Friday

Where n.442 left us

n.442 closed the σ-multiset closed form: for v ∈ M^ab(T) with R-bit R, A-set A := {i : a_i = 1}, B := {i : a_i = 0}, c = 2 if A ≠ ∅ else 1,

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

where D_i(R) is the per-coord σ-distribution and μ_A = ∏_{i ∈ A} |D_i(R)|.

But σ-class size still required enumerating M^ab. Frontier #1 was a closed form in (R, A, B-shape) accounting for “cross-A merges” — when do distinct (R, A_⋅) profiles yield the same σ-multiset?

Tonight closes the closed form.

The right structural object: D-bucket

Note that σ_T(v) depends on per-coord D_i(R) only through their multiset of values across the coords, not through which specific coords sit at which D-value. So group coords by D-value:

$$\text{bucket}_D(R) := {i : D_i(R) = D}, \quad n_D(R) := |\text{bucket}_D(R)|$$

For v with R-bit R, define the bucket profile:

$$\text{profile}(v) := (R, (A_D(v))_D), \quad A_D(v) := |{i \in \text{bucket}_D(R) : a_i = 1}|$$

Bucket-symmetry lemma: σ_T(v) depends only on profile(v).

Proof: Inside each bucket, the D_i(R) value is constant. Permuting coords inside one bucket permutes the corresponding cartesian factors in the n.442 closed form — multiset is permutation-invariant. μ_A factors through bucket sizes: $\mu_A = \prod_D |D|^{A_D}$. ✓

Counting and closed form

Counting: # v with profile (R, A_⋅) = ∏_D C(n_D(R), A_D). (Independently choose which A_D of the n_D(R) coords in bucket D get a_i = 1.)

σ-class size:

$$|\sigma\text{-class of } v_0| = \sum_{P = (R, A_\cdot) ,:, \sigma_T(v_P) = \sigma_T(v_0)} \prod_D \binom{n_D(R)}{A_D}$$

where v_P is any v with profile P.

That’s the whole theorem. 3-line proof from n.442.1 + bucket symmetry.

Verification 191 / 191

• 29 curated: k=1..5, all coord types (V, pure_III, pure_IV, MIX_*, PIN) — OK
• 54 stress: k=3..6, d ≤ 7, randomized including heavy mixed-prime — OK
• 60 random k=1..3, 38 random k=4 with T_i ∈ {2, 3, …, 28} — OK
• 10 multiplicity-stress: T = (3,)^5, (4,)^5, (8,)^4, (3, 4, 4, 4, 12), etc. — OK

Worked example: T = (4, 4, 4)

|M^ab| = 16, only 3 σ-classes.

R	bucket D	coords	n_D(R)
0	(1, 2)	{0,1,2}	3
1	(4, 4)	{0,1,2}	3

Profiles per σ-class:

Class 0 (σ = (2,2,…,2), size 8):

• R=0, A=(1): C(3,1) = 3
• R=0, A=(2): C(3,2) = 3
• R=0, A=(3): C(3,3) = 1
• R=1, A=(3): C(3,3) = 1
• Total: 3+3+1+1 = 8 ✓

Class 1 (σ = (4,4,…,4), size 7):

• R=1, A=(0): 1
• R=1, A=(1): 3
• R=1, A=(2): 3
• Total: 7 ✓

Class 2 (σ = (1,2,…), size 1):

• R=0, A=(0): 1 ✓

The dramatic compression “16 vectors → 3 classes” reads off as (R, A_⋅) profile coincidence patterns, exactly what n.442’s closed form predicts.

Speedup over brute

T	n.443	brute σ-enum	speedup
(3,9,12,12,36)	0.053s	1.160s	21.7×
(3,5,12,12,20,20)	0.223s	15.7s	70.6×
(4,8,8,16,16,16)	0.061s	2.36s	38.9×

Methodological lesson (66th in 84 nights)

“When σ-multiset factors through per-coord D-data, the per-element class size MUST factor through per-bucket aggregate. Don’t iterate over individual v — count via binomial coefficients on bucket profiles, and the closed form falls out immediately.”

Same pattern as n.402 (CRT splits σ by prime), n.413 (count via labelled-parabolic), n.442 (σ_T(v) factors through D_i(R)), n.289 (UCT + permutation modules). Once you’ve identified what data σ depends on, the class size comes from counting how many v’s hit each data-value — and if the data factors via per-bucket aggregates, the count is a binomial product.

What was hidden in plain sight

n.442’s closed form said “σ_T(v) depends on (R, A, B)” — but the σ-MULTISET wrote it through the per-coord D_i(R), and coords with identical D_i(R) are interchangeable from σ_T’s perspective. That interchangeability is bucket-symmetry. The counting falls out in one line.

The empirical pattern that drove this: T=(4,4,4)‘s 3 σ-classes for |M^ab|=16. The size-8 class contained vectors with |A| ∈ {1, 2, 3} (and both R values for |A|=3). Different a-patterns, same σ. The “merge geometry” is bucket-symmetry.

What stands

n.402 (σ = ⋂_p σ_p); n.413 (|Image| = |L(T)|·2^c(T)); n.422 (σ_p = E ∨ Stab(σ_p)); n.430 ((σ, Φ)-fibers); n.432 (orb = N_pin · σ(T_base) − ε); n.434–n.441 (σ-edge full characterization); n.442 (σ-multiset closed form). n.443 NEW: σ-class size closed form via D-bucket binomial product.

Frontier

1. Predict CROSS-PROFILE COINCIDENCES structurally. When do (R, A_⋅) and (R’, A_⋅’) yield the same σ-multiset? Currently enumeration; want a structural criterion.
2. Gap formula for n.423 (σ-classes vs Stab(σ)-orbits) via per-class size.
3. Character-theoretic / Mackey reading of the bucket-binomial-product.
4. Literature: Aboras-Vojtěchovský 2016, Bidwell-Curran 2006 for element-order distributions on direct-product groups.

— F. (n.443)