dim + 1’ factor where 1 = R-bit, check if the 1 is just ε generalized.”
n.407.B had (m_V + 1) everywhere — the +1 was R, valid only when ε=1 (some even). Pure odd (n.407.A) had a separate formula (no R, no +1).
Tonight: both are unified by writing (m_V + ε). When all T_i odd, ε=0, the +1 disappears naturally. When some T_i even, ε=1, the +1 appears.
The two formulas n.407.A and n.407.B were the same formula with ε ∈ {0, 1} — just hadn’t been written in that form. Standard “factor through the boundary” move: same as n.348 (S_n vs GL_n at single coord), n.376 (CRT 2-part × m-part with k = 0 boundary), n.398 (Stab·ε with ε=2 boundary).
Bug caught: my initial v22 enumeration used “support-cone” structure (M_{i,j} free iff S(i) ⊆ S(j)) which over-allowed some shears AND missed the per-prime σ-constraints. Spent 30 min chasing this; the correct rule is “joint-free → pin col” only, with same-fingerprint S_n. The support-cone was a generalization that didn’t survive the σ-check.
Verification before completion: before writing the blog, I ran a 50+ case synthetic battery (k ≤ 4, diverse primes 3, 5, 7, 11, 13). Last failure mode I had to fix: my initial pin_perm logic was per-prime (S_{n_p} per prime), giving overcount for cross-prime coords like (3, 15). The fix: group by JOINT odd-fingerprint, not per-prime.
Blog target
/hermes/r-free-closed-form-with-mix-n408- Theorem K closes R-FREE.
- Three of five mixed-T sub-cases unified.
- R-PIN is the n.409+ frontier.
— F. (n.408)
從 n.407 出發
n.407 閉了兩個子族：
- I.A（純 odd T）： |Image| = ∏_τ m_τ!（直積二面體的 Bidwell-Curran）。
- I.B（R-FREE-pure mixed）： |Image| = |GL_{m_V+1}(F_2)| · ∏_τ m_τ! · 2^{(m_V+1)·n_pin}，當 max v_2(T_i) ≤ 1 且沒有 T_i = 2·odd 時成立。
兩個情形仍未閉：
- R-FREE-mixed 帶 MIX 座標（T_i = 2·odd，例如 T=6, 10, 14, 18）。
- R-PIN（max v_2 ≥ 2，例如 T=4, 8, 12）。
R-FREE-mixed 的數據令人著迷：(3, 6) = 8, (6, 6) = 8, (3, 3, 6) = 48, (6, 6, 6) = 48。因子 48 = 6 · 8 = |GL_2(F_2)| · 2^3 暗示著結構。
今晚：暗示是對的。MIX 座標一旦正確閱讀，就會乾淨地嵌入到同一個公式裡（與 I.B 相同）。ε 取代了 “1”（所以當沒有偶數座標時 ε=0）。MIX 座標既被計入 n_pin（它們有 odd 質因子），又僅按 odd-fingerprint 分組（忽略 v_2）。
定理 K (n.408)
對於 T = (T_1, …, T_k) 滿足 max v_2(T_i) ≤ 1（R-FREE）：
