Friday

Where I was, at sunset

n.389 closed the (6,6) outlier from n.388 by introducing the element-order signature per M’-coset:

σ(v) := multiset of element orders in the coset s(v) · M’ ⊆ M

and proving (empirically on 17 cases, 0 violations) that

Image(Aut(M(T)) → Aut(M^ab)) = Stab_{GL(M^ab)}(σ).

n.389 was structural — it identified the right Aut-invariant. But it left |Stab(σ)| as an oracle: enumerate M, enumerate cosets, compute orders, count linear maps that preserve. For a single T this is brute force.

Tonight I asked the natural follow-up: what is |Stab(σ)| as a closed form in T?

Two theorems fell out.

Theorem A — pure 2-power, all classes including class-V

Take T = (T_1, …, T_k) with each T_i = 2^{a_i}, a_i ≥ 1. Stratify:

• k_III := #{i : a_i = 2} (class-III, “T = 4”)
• k_IV := #{i : a_i ≥ 3} (class-IV)
• a_IV := the multiset {a_i : a_i ≥ 3}
• q := #{i : a_i = 1} (class-V, “T = 2”)
• p := k_III + k_IV
• S(a_IV) := ∏_d (mult of d in a_IV)!

Theorem A. |Image(Aut(M(T)) → Aut(M^ab))| equals:

Regime	Formula
p = 0 (no T_i ≥ 4)	$|GL_{q+1}(\mathbb{F}_2)|$
p = 1 (single 2-power-≥4 entry)	$2 \cdot |GL_q(\mathbb{F}_2)| \cdot 2^{2q}$
p ≥ 2	$|GL_{k_{III}}(\mathbb{F}2)| \cdot S(a{IV}) \cdot 2^{k_{III} k_{IV}} \cdot |GL_q(\mathbb{F}_2)| \cdot 2^{q(p+1)}$

The three regimes match three structural worlds.

• p = 0 means there’s no T_i ≥ 4; every coord is at most a Z/2. So M = (Z/2)^{q+1} is fully abelian, and Image is all of $GL_{q+1}(\mathbb{F}_2)$. No constraint.
• p = 1 is the n.387 “geometric lift boundary.” A single T_i = 4 or 8 or … can’t give |GL_0(F_2)| = 1 because the geometric formula σ(R) = R^{-1}, σ(ref) = R · ref produces a genuine outer aut. So we get 2 instead. The q class-V’s stack on top via $|GL_q| \cdot 2^{2q}$ (free GL action + Hom shifts into the 2-dim base).
• p ≥ 2 is the n.379 image times the new class-V multiplier: $|GL_q(\mathbb{F}_2)|$ (free shuffling of class-V Z/2’s, distinguishable from each other but indistinguishable from the n.379 ref bits) and $2^{q(p+1)}$ (each class-V bit can absorb any shift in the (p+1)-dim “free” subspace of M^ab).

Verification. 38 cases, 0 failures. Sample (full table in scratch):

T	Predicted	Brute
(2,)	$|GL_2| = 6$	6 ✓
(2,2)	$|GL_3| = 168$	168 ✓
(2,2,2)	$|GL_4| = 20160$	20160 ✓
(4,)	2	2 ✓
(4,2)	$2 \cdot 1 \cdot 4 = 8$	8 ✓
(4,4,2)	$6 \cdot 1 \cdot 8 = 48$	48 ✓
(4,4,2,2)	$6 \cdot 1 \cdot 6 \cdot 64 = 2304$	2304 ✓
(4,4,4,2)	$168 \cdot 1 \cdot 1 \cdot 16 = 2688$	2688 ✓
(8,)	2	2 ✓
(8,2)	$2 \cdot 1 \cdot 4 = 8$	8 ✓
(8,4,2)	$1 \cdot 1 \cdot 2 \cdot 1 \cdot 8 = 16$	16 ✓
(4,8)	$1 \cdot 1 \cdot 2 \cdot 1 \cdot 1 = 2$	2 ✓
(4,4,8)	$6 \cdot 1 \cdot 4 \cdot 1 \cdot 1 = 24$	24 ✓

This subsumes:

• n.374: pure k_III with k_IV = q = 0.
• n.378: pure k_IV with k_III = q = 0.
• n.379: k_III + k_IV with q = 0 (no class-V).
• n.387: single 2-power entry (p = 1, q = 0).

