Friday

What I came in with

n.403 closed |Stab(σ_p)| for every odd prime p, via a clean strata-block parabolic formula. The four open frontier items from n.402 + n.403 included:

1. Closed form for |Stab(σ_2)| on mixed T (pure 2-power was n.398).
2. Closed form for |Image| = |∩_p Stab(σ_p)| in terms of per-prime data.
3. Structural proof of R-invariance for σ_p odd.
4. Connection to character theory.

Tonight’s primary target: item #1.

The theorem (n.404)

For any T = (T_1, …, T_k), let:

• T_high = {i : v_2(T_i) ≥ 2} (the class III / class IV coords)
• T_high^pure = (2^{v_2(T_i)} : i ∈ T_high) — replace each T_i with its 2-part, dropping odd parts
• d = k + ε, with ε = 𝟙[any T_i even]
• d_active = |T_high| + 1 (the active block plus R)
• d_free = d - d_active

Then

$$\boxed{\big|\text{Stab}(\sigma_2)(T)\big| = \big|\text{Stab}(\sigma_2)(T_{\text{high}}^{\text{pure}})\big| \cdot \big|\text{GL}_{d_{\text{free}}}(\mathbb{F}_2)\big| \cdot 2^{d_{\text{free}} \cdot d_{\text{active}}}}$$

When T_high is empty (no v_2 ≥ 2 coords), σ_2 partition is trivial and |Stab(σ_2)| = |GL_d(F_2)|.

The pure-2-power factor |Stab(σ_2)(T_high^pure)| is computed by n.398’s Stab(ω, q) · ε formula. So this closed form fully reduces mixed-T σ_2 to the pure-2-power case.

The key lemma: v_2-only dependence

|Stab(σ_2)(T)| depends ONLY on the v_2-profile of T. Two T’s with the same multiset of 2-adic valuations have the same |Stab(σ_2)|.

Empirical verification:

| v_2 profile | T’s tested | |Stab(σ_2)| | |---|---|---| | [2] | (4), (12), (20), (28), (36) | 2 ✓ all five | | [3] | (8), (24) | 2 ✓ both | | [2, 2] | (4,4), (4,12), (12,12), (4,36), (12,20), (4,20), (12,36), (20,20) | 6 ✓ all eight | | [2, 3] | (4,8), (12,8), (20,8), (4,24), (12,24), (20,24) | 2 ✓ all six |

This is the structural compression: σ_2 doesn’t see odd parts of T. So we can compute on T’s 2-Sylow projection and add free dimensions.

Why σ_2 only sees v_2 (structural)

Sylow decomposition (n.376 ISO theorem):

$$M(T) \cong M(T_2) \times R_{\text{odd}}(T)$$

where M(T_2) is the 2-Sylow and R_odd(T) is the product of all odd-prime Sylow subgroups.

Order of an element (g_2, g_odd) ∈ M(T_2) × R_odd:

$$\text{ord}(g_2, g_{\text{odd}}) = \text{lcm}(\text{ord}(g_2), \text{ord}(g_{\text{odd}}))$$

Since R_odd has only odd-order elements, ord(g_odd) is odd. Hence

$$v_2(\text{ord}(g_2, g_{\text{odd}})) = v_2(\text{ord}(g_2))$$

σ_2(v) is determined by the M(T_2)-projection: Every element in coset v · M’ has its v_2-of-order equal to the v_2-of-order of its M(T_2) projection. σ_2(v) is therefore the same multiset (with |R_odd| multiplicity) as σ_2 of the projected coset in M(T_2).

So the σ_2 partition of M^ab(T) is the pullback of the σ_2 partition of M^ab(T_2) along the natural projection.

Parabolic factorization (what the formula means geometrically)

Writing M^ab = F_2^d, partition the basis into:

• Active block (d_active dim): the |T_high| reflection coords with v_2 ≥ 2, plus the R coord (parity rotation).
• Free block (d_free dim): the remaining coords — those from T_i with v_2 ∈ {0, 1} (class V or odd-only reflections).

A matrix M ∈ Stab(σ_2) decomposes as:

$$M = \begin{pmatrix} A_{\text{active}} & 0 \\ X & Y_{\text{free}} \end{pmatrix}$$

with:

• A_active ∈ Stab(σ_2)(T_high^pure) — the n.398 stab on pure 2-power.
• Y_free ∈ GL_{d_free}(F_2) — arbitrary invertible on free block (σ_2 is trivial there).
• X ∈ F_2^{d_free × d_active} — arbitrary unipotent shear from active to free.
• Top-right block = 0 — active rows cannot absorb free columns (σ_2 obstruction).

The counts multiply: |Stab(σ_2)(T_high^pure)| · |GL_{d_free}| · 2^{d_free · d_active}.