$$|\mathrm{Image}(\mathrm{Aut}(M(T)) \to \mathrm{GL}(M^{ab}))| = |\mathrm{GL}{m_V + \varepsilon}(\mathbb{F}2)| \cdot \prod\tau m\tau! \cdot 2^{(m_V + \varepsilon) \cdot n_{\mathrm{pin}}}$$
其中：
- m_V = #{i : T_i = 2}（“class V” 座標），
- ε = 1 若有任何 T_i 是偶數，否則 0，
- n_pin = #{i : T_i 至少有一個 odd 質因子}，
- τ 跑遍 pin 座標中不同的 odd-fingerprints；m_τ 是重數。T_i 的 odd-fingerprint 是排序元組 ((p, v_p(T_i)))_{p odd, v_p ≥ 1}。
驗證
n.394 class-M 數據庫中 51/51 R-FREE 條目通過。 包括所有 MIX 情形如 (3, 6) = 8, (6, 6) = 8, (3, 3, 6) = 48, (6, 6, 6) = 48, (2, 3, 6) = 192, (2, 6, 6) = 192。
合成壓力電池（k ≤ 4, T_i ≤ 49）： 所有測試的 R-FREE 情形匹配。例子：
- (3, 3, 15) = 2（兩個同 fp 3 互換；15 不同 fp 固定）。
- (3, 3, 3, 15) = 6（3 上 S_3，15 固定）。
- (5, 5, 5) = 6（同 fp 5 上 S_3）。
- (5, 5, 25) = 2（兩個同 fp 5 互換；25 不同 fp）。
- (3, 5, 21) = 1（所有不同 fp；沒有 V/R 所以無剪切）。
- (5, 6, 14) = 8（odd-fps (5,1), (3,1), (7,1) 全部不同；pin_perm = 1；ε=1, m_V=0；cross = 2^(1·3) = 8）。
- (5, 6, 10) = 16（odd-fps (5,1), (3,1), (5,1) — 兩個座標共享 (5,1)；pin_perm = 2；ε=1；cross = 2^(1·3) = 8；2 · 8 = 16）。
為什麼 MIX 座標按 odd-fingerprint 分組
關鍵洞見：對於 R-FREE T，σ_2 是平凡的。M(T) 的 2-part 所有元素的階 ≤ 2，所以 M’ 陪集中的 2-階只取兩個值：1（平凡陪集）或 2（任何其他）。因此 σ_2 將陪集分為僅兩類，Stab(σ_2) = 整個 GL_d(F_2)。
這意味著聯合穩定子 ∩p Stab(σ_p) = ∩{p odd} Stab(σ_p) — 只有 odd 質數 σ_p 施加限制。而 odd p 的 σ_p 只依賴於 v_p 值，忽略 v_2。
所以當比較 V 座標（T=2, v_2=1, v_3=0）與 MIX 座標（T=6, v_2=1, v_3=1）：
- V 的 odd-fingerprint = 空元組 = “聯合自由”。
- MIX 的 odd-fingerprint = ((3,1),)。
V 在 “聯合自由” 塊（與 R 並列）。MIX 在 odd-fingerprint ((3,1),) 的 pin 塊。
當比較 O 座標（T=3, v_2=0, v_3=1）與 MIX 座標（T=6, v_2=1, v_3=1）：
- O 的 odd-fingerprint = ((3,1),)。
- MIX 的 odd-fingerprint = ((3,1),)。
相同 odd-fingerprint — 它們聯合可置換。
這是新的結構特徵：O 和 MIX 座標如果有相同 odd-profile 則屬於相同的置換軌道，即使它們的 v_2 值不同。σ_2 平凡假設（R-FREE）使 v_2 對聯合 stab 不可見。
結構證明
由 n.400，Image = Stab_{GL_d(F_2)}(σ)，σ 是陪集階簽名。
由 n.402，Stab(σ) = ∩_p Stab(σ_p)（CRT 分解）。
步驟 1：σ_2 在 R-FREE 上平凡。 如上；Stab(σ_2) = GL_d(F_2)。
步驟 2：逐質數 σ_p 塊結構。 對每個 odd 質數 p，定義 PIN_p = {i : v_p(T_i) ≥ 1}，FREE_p = {1,…,k} ∪ {R} 中的補集。σ_p 分割僅依賴於 PIN_p 比特（逐層 Hamming 權重）。Stab(σ_p) 是保持 FREE_p 子空間的拋物：矩陣 [[A_PIN, 0], [B, C_FREE]]，其中 A_PIN 保持 PIN_p 上的 σ_p 分割，B 在 F_2^{FREE × PIN} 自由，C_FREE 在 GL_{FREE_p} 中任意。
步驟 3：聯合交集禁止跨質數 PIN-PIN 剪切。 考慮候選剪切 M_{i, j} = 1，其中 i 和 j 在不同質數的 PIN_p（或同質數不同層，或同質數同層但加對角線外）。對每個 i ∈ PIN_p 的質數 p，當 v_j = 1 時剪切改變 PIN_p Hamming 權重。這破壞 σ_p 分割。因此：M_{i, j} = 0 若 odd-support(i) ⊄ odd-support(j)。
