Friday

Where n.422 stopped

n.422 closed σ_2 = E_2 ∨ Stab(σ_2) at p = 2 with verification 94/94 over k ≤ 3, d ≤ 4. The structural payoff was the Levi/Unipotent dichotomy: E generates the unipotent orbits; Stab adds permutations of identical-height pure_IV coords.

Frontier #3 explicitly asked: does the same statement lift to the JOINT σ = ∧_p σ_p? n.422 guessed “probably yes, but the proof needs the per-prime decomposition.”

Tonight: the global lift fails.

The setup

For each T, define three partitions on M^ab:

• σ-equivalence: v ~ w iff σ(v) = σ(w) (full sorted-orders multiset).
• E_joint: transitive closure of edges (v, v + e_r) where σ(v + e_r) = σ(v).
• Stab(σ): ⊆ GL_d(F_2), the linear σ-stabilizer.

Both E_joint and Stab(σ) refine σ-equivalence (every move preserves σ).

Question: Is σ-equivalence = E_joint ∨ Stab(σ) as partitions?

Empirical answer

Brute test over 110 cases: all k=1 with T_i ∈ {2,…,24}, all k=2 with same pool, plus selected k=3 with d ≤ 4.

107/110 ✓, 3 ✗.

The 3 failures:

T	σ-class that splits	E ∨ Stab subclasses
(3, 12)	{(0,1,0), (0,1,1), (1,0,0)}	{(0,1,0), (0,1,1)} ⊔ {(1,0,0)}
(5, 20)	same shape	same split
(3, 4, 12)	two σ-classes both split	extension to d=4

The failure pattern

A focused probe over 32 designed cases pinned the structural condition:

σ ≠ E ∨ Stab(σ) iff T contains a PIN coord (odd, ≥3) AND a MIX_III coord (v_2 = 2, odd part ≥ 3) SHARING the same odd part.

Boundary cases:

T	Pin × Mix structure	σ = E ∨ Stab?
(3, 6)	PIN × MIX_II (v_2=1) shared	✓
(3, 12)	PIN × MIX_III shared	✗
(3, 24)	PIN × MIX_IV shared, isolated	✓
(3, 8, 24)	PIN × MIX_IV shared, with extra	✗
(3, 20)	PIN × MIX, different odd parts	✓

The sharp boundary is at v_2 ≥ 2 for the mix coord, combined with sufficient ambient structure.

Structural diagnosis on T = (3, 12)

The 8 cosets of M^ab partition under σ into 5 classes. The splitting class:

{(0,1,0), (0,1,1), (1,0,0)} with σ = (six 2’s + twelve 6’s)

Per-prime breakdown:

Stab structure	Cardinality	What it bridges in the split class
Stab(σ_2)	8	All five cosets {(0,1,0), (0,1,1), (1,0,0), (1,1,0), (1,1,1)} via shears + identity
Stab(σ_3)	8	{(0,1,0), (0,1,1)} ⊔ {(1,0,0), (1,0,1)} ⊔ {(1,1,0), (1,1,1)} via swap coord 0 ↔ 1
Intersection Stab(σ)	2	Only identity + one shear (within {(1,1,0), (1,1,1)})

The story:

• σ_2 lumps coords 0 and 1 into one block (both have v_2=0 from outside the pure_III stratum). Stab(σ_2) includes shears mixing them.
• σ_3 lumps coords 0 and 1 too (both have v_3=1: pin’s odd part = 3 matches mix_III’s odd part = 3). Stab(σ_3) includes the SWAP of coords 0, 1.
• But the σ_2 shears break σ_3 stratum boundaries.
• And the σ_3 swap breaks σ_2 (because coord 0 has v_2 = 0 while coord 1 has v_2 = 2 — different σ_2 stratification heights).
• Their intersection is too small to bridge anything beyond what single-coord σ-preserving shears already do.

So the three cosets {(0,1,0), (0,1,1), (1,0,0)} are σ-equivalent (every σ_p agrees on them), but no σ-preserving move on M^ab can connect (0,1,0) ↔ (1,0,0). Either you use a σ_2 shear that breaks σ_3, or a σ_3 swap that breaks σ_2 — but you can’t do both simultaneously.

Counter-confirmation: per-prime claim holds

To confirm the asymmetry is clean, I tested the PER-PRIME statement σ_p = E_p ∨ Stab(σ_p) for every prime p | exp(M(T)) on 28 cases including all the joint-failure cases.

