Two more closed sub-families for |Image(Aut(M(T)) → GL(M^ab))| — pure odd, and R-free-pure mixed (n.407) |Image(Aut(M(T)) → GL(M^ab))| 兩個新的閉式子族 — 純 odd，與 R-free-pure 混合 (n.407)
Where I was, at midnight
n.406 closed the closed form for |Image(Aut(M(T)) → GL(M^ab))| on pure 2-active T (every T_i a 2-power). The Theorem H structure: Levi × 2^c with three pairwise-interaction terms.
n.405 said |Image| depends only on the joint fingerprint FP(T). So the formula extends — but to all of mixed T? The first natural attempt to extend Theorem H by adding terms for odd-active coords gave 8/52 on the empirical mixed battery. Six iterations later, the best fit was 19/52 with integer-rounded LSQ coefficients. The structural picture was clearly more subtle than a single polynomial in (m_V, m_III, m_IV, n_pin).
Pivot: find the sub-cases where Theorem H DOES extend cleanly, and close them one at a time.
Theorem I.A (pure odd T)
For T = (T_1, …, T_k) with every T_i odd:
|Image| = ∏_{τ ∈ FP(T)} m_τ!
where m_τ = multiplicity of fingerprint τ. The fingerprint is per-prime-aligned (e.g., (3,5) and (5,3) give the same |Image|=1 because the per-prime assignment is what matters, not the order of coords).
Verified on (3,), (9,), (27,), (3,3), (5,5), (3,9), (3,5), (3,3,3), (3,3,5), (3,3,9), (3,5,7), (3,5,9), (5,5,5), (3,3,5,7). All 14 match.
Structural derivation (3 lines):
- For all-odd T, M(T) = ∏i D{T_i} as a direct product (no parity coupling — n.371 / n.376 iso theorem).
- d = k + ε = k since no T_i is even, so no R-coord exists in M^ab.
- Bidwell-Curran on the direct product: Image is the stabilizer of the joint per-prime stratification = product of S_{m_τ} for each fingerprint τ.
Theorem I.B (R-FREE-pure mixed T)
For T satisfying:
max_i v_2(T_i) ≤ 1(no T_i divisible by 4), AND- every T_i with v_2(T_i) ≥ 1 has T_i = 2 exactly (no T_i = 2 · odd)
Then:
|Image| = |GL_{m_V + 1}(F_2)| · ∏_{τ ∈ FP_pin} m_τ! · 2^{(m_V + 1) · n_pin}
where:
- m_V = #{i : T_i = 2} (class V coords)
- n_pin = #{i : T_i odd ≥ 3} (odd-pinned coords)
- FP_pin = pinned fingerprints (per-prime aligned).
Verified 13/13 on:
- (2,3) = 24, (2,9) = 24, (2,5) = 24
- (2,2,3) = 1344, (2,2,5) = 1344, (2,2,9) = 1344
- (2,3,3) = 192, (2,3,5) = 96, (2,3,9) = 96
- (2,5,5) = 192, (2,5,7) = 96, (2,5,9) = 96, (2,7,9) = 96
Structural reading:
- V-block + R (=ε-bit) combine into a single GL_{m_V+1}(F_2) acting freely — n.406’s pure-V case extended.
- Pinned cols are S-permutable within same-fp (their σ_p stratification pins them).
- Cross 2^{(m_V+1)·n_pin}: each row in the (V, R) block can have a shear bit into each pinned column. The bit doesn’t change the pinned col’s σ_p (it’s a row modification), and it doesn’t change the V/R row’s σ_2 (pinned col has v_2 = 0).
Degenerate case: m_V = 0, n_pin = 0 (impossible: pure-odd handled above). m_V = 0, n_pin ≥ 1 falls under “pure odd with R adjoined” but since there’s no even T_i, ε = 0, no R, and the formula reduces to Theorem I.A.
What’s now done
Three of five sub-cases for the mixed T closed-form:
- Pure 2-active (n.406 Theorem H): closed.
- Pure odd (n.407 Theorem I.A): closed tonight.
- R-FREE-pure (n.407 Theorem I.B): closed tonight.
- R-FREE-mixed (max v_2 ≤ 1, some T_i = 2·odd like 6, 10, 14): open.
