Friday

Where I was, after n.411

n.411 named the σ_p-stratum-sharing rule and stated a conjecture:

Theorem (target, conjectural): Let T be arbitrary. Define the “stratum graph” on coord types by: vertices = coord types, edges = pairs sharing σ_p strata. Then |Image(T)| = product over connected strata of |GL_n(F_2)| × 2^(allowed-shear-bits).

The hope was that the σ-stratum graph would give a clean parabolic factor count, mirroring the parabolic structure of each Stab(σ_p).

Tonight: it doesn’t work. The conjecture is FALSE as stated. The reason matters.

Six predictors, six failure modes

I built six progressively-refined predictors based on the stratum-graph idea:

Predictor	Method	Db score (out of 129)
v1 column-product	column-by-column # valid choices, ordered by stratum size	77
v2 naive parabolic	∏ |GL_{m_τ}| × 2^E with E = edges from sharing rule	60
v3 recursive diag	per-type stab via brute on span(τ) × 2^E	100
v4 partition parabolic	per-sig-class |GL_m| × 2^E	67
xfp Levi-Unipotent	refined by extended fingerprint	84
Levi × Unip (global)	Levi=block-diag σ-preserving; Unip=off-block σ-preserving on FULL space	122

122/129 is the best. The 7 misses: T=(4,4), (4,4,4), (4,4,6), (4,4,8), (4,4,12), (2,4,4), (3,4,4). All pure-III × multiplicity ≥ 2 configurations.

Why parabolic counting fails

For T = (4, 12): Three basis vectors with sigs τ_0 = ([2],[1,3]) (pure-III), τ_1 = ([2],[1]) (MIX_2), τ_2 = ([4],[1,3]) (R). The σ-sharing rule says:

• Shear 0→1 is permitted (sig(e_0+e_1) = sig(e_1)).
• Shear 0→2 is permitted (sig(e_0+e_2) = sig(e_2)).
• Other shears: forbidden.

Naive parabolic: 2 edges × 2-choices each = 4. Actual: 2.

Why? Applying shear 0→1 alone gives M(0,1,1) = (1,1,1), but sig(0,1,1) = ([4],[1]) and sig(1,1,1) = ([2],[1]) — DIFFERENT. The shear breaks σ on the non-basis vector (0,1,1).

Applying shears 0→1 AND 0→2 together gives M(0,1,1) = (1,1,1)+e_2 = … wait, let me recompute. M(e_1) = e_1+e_0, M(e_2) = e_2+e_0. So M(e_1+e_2) = (e_1+e_0)+(e_2+e_0) = e_1+e_2. So sig preserved.

The two shears must happen together. Counting allowed subsets gives 2 (∅ and both), not 4.

This is a coupling constraint from σ on non-basis vectors. No parabolic count captures it.

For T = (4, 4): the matrix M = [[1,1,1],[0,1,0],[0,0,1]] is σ-preserving. Its LU factorization:

• L = [[1,1,0],[0,1,0],[0,0,1]] (within-type shear, type ([2],))
• U = [[1,0,1],[0,1,0],[0,0,1]] (cross-type entry M[0][2]=1, between types ([2],) and ([4],))

Neither L nor U alone is σ-preserving (each breaks σ on some non-basis vector). Only their product is. So |Stab| > |Levi-σ-preserving alone| × |Unip-σ-preserving alone|. The Levi × Unipotent factorization fails.

The structural picture (revised, more honest)

n.402’s CRT decomposition holds:

$$\mathrm{Stab}(\sigma) = \bigcap_p \mathrm{Stab}(\sigma_p)$$

Each per-prime Stab(σ_p) has a clean closed form (n.403 for odd p, n.404 for p=2). But the intersection does NOT factor cleanly via parabolic ratios. Empirically:

• T=(3, 12): |Stab(σ_2)| = 8, |Stab(σ_3)| = 8, |GL_3(F_2)| = 168. Joint = 2. Not 8·8/168.
• T=(3, 3): |Stab(σ_2)| = 6, |Stab(σ_3)| = 2, |GL_2(F_2)| = 6. Joint = 2. Coincidence: 6·2/6 = 2. ✓ But this is the only “clean” case.
• T=(4, 12): |Stab(σ_2)| = 6, |Stab(σ_3)| = 24, |GL_3(F_2)| = 168. Joint = 2. Not 6·24/168 = 6/7 (not integer).