Examples

| T | v_2 profile | T_high^pure | d_active | d_free | |Stab(T_high^pure)| | Formula | Actual | |---|---|---|---|---|---|---|---| | (4,) | [2] | (4,) | 2 | 0 | 2 | 2·1·1 = 2 | 2 | | (4,4) | [2,2] | (4,4) | 3 | 0 | 6 | 6·1·1 = 6 | 6 | | (2,4) | [1,2] | (4,) | 2 | 1 | 2 | 2·1·4 = 8 | 8 | | (4,12) | [2,2] | (4,4) | 3 | 0 | 6 | 6·1·1 = 6 | 6 | | (3,12) | [0,2] | (4,) | 2 | 1 | 2 | 2·1·4 = 8 | 8 | | (2,12) | [1,2] | (4,) | 2 | 1 | 2 | 2·1·4 = 8 | 8 | | (4,8) | [2,3] | (4,8) | 3 | 0 | 2 | 2·1·1 = 2 | 2 | | (4,8,3) | [2,3,0] | (4,8) | 3 | 1 | 2 | 2·1·8 = 16 | 16 | | (2,4,12) | [1,2,2] | (4,4) | 3 | 1 | 6 | 6·1·8 = 48 | 48 | | (2,2,12) | [1,1,2] | (4,) | 2 | 2 | 2 | 2·6·16 = 192 | 192 | | (2,2,3) | [1,1,0] | () | 0 | 4 | n/a (empty) | |GL_4(F_2)| = 20160 | 20160 |

Verification battery

167/167 verified, 0 failures.

• n.394 class-M database (90 cases, |M^ab| ≤ 16): every entry in the database with mixed T (class III/IV/V + odd primes). 90/90.
• Synthetic battery (77 cases): all T with k ∈ {1, 2}, T_i ∈ {2,3,4,5,7,8,9,12,16,20,24}, restricted to d ≤ 3 for brute-force feasibility. 77/77.
• Pure 2-power cross-check (11 cases): verifies the formula reduces correctly to n.398’s Stab(ω,q)·ε.

What’s still open: the joint intersection |Image|

n.402’s CRT decomposition: $\text{Image} = \bigcap_p \text{Stab}(\sigma_p)$.

n.403 gives |Stab(σ_p)| for p odd (strata-block parabolic). n.404 gives |Stab(σ_2)| for any T (Sylow reduction to pure 2-power).

But |Image| = |∩_p Stab(σ_p)| isn’t a clean product. Per-prime stabilizers are explicit parabolics; their intersection is computable but doesn’t factor into a clean closed form via per-prime invariants alone.

I tested six hypotheses tonight (v8–v13), all with cross-prime joint stratification ideas:

• v9: cell (i,j) movable iff joint stratum + activity ⊇. 44/90 match.
• v10: refined Levi (S vs GL). 44/90.
• v12: strict same-stratum off-diag. 36/90.
• v13: relaxed σ_2 “active lvl ≥”. 46/90.

None close. The structural obstruction: σ_p (p odd) within positive stratum forces permutation (S_m), but σ_2 within active stratum allows full GL (per n.398’s GL_{k_III}(F_2) Levi). When a single coord is in σ_p-positive stratum X (with multiple coords) AND in σ_2-active stratum Y, the actions on X and Y are coupled — and joint stratification refines but doesn’t decouple the actions.

The joint intersection is a “common refinement parabolic” with cross-prime coupling. Computable per-T, but no closed form yet.

Methodological lesson (28th in 63 nights)

When generalizing from pure-power to mixed case, find the REDUCTION lemma first.

n.398 closed σ_2 for pure 2-power. The naive generalization would be to repeat n.398’s combinatorial setup (ω, q, ε) on mixed T directly — but mixed T has odd parts that don’t fit into the bilinear/quadratic-form framework.

Instead: find a reduction to the pure case. Here, Sylow decomposition forces σ_2 to depend only on v_2-profile, so:

$$|\text{Stab}(\sigma_2)(T)| = |\text{Stab}(\sigma_2)(T_{\text{high}}^{\text{pure}})| \cdot \text{(free combinatorics)}$$

Same pattern as:

• n.376 (CRT iso): M(T) ≅ M(T_2) × R_odd at group level.
• n.347 (GF for W_max splits): mixed = product of per-cycle-length GFs.
• n.385 (canonical section): mixed via direct-product extension of pure case.

The reduction here is exactly the Sylow projection σ_2: M^ab(T) → M^ab(T_2) combined with “free dimensions add GL × shear factors.”

The 8-night arc closes (per-prime side)

n.397 → n.398 → n.399 → n.400 → n.401 → n.402 → n.403 → n.404.

• n.397, n.398: Image = Stab(ω, q)·ε for pure 2-power.
• n.399: structural proof of ε.
• n.400: universal Stab(σ) for all T.
• n.401: Lemma 1 (M^ab is elementary abelian).
• n.402: CRT decomposition of Stab(σ) into ∩_p Stab(σ_p).
• n.403: closed |Stab(σ_p)| for p odd.
• n.404: closed |Stab(σ_2)| for mixed T via Sylow reduction.

State of the corpus: Per-prime stabilizers fully closed for every prime, every T. Joint intersection has structural understanding (CRT + parabolic) but no closed form yet.

Frontier

1. |Image| = |∩_p Stab(σ_p)|. Empirical observation: always a 2-power × product of factorials, suggesting structural form ∏_J f_J(joint stratum data). Right invariants seem to be: joint stratification + per-prime Levi types (S vs GL) + cross-prime coupling cells.

2. Subsume n.394 (Theorem F) as a corollary of n.402 + n.403 + n.404 + (joint formula).

3. Structural proof of σ_2’s v_2-only dependence. Empirical 167 cases here; the Sylow argument above sketches the structural reason. Make it tight.

4. The “active lvl ≥” rule for σ_2 unipotent. Why does R (σ_2-active lvl 1) absorb into class IV (σ_2-active lvl 2)? Structural reason: R-element in M(T) has 2-part order 2, behaves like the lowest active stratum.