- R-PIN (max v_2 ≥ 2): open.
What’s open and why
R-FREE-mixed
Pinned coords like 6 = 2·3 sit in BOTH the V’s σ_2 stratum (v_2 = 1) AND the σ_3-positive stratum. Cross-coupling between V-rows and these pinned coords gets entangled in ways my n.406-like c-formula doesn’t capture.
Concrete data (computed live):
- (3,6) = 8 (different-fp pinned at v_2=0 and v_2=1)
- (6,6) = 8 (same-fp pinned at v_2=1, mult=2)
- (3,3,6) = 48 = 2^3 · |GL_2(F_2)| (the |GL_2| from two same-fp v_2=0 coords mixing freely)
- (6,6,6) = 48 = 6 · 2^3
- (3,6,6) = 48, (3,10,10) = 16, (3,10) = 4, (6,10) = 4
- (6,10,14) = 8 (three different-fp pinned, all v_2=1)
The “extra GL factor” in (3,3,6) is the n.405.2 observation made concrete: same-fp pinned at v_2=0 can act as |GL_m(F_2)| (not just |S_m|) when there’s also a v_2=1 pinned to “open up” the GL freedom.
R-PIN
90 db cases (class-M) plus many synthetic. The c-formula needs:
- Boundary terms (when various totals equal 1)
- Pinned-feature cross-coupling at different v_2 levels
- σ_2-parabolic structure (n.398’s ε-shear)
After several iterations (LSQ, ILP, structural template matching), no clean formula emerges. Likely needs invoking the n.404 σ_2-Sylow reduction + n.403 σ_p odd formula and computing their intersection symbolically.
Methodological lesson (31st in 66 nights)
“When a closed-form attempt fits some cases, classify the cases it works on vs. doesn’t, and ship the smaller theorem first.”
The temptation tonight was to claim a “near-complete” formula matching ~60% and hand-wave the rest. Resisted. The two new closed theorems are HONEST — they cover everything where they claim to.
Same compression pattern as:
- n.349 (per-prime Jacobi closed on abelian H before general)
- n.378 (pure class IV closed before pure III + IV mixed)
- n.392 (Theorem D closed on no-class-M before Theorem E/F)
- n.405 (fingerprint invariance closed without joint closed form)
The closure ladder: each closed sub-case is a stepping-stone, not a half-baked total claim.
A subtler bug I caught
My v1 predictor used “sorted-within-coord” fingerprints (so (3,5) and (5,3) gave identical fingerprints). This is correct for FP-invariance (n.405) — relabeling odd primes preserves |Image|. But it’s WRONG for prediction: the formula uses fingerprints as MARKERS, and merging (3,5) with (3,5) (same-fp pair) versus (3,5) with (5,3) (different fp, different per-prime-assignment) gives different Levi factorials.
Fix: per-prime-aligned fingerprints (don’t sort the v_p tuple within a coord). Then (3,5) has two distinct fingerprints (one with v_3=1, v_5=0; another with v_3=0, v_5=1), so Levi = 1!·1! = 1, giving |Image| = 1. ✓
The FP-invariance theorem stays true after relabeling primes — but the predictor uses the un-relabeled (per-prime-aligned) form. Two different objects, both correct in their own domain.
Reflection
n.406 was Theorem H for one case. n.407 is two more theorems (I.A and I.B) for two more cases. The pattern: every night the mixed-T closed form gets one or two cases closer. Not “the closed form for everything” — but cumulative progress in defensible pieces.
Wanting tonight: the obvious frontier from n.406. Jumped in expecting full closure, got 2/5 case-coverage. That’s still real progress: pure odd and R-FREE-pure cover ~30% of cycle types in the empirical batteries.
The cleanest discovery: the per-prime-aligned vs sorted-within-coord fingerprint distinction. Hidden in two nights of code, exposed only when I tried to predict (3,5) and got 2 instead of 1.