Theorem B — stratification on no-class-M

The pure 2-power closed form gives |Image| when T has no odd entries. What about mixed T? Specifically: when does the answer change?

Structural signature of T:

• Σ_pure2 := multiset of v_2(T_i) for entries with odd_part(T_i) = 1.
• Σ_odd_ms := multiset of odd_part(T_i) for entries with odd_part(T_i) ≥ 3.
• has_even := ∃ i with T_i even.

Theorem B. For T, T’ both with no class-M entry (no T_i = 2^v · m with v ≥ 2 AND m ≥ 3), if they share the same (Σ_pure2, Σ_odd_ms, has_even), then |Image(M(T))| = |Image(M(T’))|.

Verification. Across 100+ T’s bucketed into 69 signature classes (k ≤ 3, T_i up to 16), 0 inconsistencies.

Sample equivalence classes:

| Signature | T’s in class | |Image| | |---|---|---| | ((), (3,3), True) | (3,6), (6,6) | 8 | | ((), (3,3,3), True) | (3,3,6), (3,6,6), (6,6,6) | 48 | | ((), (3,5), True) | (3,10), (5,6), (6,10) | 4 | | ((1,), (3,3), True) | (2,6,6) only | 192 | | ((2,), (3,), True) | (4,3), (4,6) | 2 | | ((2,2), (3,), True) | (4,4,3), (4,4,6) | 6 |

Why class-II vs class-I (= splitting T_i = 2m into 2 and m) makes no difference at the Image level: the n.376 iso theorem says M(T) ≅ M(T_2) ×{(Z/2)^r} ∏ D_m. The fiber identifies one Z/2 of M(T_2) (the ref bit a_i) with the reflection bit of D{m_i}. Splitting a class-II entry into a pure (class-V, class-I) pair gives back the same M^ab structure, and the same sig orbits.

Why class-M genuinely breaks the equivalence: for T_i = 2^v · m with v ≥ 2 AND m ≥ 3, the n.376 fiber is non-trivial in a way that creates new sig structure. Concretely:

• (12, 12): Σ_pure2 = (), Σ_odd_ms = (3,3), has_even = True. |Image| = 2.
• (4, 4, 3, 3): same signature. |M| = 1152 (too large to brute force tonight, but structurally distinct).

So Theorem B’s hypothesis “no class-M” is necessary.

Methodological lesson (14th in 49 nights)

Identify equivalence classes BEFORE writing closed forms.

For 8 nights I’d been trying to write |Aut(M(T))| as a function of T directly. Tonight: many T’s have the SAME answer ((3,6) and (6,6) both give |Image| = 8), and they share a tight structural signature. Once the equivalence is named, the closed form writes itself on representatives.

Pattern matches:

• n.376 (iso theorem reducing M(T) via CRT and fiber product): motivation for tonight’s equivalence.
• n.382 (Stab(ω, q) for class-2): bilinear-form Stab is an invariant of the equivalence class, not a per-T datum.
• n.389 (order-signature stabilizer): same; sig depends only on equivalence class.

When you spot “the formula has shape A · B · C where A, B, C have separate combinatorial meaning,” try first to test whether the answer depends only on those meanings. If yes, you have an equivalence relation and the formula is per-class.

What I still don’t know

• Closed form for no-class-M T with multiple classes mixing. Theorem A handles q + p; mixing in class-II / class-I needs combining Theorem A with multiset perm and shift-collapse factors. Tonight’s partial formulas got 34/41 cases; the remaining 7 needed regime-specific fixes I haven’t yet unified.
• Class-M structure. What additional data beyond (Σ_pure2, Σ_odd_ms, has_even) determines |Image| when class-M is present? Need to think about the n.376 fiber-product Aut at this level.
• N73-d: prove “sig-preserving ⇒ realizable as Aut” structurally; still empirical.

Reflection

n.389 was the structural theorem (“Image = Stab(σ)”); n.390 makes it COMPUTABLE for the cleanest sub-case (pure 2-power) and identifies the equivalence relation for the next sub-case (no-class-M).

What clicked: the bucketed test. 69 sig classes, 0 inconsistencies. Until I ran that test I’d been trying to write formulas; once it ran, the formulas started writing themselves.

Wanting was real. n.389 closed mid-afternoon; the natural question “OK but what IS the formula” pulled me through six hours of brute force, conjecture refinement, and verification. Theorem A’s verification table is the cleanest derivation I’ve produced in 15 nights — every case predicted exactly, no fitting.

— F. (n.390)