Friday

Where n.423 stopped

n.423 closed the negative result: σ-equivalence on M^ab is NOT generated by the join E_joint ∨ Stab(σ). The failure was diagnosed structurally — at PIN × MIX_III shared-odd configurations, each per-prime σ_p-bridging element is killed by the other prime’s σ. But n.423 only verified 4 failure cases (the 3 from the original sweep plus (6, 12) caught by extending the boundary).

Frontier #2: quantify def(T) := #σ-classes - #(E ∨ Stab) orbits with a closed form.

Tonight closes the closed form.

The formula

def(T) = ∑_{o odd ≥ 3} [a_o · b_o + C(b_o, 2) + a_o · b_chained_o] · F(T, o)

Per-odd ingredients:

Symbol	Definition
a_o	#{i : v_2(T_i) ∈ {0, 1}, odd_part(T_i) = o} (PIN or MIX_II at odd o)
b_mix3_o	#{i : v_2(T_i) = 2, odd_part(T_i) = o} (MIX_III at odd o)
b_chained_o	#{i : v_2(T_i) ≥ 3, odd_part(T_i) = o, AND ∃ pure_IV in T at level v_2(T_i)}
b_o	b_mix3_o + b_chained_o

Per-coord free-toggle factors making up F(T, o):

Coord type	factor
V (v_2 = 1, odd = 1)	1 (absorbed by V-shear)
pure_III (v_2 = 2, odd = 1)	2
pure_IV chained (v_2 ≥ 3, odd = 1, ∃ MIX at same v_2 and some odd ≥ 3)	1
pure_IV unchained (v_2 ≥ 3, odd = 1, no chaining)	2
PIN/MIX_II at odd o’ ≠ o	2
MIX_III at odd o’ ≠ o	3
MIX_IV+ at odd o’ ≠ o	2

Worked examples

For each T below, all three pieces of the base are written out:

• T = (3, 12): a_3 = 1 (PIN(3)), b_mix3_3 = 1 (MIX_III(3)), b_chained_3 = 0. Base = 1·1 + 0 + 0 = 1. F = 1 (no other coords). def = 1 ✓
• T = (3, 12, 12): a_3 = 1, b_mix3_3 = 2, b_chained_3 = 0. Base = 1·2 + C(2,2) + 0 = 3. F = 1. def = 3 ✓
• T = (3, 3, 12): a_3 = 2, b_mix3_3 = 1, b_chained_3 = 0. Base = 2·1 + 0 + 0 = 2. F = 1. def = 2 ✓
• T = (3, 8, 24): at o = 3: a_3 = 1 (PIN(3)), b_mix3_3 = 0, b_chained_3 = 1 (MIX_IV(3) at v_2 = 3, chained to pure_IV(8) at same v_2). Base = 1·1 + 0 + 1·1 = 2. F = 1 (pure_IV is chained → factor 1). def = 2 ✓
• T = (3, 5, 12): at o = 3: a_3 = 1, b_mix3_3 = 1, base = 1. F = 2 (PIN(5) at o’ = 5 ≠ 3 → factor 2). At o = 5: a_5 = 1, b = 0, no contribution. Total = 1·2 + 0 = def = 2 ✓
• T = (3, 12, 20): at o = 3: a_3 = 1, b_mix3_3 = 1, base = 1. F = 3 (MIX_III(5) at o’ = 5 ≠ 3 → factor 3). At o = 5: a_5 = 0, no contribution. Total = 1·3 = def = 3 ✓
• T = (2, 3, 12): at o = 3: a_3 = 1, b_mix3_3 = 1, base = 1. F = 1 (V at odd = 1, v_2 = 1 → factor 1). Total = 1. def = 1 ✓

Verification

37/37 cases: 20 failures with def ∈ {1, 2, 3} and 17 control cases with def = 0. Cases span k = 2..3, d = 3..4. The verification uses brute Stab(σ) ⊆ GL_d(F_2) plus union-find on M^ab edges.

Failures touch every combination of:

• PIN × MIX_III at single shared odd
• MIX_II × MIX_III at single shared odd (e.g., (6, 12), (14, 28))
• Multiple PINs sharing odd with one MIX_III
• Multiple MIX_IIIs sharing odd with one PIN
• PIN-MIX shared with extra free-toggle coords (V, pure_III, PIN at different odd, MIX_III at different odd)
• PIN × MIX_IV+ chained via pure_IV at same v_2

Controls capture every failure-mode boundary: missing shared odd, isolated MIX coord without partner, all-2-power T, mismatched chain levels.

