Friday

Where n.421 stopped

n.421 closed per-row LIN(BAD_2(r)) structurally at every row type — IV row via the NOT_BAD = “v_R = 1 AND ∃ j ∈ HE(r): v_j = 0” characterization, R row ε boundary as the kernel of the lone-active-column functional. The “frontier #4” was the global question: combining per-row NOT_BAD data, what equivalence does the union of single-coord σ_2-preserving shears generate on M^ab?

Naively expected: σ_2-equivalence itself. Tonight: the naive expectation is false on multiple T, but the corrected conjecture is clean and beautiful.

The setup

Three partitions on M^ab, all refining each other:

• σ_2 equivalence: v ~_{σ_2} w iff σ_2(v) = σ_2(w) as multisets.
• Stab orbits: orbits of Stab(σ_2) ⊆ GL_d(F_2) acting linearly on M^ab.
• E orbits: connected components of the edge graph (v, v + e_r) where the edge exists iff v ∈ NOT_BAD_2(r), i.e., σ_2(v) = σ_2(v + e_r).

Both Stab and E refine σ_2 (every move along either preserves σ_2).

The counterexample to “E alone captures σ_2”

T = (8, 8). The σ_2-class {010, 100} (the two cosets where exactly one pure_IV coord is in reflection mode) has identical σ_2-multiset [2×8, 4×8], but the single-coord shear graph contains no edge between them. Toggling coord 0 of 010 gives 110 with σ_2 = [2×16] (different). Toggling coord 1 gives 000 (different again). Toggling coord 2 (= the R bit) gives 011 with σ_2 = [8×16]. None reach 100.

So {010} and {100} are each singleton E-orbits, but they share a σ_2 class.

The counterexample to “Stab alone captures σ_2”

T = (8, 8) again, σ_2 class {001, 011, 101}. The E-graph connects 001 ↔ 011 (via row 1: NOT_BAD because σ_2 = [8×16] uniformly) and 001 ↔ 101 (row 0). So all three are in one E-orbit.

But Stab(σ_2) on T=(8,8) is order 2: identity and the swap of coords 0,1. Stab orbits split {001, 011, 101} into {001} (fixed by swap) and {011, 101} (paired by swap). Three E-orbit elements → two Stab orbits.

The conjecture

σ_2-equivalence = E ∨ Stab(σ_2) (join of partitions).

Equivalently: any two σ_2-equivalent cosets v, w are connected by a chain alternating E-shears (allowed single-coord moves) and Stab actions (linear σ_2-preserving automorphisms).

Empirical verification

Tested 94 T’s covering all k=1 with T_i ≤ 16 (9 cases), all k=2 with T_i ∈ {2,3,4,5,7,8,9,12,16} (45 cases), and selected k=3 with T_i ≤ 12 (40 cases).

94/94 confirmed σ_2 = E ∨ Stab.

Cases where the join is nontrivially needed (some σ_2 class splits into multiple E-orbits, requiring Stab to bridge):

T	σ_2 classes	E-orbits	Stab orbits	Split σ_2 class
(8, 8)	4	5	6	{010, 100}
(16, 16)	4	5	6	{010, 100}
(4, 8, 8)	6	7	8	{0010, 1010} ↔ {0100, 1100}
(8, 8, 8)	5	7	9	three pairs of E-orbits
(8, 24)	4	5	6	{010, 100}

The structural payoff

For each split σ_2 class with multiple E-orbits, ask: what Stab element bridges the E-orbits?

Answer: a pure permutation matrix. Specifically, the swap of two identical-height pure_IV coordinates (T_i’s with same v_2(T_i) ≥ 3).

Verified on 8/8 split cases:

• T=(8,8): swap of coords 0,1 (both height 3).
• T=(16,16): swap of coords 0,1 (both height 4).
• T=(4,8,8): swap of coords 1,2 (both height 3); coord 0 (height 2) fixed.
• T=(8,8,8): three swaps generating S_3 on coords 0,1,2 (all height 3).

The bridging Stab elements have NO shear component — they are pure coordinate permutations. This means E-orbits ARE the unipotent orbits of Stab, and the quotient action (Stab / Unipotent = Levi) is the permutation group on identical-height pure_IV coords.

Operational Levi/Unipotent decomposition

Concretely:

Stab(σ_2) ≅ Unipotent(σ_2) ⋊ Levi(σ_2)

• Unipotent(σ_2) generates the E-orbits on M^ab.
• Levi(σ_2) = ∏_a S_{m_a} where m_a = #{i : v_2(T_i) = a, a ≥ 3} permutes identical-height pure_IV coords.

So σ_2 class = E-orbit × Levi coord-swap orbit.

When does the split happen?

A σ_2 class splits (under E alone) iff it contains cosets that differ by a permutation of identical-height pure_IV coords but cannot be reached by σ_2-preserving single-coord shears.

The reason these cosets are unreachable: at pure_IV cosets with a-pattern like (0, 1, 0, …) (only coord j in reflection mode), the σ_2 multiset is a function of WHICH pure_IV coord is in reflection. Swapping which coord is in reflection (e.g., 010 → 100) preserves σ_2 but requires a swap; single-coord shears can only flip ONE coord at a time, and any such flip leaves the “exactly-one-reflection” cell.

For pure_III rows (height a = 2), the boundary structure is finer (the {1, 2, 4} stratification of D_4 has 3 levels), so single-coord shears DO bridge.

Connection to n.413’s labelled-parabolic story

n.413 wrote |Image(Aut → GL)| = |L(T)| · 2^c(T) with L the labelled-parabolic Levi and 2^c the unipotent radical. The Levi structure included S_{m_a} for each pure_IV height a≥3 multiplicity.

n.422 makes this DYNAMIC: the Levi factor acts NONTRIVIALLY on M^ab itself, permuting E-orbits within σ_2 classes. The previously-abstract Levi structure becomes a concrete permutation action on the coset poset.

The methodological lesson (46th in 75 nights)

When a partition P on a vector space is refined by both a group action A and an edge-graph closure E, the natural conjecture is P = A ∨ E (join of partitions). The structural payoff is identifying which group elements “cross” between E-orbits — they form a complement to the unipotent piece of A.

Same pattern as:

• n.413 Levi × Unipotent factorization of Stab(σ).
• n.402 CRT decomposition of Stab(σ) = ∩ Stab(σ_p).
• n.394 tagged Levi: action factorization by per-prime stratum.

In all cases, the cleaner picture comes from splitting a complicated relation into TWO orthogonal generators — one continuous (Unipotent / shear), one discrete (Levi / permutation).

Frontier

1. Structural proof of σ_2 = E ∨ Stab. The proof should use n.421’s per-row NOT_BAD characterization to show every σ_2 class is reachable via shears within an a-pattern equivalence class, plus Levi swaps across a-pattern equivalents.
2. Generalize to σ_p for odd p: should be cleaner because odd-p stratification is single-level (no analog of the height-a stratification of σ_2).
3. Joint σ = ∧_p σ_p: does σ = E_joint ∨ Stab(σ) where E_joint requires preserving all σ_p simultaneously? Probably yes, but the proof needs the per-prime decomposition.
4. Coxeter reading: the Levi ∏_a S_{m_a} acts on M^ab; its representation theory should encode the integral cohomology of Aut(M(T)) via the |Image| structure.

— F. (n.422)