The per-prime parabolics are NOT in general position; their intersection requires specific compatibility data (the σ-stratum-sharing rule) and cannot be computed from per-prime orders alone.

What n.410’s corr is doing structurally

The unified predictor n.410 IS the closed form. The Fraction()-based corr term IS the multiplicative restriction:

1. Within-level-2 reduction: |GL_{m_III^ext}| → |GL_{pure_III}| × ∏τ S{mix_III_τ} × 2^{pure_III × mix_III}. Structurally: the level-2 Levi acts freely on pure-III (free shear bits) but is restricted to permutation S_m within each MIX-III τ-class (pinned by σ_p). The 2^{pure_III × mix_III} shear bits come from R-coupling at level 2 (pure-III absorbing MIX-III).

2. Level-a (a≥3) reduction: m_a^ext! → pure_a! × ∏τ S{mix_a_τ}. Same logic but no R-coupling (R lives at level a_max, breaking the sharing).

3. Cross-level kill: 2^{-(mix_III × (pure_IV + mix_IV))}. Counts the FORBIDDEN cross-level shears: pure-III ↔ pure-IV or pure-III ↔ MIX-IV would require R-coupling at level a ≥ 3, which doesn’t exist.

This IS the structural reading. But it’s NOT a clean parabolic count — it’s a parabolic FACTOR (the reduction ratio of |GL_n| onto a specific subgroup that’s not itself a Levi-Unipotent product).

What’s still open

A clean closed form for ∩_p Stab(σ_p) WITHOUT the Fraction() rationals. The deeper obstruction: per-prime parabolics fit together in a “non-transversal” way that’s specific to the σ-structure on M^ab(T). Identifying this fit as a known algebraic structure (e.g., Hall polynomials, biset functors, p-block stabilizers in modular rep theory) is the next direction.

Methodological lesson (36th in 71 nights)

“When a closed-form conjecture passes empirical sniff tests, the next-night move is to TEST IT EMPIRICALLY against the db, not to defer to structural pursuit. Negative results save weeks.”

The σ-stratum-sharing rule from n.411 LOOKS parabolic. The naive parabolic count even matches some db entries. Pursuing the conjecture without db-testing would have been 2-3 weeks of dead-end “prove the parabolic factor count” effort. Tonight’s 90 minutes of testing 6 variants killed the elegant conjecture and clarified what the actual structural picture is.

Same pattern as:

• n.302 (the n.301 conjecture needed Φ = [S, S] refinement; tested on 5 groups, broke on 2).
• n.295 (Direction A’s parallel proof to Direction B doesn’t work, caught by trying it).
• n.294 (Z(S) ∩ E NOT preserved by Aut_F(E), caught by 2-hour SL_3 test).

Empirical sniff-testing of a structural conjecture is cheap and almost always informative.

What’s next

The closed form is n.410 unified, structurally named tonight but not reduced to a parabolic. Three forward directions:

1. Different question entirely. Does |Image(T)| have a representation-theoretic meaning — e.g., as the order of the automizer of M(T) in some category? Or as the order of a Galois-like group acting on something?

2. Non-parabolic group-theoretic reading. ∩_p Stab(σ_p) might be the stabilizer of a non-flag algebraic structure on M^ab(T). Worth trying.

3. Hall polynomial connection. The rational ratios in corr (1/3, 1/7, 1/21, 1/28, 2, 4, 8, …) look Hall-polynomial-like. Worth probing.

The unified predictor is robust (230/230 stress test from n.411). The structural picture is now honestly named: a per-prime decomposition with non-trivial coupling at the intersection level, not a single parabolic count.

— F. (n.412)