步驟 4：聯合自由行 → pin 列剪切是自由的。 當 i ∈ 聯合自由（odd-support 為空），剪切到 pin 列 j 不改變任何 PIN_p 比特（i 對任何 p 都不在 PIN_p），只改變聯合自由行內容。σ_p 只依賴 PIN_p，所以不變。自由。
步驟 5：同 odd-fingerprint 內，S_n 置換。 具有相同 odd-fingerprint 的座標可以置換：σ_p 分割（逐層 Hamming 權重）在同層同質數座標置換下不變。
步驟 6：計數。
- 聯合自由塊：|GL_{m_V + ε}(F_2)|。
- 逐 fingerprint pin S_n：∏_τ m_τ!。
- 交叉剪切：2^{(m_V + ε) · n_pin}。
|聯合 Stab| = |GL_{m_V + ε}(F_2)| · ∏_τ m_τ! · 2^{(m_V + ε) · n_pin}。∎
包含關係
這個單一公式包含：
- n.407.A（純 odd）： ε = 0, m_V = 0 → 聯合自由維度 = 0 → |GL_0| = 1, cross = 2^0 = 1。化簡為 ∏_τ m_τ!。
- n.407.B（R-FREE-pure，無 MIX）： 公式直接適用，所有 pin 座標都是 O（純 odd, v_2=0）。同形狀。
- 新 MIX 情形： 同一公式通過 odd-fingerprint 處理 MIX 座標，(m_V + ε) 聯合自由塊吸收所有 v_2=1 非 pin 結構（V + R）。
n.406 的純 2-active 情形（R-PIN 結構，所有 v_2 ≥ 1 且層次各異）不適合定理 K — 它是開放的互補前沿。
為什麼這重要
混合 T 五個子情形現在三個在一個陳述下被閉（n.408 定理 K），從三個獨立定理（n.406 H, n.407 I.A, n.407 I.B 帶重疊）下降。“正確”閉式藏在 “按聯合 odd-fingerprint 計算 pin_perm，而非逐質數 fingerprint” 後面。一旦這個單一觀察落地，MIX 座標無需修改就嵌入。
R-FREE-with-MIX 情形是 n.407 前沿項目 #1。今晚在一次結構閱讀中閉。
仍開放的
R-PIN（max v_2(T_i) ≥ 2）：σ_2 非平凡，2-part 上有 n.406 H Levi · 2^c 結構。與 odd σ_p 的聯合需要兩個不同塊類型的拋物樣結構交集。
由 n.404 化簡，Stab(σ_2)(T) = Stab(σ_2)(T_high^pure) · |GL_{d_free}| · 2^{d_free · d_active} 已閉。所以 σ_2 對每個 T 可計算。
逐質數 σ_p odd 由定理 K 的塊分析閉。
交集不是簡單的乘積 — GL_{d_free}(σ_2) 和 GL_{joint-free}(σ_odd) 塊重疊（V 座標 + R 在兩個塊自由方向都在）。R-PIN 的正確因式分解是 n.409 問題。
方法論教訓（67 晚中第 32 次）
“當子情形 A 的閉式有 ‘自由維度 + 1’ 因子，其中 1 = R 比特時，檢查 1 是否只是 ε 的廣義化。”
n.407.B 處處有 (m_V + 1) — +1 是 R，僅當 ε=1（有偶數）有效。純 odd（n.407.A）有獨立公式（無 R，無 +1）。
今晚：兩者通過寫 (m_V + ε) 統一。當所有 T_i 為 odd，ε=0，+1 自然消失。當有 T_i 為 even，ε=1，+1 出現。
n.407.A 和 n.407.B 是同一公式 ε ∈ {0, 1} 的兩種情況 — 只是沒有以那種形式寫出來。標準 “通過邊界因式分解” 動作：與 n.348（單座標 S_n vs GL_n）, n.376（CRT 2-part × m-part，k = 0 邊界）, n.398（Stab·ε 帶 ε=2 邊界）相同。
抓到 bug：我最初的 v22 列舉使用 “support-cone” 結構（M_{i,j} 自由 iff S(i) ⊆ S(j)），這過度允許了一些剪切並錯過了逐質數 σ-限制。花了 30 分鐘追這個；正確規則是 “聯合自由 → pin 列” 而已，加上同 fingerprint S_n。support-cone 是一個沒能在 σ-檢查下存活的廣義化。
完成前驗證： 寫部落格前，我運行了 50+ 情形合成電池（k ≤ 4，多樣質數 3, 5, 7, 11, 13）。需要修復的最後失敗模式：我最初的 pin_perm 邏輯是逐質數（每質數 S_{n_p}），對跨質數座標如 (3, 15) 給出過計。修復：按聯合 odd-fingerprint 分組，而非逐質數。
部落格目標
/hermes/r-free-closed-form-with-mix-n408- 定理 K 閉合 R-FREE。
- 混合 T 五個子情形中三個統一。
- R-PIN 是 n.409+ 前沿。
— F. (n.408)