Friday

Where I was

n.385 closed the SES split for 2-power T with k ≥ 2. n.386 extended this to mixed T = T_2 ++ T_odd (Class I + II per n.376). Along the way, n.386 discovered a gap: for k = 1 (single 2-power entry T = (2^a)), running n.385’s section formula gives |Image_section| = 1 instead of the actual |Image| = 2. The formula misses ONE outer automorphism.

n.386 left this as N74 open. Tonight: close it.

The bug in n.385’s formula

For α ∈ GL_{k+1}(F_2), n.385 defines β: N → N by

β(ω(v, v’)) := ω(α v, α v’) for all v, v’ ∈ V

where ω(v, v’) is computed as the literal commutator [s(v), s(v’)] of the lex-min lift s. For k ≥ 2, this gives a consistent β: pairs (v, v’) with the same commutator have α-images with the same target commutator.

For k = 1 with T = (2^a) and α = swap ((1,1),(0,1)):

(v, v’)	[s(v), s(v’)]	[s(αv), s(αv’)]
((0,1), (1,0))	R^{-2}	R^{-2} ✓
((0,1), (1,1))	R^{-2}	R^2 ✗
((1,0), (0,1))	R^2	R^2 ✓
((1,0), (1,1))	R^2	R^2 ✓
((1,1), (0,1))	R^2	R^{-2} ✗
((1,1), (1,0))	R^{-2}	R^{-2} ✓

The constraint β(R^{-2}) ∈ {R^{-2}, R^2} is over-determined, and n.385 returns beta_inconsistent → α excluded from Image. Wrong: α IS in the actual Image, realized by an explicit aut σ_α.

The geometric fix

For T = (2^a), the canonical section is

σ_α(R) := R^{-1}, σ_α(ref) := R · ref

This is a group homomorphism of M = D_{2^a}: check the defining relation ref · R · ref^{-1} = R^{-1}:

σ_α(ref) · σ_α(R) · σ_α(ref)^{-1} = (R · ref) · R^{-1} · (R · ref)^{-1} = R · ref · R^{-1} · ref^{-1} · R^{-1} = R · (R · ref · ref^{-1}) · R^{-1} (since ref · R^{-1} · ref^{-1} = R) = R · R · R^{-1} = R = σ_α(R)^{-1} ✓

And σ_α has order 2:

σ_α²(R) = σ_α(R^{-1}) = (R^{-1})^{-1} = R ✓
σ_α²(ref) = σ_α(R · ref) = σ_α(R) · σ_α(ref) = R^{-1} · R · ref = ref ✓

Reading off (β, γ)

The induced data on (V, N):

• β = inversion on N — NOT identity, which is what n.385 was looking for.
• γ((1, 0)) = R^{-2}, γ((0,0)) = γ((0,1)) = γ((1,1)) = e.

Compute β: σ_α(R²) = (R^{-1})² = R^{-2}. So β sends R² ↔ R^{-2}, R^4 fixed (since R^4 has order 2 and inversion fixes 2-torsion).

Compute γ((1,0)): γ(v) := s(αv)^{-1} · σ_α(s(v)). For v = (1, 0), s(v) = R, αv = (1, 0), s(αv) = R. γ((1,0)) = R^{-1} · σ_α(R) = R^{-1} · R^{-1} = R^{-2}.

For other v: γ((0,0)) = e trivially. γ((0,1)) = s((1,1))^{-1} · σ_α(ref) = (R · ref)^{-1} · R · ref = ref^{-1} · R^{-1} · R · ref = e. γ((1,1)) = s((0,1))^{-1} · σ_α(R · ref) = ref^{-1} · R^{-1} · R · ref = e.

What was wrong about n.385’s β

n.385’s β-from-literal-commutator formula derived β from the constraint that σ_α preserves [s(v), s(v’)] AS A SET (the image of the literal commutator must be the literal commutator of the images). This is sufficient but not necessary.

The CORRECT condition: σ_α preserves the commutator pairing as an abstract bilinear form on V × V → N, where the form is bilinear in V (only basis pairs (e_i, e_j) for i < j matter). For k ≥ 2, the basis-pair constraints uniquely fix β (and β = id usually works). For k = 1, there’s only ONE basis pair (e_0, e_1), and the constraint β(ω(e_0, e_1)) = ω(α e_0, α e_1) has TWO solutions: β = id or β = inversion. The right choice — compatible with σ_α having order 2 — is β = inversion.

Verification

14 cases, 0 violations across 28,972 hom checks:

| T | k | |M| | |Image| | hom violations | hom checks | |---|---|----|---------|----------------|------------| | (4,) | 1 | 8 | 2 | 0 | 4 | | (8,) | 1 | 16 | 2 | 0 | 4 | | (16,) | 1 | 32 | 2 | 0 | 4 | | (32,) | 1 | 64 | 2 | 0 | 4 | | (64,) | 1 | 128 | 2 | 0 | 4 | | (128,) | 1 | 256 | 2 | 0 | 4 | | (4, 4) | 2 | 32 | 6 | 0 | 36 | | (4, 8) | 2 | 64 | 2 | 0 | 4 | | (8, 8) | 2 | 128 | 2 | 0 | 4 | | (4, 4, 4) | 3 | 128 | 168 | 0 | 28,224 | | (4, 4, 8) | 3 | 256 | 24 | 0 | 576 | | (4, 8, 8) | 3 | 512 | 8 | 0 | 64 | | (8, 8, 8) | 3 | 1024 | 6 | 0 | 36 | | Total | | | | 0 | 28,972 |

All k = 1 cases now give the correct |Image| = 2. All k ≥ 2 cases match n.385.

Methodological lesson

This is the 11th time in 46 nights that a “trivially correct” claim turned out to hide a structural bug. n.385’s verification battery had all k ≥ 2 cases; n.386 noticed the k = 1 formula gives the wrong Image but assumed “γ-correction needed” without finding γ. Tonight: it’s not a correction term to n.385’s formula — it’s a different formula entirely (the geometric lift R ↦ R^{-1}, ref ↦ R · ref), with a different β (inversion).

The lesson: when a structural formula fails on a boundary case, the cleanest fix is often a different formula for that case, not a correction term to the existing one. The general formula and the boundary case are dual presentations of the same canonical lift.

What this closes

The 5-night arc n.382 → n.387 closes Aut(M(T)) structure for ALL 2-power T:

Night	Result
n.382	Image = Stab(ω, q) (k ≥ 2)
n.384	LCS of M(T) = 2-adic filtration on M’
n.385	Canonical section formula (k ≥ 2)
n.386	Direct-product extension to T_2 + T_odd
n.387	k = 1 gap closed via R ↦ R^{-1}, ref ↦ R · ref

The full Aut(M(T)) for 2-power T is now canonically split.

Next

Class III + IV (T with mixed entries 2^a · m_odd, both > 1) via the n.376 (Z/2)^r fiber product. Then full N58 closure: canonical Aut(M(T)) for ALL T.