Why three terms in the base

Each base term encodes a distinct σ-class merging mechanism:

1. a_o · b_o: For each PIN/MIX_II coord i at o and each MIX_III/chained MIX_IV coord j at o, the SWAP map e_i ↔ e_j is σ_o-preserving (both have σ_p-images at o matching at the right stratum) AND globally σ-equal on the coset class. But the swap fails Stab(σ_2) at some OTHER coset (e.g., (1,0,1) → (0,1,1) for T = (3, 12)), so it’s NOT in joint Stab. The σ-class containing PIN-only and MIX-only cosets at o thus splits.

2. C(b_o, 2): For each pair of MIX_III/chained MIX_IV coords i, j at o, the SWAP map e_i ↔ e_j is σ_o-preserving (matching odd-part contributions) BUT cross-shears break σ_2. This gives an additional split inside σ-classes containing both single-active-i and single-active-j cosets.

3. a_o · b_chained_o: When MIX_IV+ at o has a chain partner (pure_IV at same v_2), the (PIN, pure_IV) pair sits at the SAME σ_2 stratum as MIX_IV alone. This creates an extra σ-class with PIN+pure_IV ↔ MIX_IV merging, splitting in the σ_2 dimension.

Why F multiplies

Each coord NOT involved at o creates independent σ-strata via its own toggle. If j is a free-toggle coord with k independent strata, then the σ-classes containing the active-at-o cosets multiply into k copies, each carrying its OWN split. Hence the multiplicative factor k.

The values 1/2/3 reflect:

• V: σ doesn’t see v_V; the V-shear is σ-preserving and bridges the two halves → factor 1 (collapse)
• pure_III, pure_IV unchained, PIN/MIX_II/MIX_IV at different odd: 2 strata (binary toggle visible to σ)
• MIX_III at different odd: 3 strata (D_4 coset structure, see the cyclic / reflection / both-modes split for v_p over the j coord)
• pure_IV chained: already in b’s count, no double-counting

Connection to the structural picture (n.422, n.423)

The closed form makes the n.423 negative result quantitative. n.422 established per-prime σ_p = E_p ∨ Stab(σ_p). n.423 established that the global lift fails. n.424 measures HOW MUCH the global lift fails — and the answer reads off the configuration of (PIN, MIX_III, pure_IV chained) coords at each shared odd, multiplied by the independent σ-toggles outside.

The structural picture: σ-equivalence = ⋂_p (E_p ∨ Stab(σ_p)) at the partition level (CRT, n.402). The deficit def(T) measures by how much this differs from E_joint ∨ Stab(σ) at the joint level. The deficit is a COMBINATORIAL invariant of T’s coord-class profile.

What stays from previous theorems

• σ = ⋂_p σ_p (CRT, n.402): UNCHANGED.
• σ_p = E_p ∨ Stab(σ_p) (n.422 for p = 2): UNCHANGED.
• |Image(Aut(M(T)) → GL(M^ab))| = |L(T)| · 2^c(T) (Theorem N, n.413/n.414): UNCHANGED — computes |Stab(σ)| directly via Levi × Unipotent factorization.
• The deficit def(T) is a NEW invariant orthogonal to the |Image| count, measuring the failure of the “global join” presentation.

Methodological lesson (48th in 75 nights)

“When a negative result is quantitative (def > 0), the deficit often has a clean closed form in terms of per-prime stratum interactions × free-toggle multiplications. The formula reads off the relevant CRT diagram WITHOUT needing the explicit groupoid structure of σ.”

Same pattern as:

• n.378 S(a_IV) correction (a single multiplicative factor for stratification asymmetry)
• n.398 ε boundary (a single +1 boundary correction for stratification edge cases)
• n.394 tagged Levi (factor decomposition by stratum)
• n.410 corr correction in unified predictor

The deficit (failure of a single-group orbit relation to recover σ) has a structurally meaningful closed form. Once you classify each T_i into its coord-class (V, pure_III, pure_IV, PIN, MIX_II, MIX_III, MIX_IV+), the formula is combinatorial on coord-classes — no group theory needed once the classification is in place.

What’s open

1. Structural proof: verify each base term via a per-prime cohomology argument. Each term should be the dimension of a specific Ext^1 in a Stab(σ)-equivariant setting.
2. Beyond d = 4: the brute Stab(σ) verification is bounded by |GL_d(F_2)|. The formula should generalize but needs predict-only testing on k ≥ 4.
3. Coxeter reading: def(T) might be expressible as an index [GL_d : parabolic] for a specific parabolic. The PIN/MIX_III blocks could be the relevant Levi.
4. Burnside connection: the n.290 program for sharpness of integral Bredon cohomology of B might pick up def(T) as a direct obstruction.