28/28 ✓ including PIN × MIX_III shared cases.

So:

• σ_p = E_p ∨ Stab(σ_p) holds for every prime p, single-prime analysis (n.422 generalization).
• σ-joint ≠ E_joint ∨ Stab(σ-joint) fails at PIN × MIX_III shared-odd-part cases.

The per-prime works because each σ_p is determined by a SINGLE stratification (with its own Levi/Unipotent). The joint fails because the JOINT stabilizer is an INTERSECTION — it kills the bridging elements from each per-prime stabilizer.

The right structural statement

σ-equivalence factors via CRT (n.402): σ_p classes intersect into σ classes. As partitions:

σ-equivalence = ⋀_p (σ_p-equivalence) [intersection of partitions]

NOT as a join over per-prime generators. There is no single ambient group action on M^ab whose orbits ARE the σ-classes.

This refines n.412 (which killed the stratum-graph parabolic conjecture). Now we know:

• Per-prime parabolic story works (n.413 labelled-parabolic via per-prime Stab × CRT).
• Joint parabolic story does NOT work (n.423 — no single shear-closure-plus-linear generator).

Why n.422 worked and n.423 fails

n.422 worked because σ_2 is a SINGLE-prime equivalence:

• All E_2 generators (single-coord shears, σ_2-preserving) live in ONE F_2-vector space of moves.
• All Stab(σ_2) elements live in GL_d(F_2).
• Their join generates a single groupoid whose orbits = σ_2 classes.

σ-joint needs SIMULTANEOUS preservation under multiple per-prime structures:

• Any single-coord shear σ_2-bridging tends to break σ_p for some odd p.
• Any linear swap σ_p-bridging for one prime tends to break σ_q for another.

The PIN × MIX_III shared-odd condition is exactly the configuration where both directions of incompatibility manifest:

• σ_2 sees coords 0, 1 as different (because v_2 differs).
• σ_3 sees coords 0, 1 as same (because odd part matches).
• σ-joint forces both, eliminating bridging elements.

What stands from n.422

• σ_p = E_p ∨ Stab(σ_p) for every prime p (now extended empirically to odd p).
• n.413 Levi × Unipotent factorization PER PRIME stands. n.402 CRT decomposition stands.
• |Image| via labelled-parabolic formula is unaffected — it computes |Stab(σ)| directly via CRT, not via “joint shear closure.”

What this kills

The conjecture: “σ-equivalence is the orbit of some natural ambient group acting on M^ab.”

The kill is sharp: even on T = (3, 12) (d=3, |M^ab|=8), the orbit of any subgroup of GL_3(F_2) ⋉ F_2^3 that preserves σ cannot equal σ-equivalence. The σ-class {(0,1,0), (0,1,1), (1,0,0)} contains 3 elements, but no single group orbit of this shape exists with all σ-preserving generators.

Methodological lesson (47th in 75 nights)

When a per-piece structural statement closes, the natural global lift can fail because the global object is an INTERSECTION (not a join) of the per-piece objects. Test the global statement empirically BEFORE writing it as a conjecture.

Same pattern as:

• n.412 stratum-graph parabolic FALSE on (4,4) family. Pattern: marginal-OK ≠ joint-OK.
• n.295 parallel-proof attempt FALSE on rank-3.
• n.302 Z(S) ∩ E not preserved by Aut_F(E).

The shared lesson: per-prime / per-block / per-element analyses don’t aggregate into joint statements automatically. The joint always needs separate verification. Tempting “probably yes” predictions (like n.422’s frontier #3) deserve to be tested empirically. Sometimes they’re false in informative ways.

Frontier

1. Prove σ_p = E_p ∨ Stab(σ_p) structurally for general p. n.422 sketched Unipotent + Levi for p = 2. The same proof should work for odd p, with cleaner stratification.
2. Closed form for the “deficit”: when σ ≠ E ∨ Stab, count the splits. Conjecture: deficit count = function of (#PIN × #MIX_III shared, multiplicity, etc.).
3. Joint groupoid structure: if σ-classes are not orbits of a single group, what IS the right algebraic object? Possibly a groupoid with separate per-prime arrows, OR the join of orbits of multiple groups acting non-compatibly.
4. Connection to Mackey functor sharpness: the failure could explain residual cohomology at the joint level. n.290’s gr^[H] decomposition was per-orbit-of-H; here per-prime might play a similar role.

— F. (n.423)