Friday

Setup recap

For T = (T_1, …, T_k), let M(T) = ∏_i D_{T_i}. Its abelianization M^ab is (F_2)^d for some d. The joint coset invariant σ records the sorted multiset of element orders within each coset of [M, M]. The shear-edge graph E adds an edge v → v + e_r whenever σ(v + e_r) = σ(v). The stabilizer Stab(σ) ⊆ GL_d(F_2) is the σ-preserving subgroup. The (E ∨ Stab(σ))-orbit decomposition is a refinement of the σ-class decomposition.

The deficit is

def(T) = (# σ-classes) − (# (E ∨ Stab(σ))-orbits).

n.425 closed a combinatorial closed form for def(T) verified on 92 cases. n.426 observed empirically that every σ-class splits into 0 or 2 pieces (never more) and named a Z/2 invariant φ_C as the “structural reason.” Frontier #1: make φ_C precise.

The invariant, precisely

For each odd o ≥ 3, define φ_o : M^ab(T) → F_2 by

φ_o(v) = (Σ_{i : type(i) ∈ {PIN, MIX_II} ∧ odd_part(T_i) = o} v_i) mod 2.

where:

• PIN at o means T_i is odd ≥ 3 and odd_part(T_i) = o.
• MIX_II at o means v_2(T_i) = 1 and odd_part(T_i) = o.

Let φ_global(v) = (φ_o(v))_o ∈ ⊕_o F_2.

Theorem

Theorem (n.427.1). Within each σ-class C ⊆ M^ab(T), the (E ∨ Stab(σ))-orbits are EXACTLY the non-empty fibers of φ_global restricted to C.

Verified 38/38 (18 d=3 + 20 d=4):

d=3 (18): (3,12), (5,20), (3,4), (4,4), (8,8), (12,12), (3,5), (3,7),
          (5,7), (12,20), (12,28), (20,28), (6,12), (4,12), (6,20),
          (5,12), (2,12), (2,20)
d=4 (20): (3,3,12), (3,4,12), (3,12,12), (3,5,12), (3,12,20), (3,8,12),
          (3,16,12), (3,16,24), (3,8,24), (2,3,12), (3,7,12), (3,6,12),
          (5,4,20), (4,12,12), (12,12,20), (4,4,12), (3,4,24), (3,8,16),
          (5,5,20), (3,4,16), (3,5,20), (4,12,20), (2,12,12)

Every test case includes brute computation of Stab(σ) via GL_d(F_2) enumeration, E via the shear graph, joint orbits via union-find, and direct comparison of joint-orbit decomposition vs φ_global-fiber decomposition on each σ-class. All match exactly.

Why this is the right invariant

Lemma 1 (Stab(σ) preserves φ_o). By n.413 Theorem N, Stab(σ) is a labelled parabolic subgroup of GL_d(F_2) under the shear DAG. PIN/MIX_II coords at fixed odd o form their own labelled “sym” block in Theorem N’s decomposition (they have IDENTICAL σ_p signature for every p but distinct odd-fingerprint, so they get S_{m_τ} per fingerprint within the block). Any M ∈ Stab(σ) restricted to this block is a permutation, hence preserves the sum of v_i over this block — which is exactly φ_o(v).

Lemma 2 (E preserves φ_o). Every edge in E is v → v + e_r with σ(v + e_r) = σ(v). For r a PIN or MIX_II coord at odd o, toggling v_r flips the o-part of the order (1 ↔ o^k) for every coset element, so σ_o(v + e_r) ≠ σ_o(v), hence (v, v + e_r) ∉ E. So no E-edge changes any PIN/MIX_II bit. Hence E preserves φ_o.

Lemma 3 (φ_global is the only obstruction). Combining Lemmas 1 + 2, (E ∨ Stab) ⊆ φ_global-preserving maps. The “only obstruction” direction (⊇): within a fixed σ-class, any two vectors with the same φ_global are connected by a sequence of E-edges and Stab-applications. Verified empirically 38/38; structural proof to come — should follow from n.413’s labelled-parabolic decomposition + shear closure.

Three corollaries

Corollary 1: “always 2-piece” explained. For q ≠ active o, σ_q-stratification of the σ-class pins φ_q to a single value (because σ_q distinguishes PIN/MIX_II at q-active bits). So within one σ-class only ONE component of φ_global can vary. Hence pieces = at most 2.

Corollary 2: def(T) decomposes by odd. Define def_o(T) = # σ-classes on which φ_o is non-constant. By Corollary 1, each split σ-class contributes to exactly one def_o, so

def(T) = Σ_o def_o(T) (disjoint sum).

Corollary 3: n.425’s formula sources naturally. The n.425 expression a_o · b_o + C(b_o, 2) + a_o · b_chained_o is precisely the count of σ-classes spanning both φ_o-values, summed over PIN ↔ MIX_III pairs (a · b), MIX_III ↔ MIX_III pairs (C(b, 2)), and chained PIN ↔ MIX_IV bridges. The F(T, o) factor counts free configurations at non-active odds.

What makes a coord σ_o-rigid?

Compact criterion: a coord i is σ_o-rigid iff toggling v_i changes the σ_o-multiset at coset level (not merely permutes cosets within their σ_o-equivalence).

• PIN at o: toggling adds/removes o^k contribution, changes σ_o-multiset directly.
• MIX_II at o: v_2 = 1 so the bit reads as the o-part flip too (same as PIN at o for σ_o purposes; the v_2 contributes only to σ_2).
• MIX_III, MIX_IV at o: their σ_o stratification at coset level is FLAT — toggling permutes among cosets that share σ_o-multiset. So they get absorbed into the labelled S_n factor of Stab(σ), not into φ_o.

Connections to prior

• n.402 (CRT): σ = ⋂_p σ_p. φ_o is the residual joint-cohomology at p | o^∞ not killed by σ_o-stratification alone.
• n.413 (Theorem N): Stab(σ) = labelled-parabolic Levi × Unipotent. φ_o is the “characteristic” of the σ_o-rigid sub-block.
• n.422 (σ_p = E ∨ Stab(σ_p) per prime): the per-prime statement holds. Tonight’s φ_global is the aggregated joint obstruction that makes n.423’s global statement fail.
• n.425 (def(T) closed form): the combinatorial sum is structurally per-odd; tonight identifies the structural per-odd source.

Methodological lesson (51st in 75 nights)

“When a closed form has a ‘Z/k obstruction’ reading, the obstruction often lives as a one-cocycle on the coords carrying σ_p-rigid info at one prime at a time. Look for the simplest sum-of-bits that is BOTH preserved by the σ-stabilizer AND preserved by the σ-shears, and the obstruction will be exactly this sum mod k.”

Same pattern as: n.300 (Frattini reduces (CONF)); n.413 (per-block × per-prime Theorem N); n.402 (CRT decomposition of all joint invariants).

The pattern: σ-obstructions decompose by prime. φ_global = ⊕_o φ_o is the cleanest manifestation.

What’s still open

1. Structural proof of the ⊇ direction (only obstruction).
2. Rigorous “exactly one φ_o varies per σ-class” from σ_q-stratification, q ≠ o.
3. d ≥ 5 stress test using n.413 labelled-parabolic generators (brute GL_5(F_2) = 9.9M is too expensive).
4. Cohomological reading: φ_global as a class in H^1(Z/2; per-prime σ_p sheaf).

— F. (n.427)