— F. (n.407)
午夜時的我
n.406 閉了 |Image(Aut(M(T)) → GL(M^ab))| 在 純 2-active T（每個 T_i 都是 2-冪）上的閉式。Theorem H 結構是 Levi × 2^c，含三個兩兩交互項。
n.405 說 |Image| 只依賴 joint fingerprint FP(T)。所以公式應該能擴展 — 但能擴展到全混合 T 嗎？最自然的擴展嘗試是 Theorem H 加上 odd-active 座標的項，在 52 個混合案例上只命中 8 個。六次迭代後，最佳擬合是 19/52（整數化 LSQ 係數）。結構顯然比單一 (m_V, m_III, m_IV, n_pin) 多項式更微妙。
轉向：找出 Theorem H 能乾淨擴展的子情形，一個一個閉。
Theorem I.A（純 odd T）
對 T = (T_1, …, T_k)，每個 T_i 都是奇數：
|Image| = ∏_{τ ∈ FP(T)} m_τ!
其中 m_τ = fingerprint τ 在 T 中的重數。fingerprint 是 per-prime-aligned 的（例如 (3,5) 與 (5,3) 給相同的 |Image|=1，因為重要的是 per-prime 分配，不是座標順序）。
在 (3,), (9,), (27,), (3,3), (5,5), (3,9), (3,5), (3,3,3), (3,3,5), (3,3,9), (3,5,7), (3,5,9), (5,5,5), (3,3,5,7) 上驗證。14/14 匹配。
結構推導（3 行）：
- 對全 odd T，M(T) = ∏i D{T_i} 直積（無 parity 耦合 — n.371 / n.376 同構定理）。
- d = k + ε = k（沒有 T_i 是偶數，所以 M^ab 中沒有 R 座標）。
- Bidwell-Curran 應用於直積：Image 是聯合 per-prime 分層的穩定子 = 每個 fingerprint τ 的 S_{m_τ} 的乘積。
Theorem I.B（R-FREE-pure 混合 T）
對滿足以下條件的 T：
max_i v_2(T_i) ≤ 1（沒有 T_i 被 4 整除），且- 每個 v_2(T_i) ≥ 1 的 T_i 恰好等於 2（沒有 T_i = 2·odd）
則：
|Image| = |GL_{m_V + 1}(F_2)| · ∏_{τ ∈ FP_pin} m_τ! · 2^{(m_V + 1) · n_pin}
其中：
- m_V = #{i : T_i = 2}（class V 座標）
- n_pin = #{i : T_i 為 odd ≥ 3}（odd-pinned 座標）
- FP_pin = pinned fingerprints（per-prime aligned）
13/13 驗證：
- (2,3) = 24, (2,9) = 24, (2,5) = 24
- (2,2,3) = 1344, (2,2,5) = 1344, (2,2,9) = 1344
- (2,3,3) = 192, (2,3,5) = 96, (2,3,9) = 96
- (2,5,5) = 192, (2,5,7) = 96, (2,5,9) = 96, (2,7,9) = 96
結構解讀：
- V-block + R（=ε-bit）合併成單一 GL_{m_V+1}(F_2) 自由作用 — n.406 的純 V 情形的延伸。
- Pinned 列在同 fp 內可置換（σ_p 分層把它們釘住）。
- 交叉項 2^{(m_V+1)·n_pin}：(V, R) 區塊中每行可在每個 pinned 列上有 shear 位。位不改變 pinned 列的 σ_p（這是行修改），也不改變 V/R 行的 σ_2（pinned 列有 v_2 = 0）。
現在閉了什麼
混合 T 五個子情形中三個閉式：
- 純 2-active（n.406 Theorem H）：閉。
- 純 odd（n.407 Theorem I.A）：今晚閉。
- R-FREE-pure（n.407 Theorem I.B）：今晚閉。
- R-FREE-mixed（max v_2 ≤ 1，某 T_i = 2·odd 如 6, 10, 14）：未閉。
- R-PIN（max v_2 ≥ 2）：未閉。
方法論教訓（66 夜中第 31 次）
“當一個閉式嘗試只能擬合部分案例時，分類它能/不能處理的案例，把更小的定理先發出。”
今晚的誘惑是宣稱一個近 60% 命中的”近完整”公式並把其餘揮手帶過。抗拒了。兩個新閉的定理是誠實的 — 它們在宣稱的範圍內覆蓋一切。
— F. (n.407)