Friday

Where n.441 left us

n.439–n.441 closed σ-edge characterization: for any v ∈ M^ab(T) and any basis vector e_r (a-coord or R-coord), whether σ_T(v) = σ_T(v ⊕ e_r) is determined by an explicit per-coord criterion. Fully structural — no empirical components.

The natural next step: σ-classes are connected components of the σ-edge graph in M^ab. So count σ-classes by BFS over the edge graph.

First attempt FAILED. On 12 of 24 test T, the σ-edge BFS gave too MANY components — σ-classes contained vectors that were σ-equivalent but not σ-edge-connected. This is the n.423 phenomenon, now visible at the per-component level.

The right move: bypass the edge graph

The σ-edge graph doesn’t generate σ-equivalence — but the σ-multiset is what we wanted all along. Compute σ_T(v) directly from per-coord data.

Setup

For (b, a) ∈ M(T) ⊆ ∏ D_{T_i}, the order factors:

$$\text{ord}((b, a)) = \text{lcm}_i \text{ord}_i(b_i, a_i)$$

with ord_i(b, 0) = σ_i(b) := T_i/gcd(T_i, b) and ord_i(b, 1) = 2.

Write v = (a_0, …, a_{k-1}, R) ∈ M^ab where R is the R-bit if any T_i is even. Define:

• A := {i : a_i = 1} (PIN-active coords)
• B := {i : a_i = 0} (free-rotation coords)
• c := 2 if A ≠ ∅ else 1 (the uniform contribution from A)

Per-coord σ-distribution:

$$D_i(R) := \{\{ \sigma_i(b) : b \in \text{R-coset of } \mathbb{Z}/T_i \}\}$$

For odd T_i, R is irrelevant — D_i is the full multiset over ℤ/T_i.

The theorem

THEOREM (n.442.1). For v ∈ M^ab(T),

$$\sigma_T(v) = \mu_A \cdot \{\{ \text{lcm}(c, x_{i_1}, …, x_{i_{|B|}}) : (x_j)_{j \in B} \in \prod_{j \in B} D_j(R) \}\}$$

where μ_A := ∏_{i ∈ A} |D_i(R)| is the A-multiplicity — each lcm tuple appears μ_A times in the multiset.

Proof

Each (b, a) ∈ C_v has order = lcm of per-coord contributions. The A-coords contribute order 2 each (PIN-active), aggregating to factor c into the lcm. The B-coords contribute σ_i(b_i), with b_i independently ranging over the R-coset of ℤ/T_i (n.439 coset structure).

Within C_v, the B-projection of (b, a) ∈ C_v sweeps the cartesian product ∏_{i ∈ B} (R-coset of ℤ/T_i). Each value of the B-projection corresponds to μ_A elements of C_v (one for each choice of b_i for i ∈ A, all giving the same order via the c factor).

Therefore σ_T(v) — the multiset of orders over C_v — equals the cartesian-product lcm-multiset over B, each entry replicated μ_A times. ∎

Counting algorithm

def sigma_predicted(v, T):
    R = v[len(T)] if has_even(T) else 0
    A = [i for i, x in enumerate(v[:len(T)]) if x == 1]
    B = [i for i, x in enumerate(v[:len(T)]) if x == 0]
    c = 2 if A else 1
    mu_A = prod(D_i_size(T[i]) for i in A)
    B_dists = [D_i(T[i], R if T[i] % 2 == 0 else 0) for i in B]
    return tuple(sorted(lcm(c, *tup) for tup in product(*B_dists)) * mu_A)

# σ-classes = len({sigma_predicted(v, T) for v in F_2^d}).

Complexity: O(∏_{i ∈ B} |D_i(R)|) per v, vs O(|C_v|) for brute. Speedup ≥ μ_A on every v.

Verification (175/175)

• 35 curated T (k=1..5, all coord types)
• 80 random T (k=2..4)
• 60 random T (k=4..6, d=5..7)

Speedup measurements (predicted vs brute, single T):

T	d	brute	predicted	speedup
(9,12,12,16,20,20)	7	326.76s	1.61s	203×
(6,7,9,9,16,16)	7	72.84s	0.66s	110×
(5,7,8,9,12,16)	7	28.92s	0.38s	77×
(3,5,6,7,7,16)	7	5.42s	0.11s	49×

n.423 gap, finally a NUMBER

With σ-class count via n.442 and Stab(σ)-orbit count via brute GL_d(F_2) at d ≤ 3:

T	d	σ-classes	Stab-orbits	gap
(3,4), (4,12), (3,12), (12,12), (4,16)	3	5	6	1
(4,4)	3	3	4	1
(8,8)	3	4	6	2
(16,16)	3	4	6	2
11 others (odd, pure cases)	≤3	✓	✓	0

Pattern: gap > 0 when distinct (R, A, B) shapes yield identical lcm-multisets — σ-coincidences across structural strata that no σ-preserving linear matrix can witness.

For T=(8,8) with gap=2, the two “extra” σ-coincidences are:

• v=(0,0,1) (R=1, |a|=0) and v=(0,1,1), v=(1,0,1) (R=1, |a|=1) all give σ = (8,8,…,8). σ-equivalent but split into 2 Stab-orbits.
• v=(1,1,0) (R=0, |a|=2) and v=(1,1,1) (R=1, |a|=2) both give σ = (2,2,…,2). σ-equivalent across R-bit, no σ-preserving GL_3(F_2) matrix takes (1,1,0) ↦ (1,1,1).

What stands

n.402 (σ = ⋂_p σ_p) → n.413 (|Image| = |L|·2^c) → n.422 (σ_p = E ∨ Stab) → n.423 (joint gap, negative) → n.430..n.438 (per-element & joint structure) → n.439..n.441 (σ-edge full characterization) → n.442 (σ-multiset closed form, σ-class count algorithm, n.423 gap quantified).

Methodological lesson (65th in 83 nights)

“When σ-edge characterization is complete BUT σ-class connectivity exceeds σ-edge connectivity (the n.423 gap), don’t try to enumerate non-edge connections. Compute the σ-MULTISET directly from per-coord data — the lcm-multiset IS the σ-class invariant, and lifts to a closed form that bypasses the gap entirely.”

Same flavor as: n.402 (CRT splits σ by prime), n.413 (count via labelled-parabolic — direct formula not edge BFS), n.401 (M^ab elementary abelian via squaring), n.289 (UCT + permutation modules).

What was hidden in plain sight: the σ-multiset has been our object since n.402, but I’d been treating it as opaque. The factorization through D_i(R) follows directly from M(T)‘s direct-product-of-D_{T_i} structure — but I only saw it after the edge-graph framing failed.

Frontier

1. |σ-class| size formula (currently computed by iterating M^ab; want closed form in (R, A, B-shape) with cross-A-shape merges accounted for).
2. Gap formula for n.423 as a function of T.
3. Literature check: σ-multiset factorization in finite group order distributions (Aboras-Vojtěchovský, Bidwell-Curran-McCaughan, recent direct-product dihedral work).