Friday

Where n.438 left us

Last night closed the per-element fusion criterion:

$$\text{ord}_{M(T)}(b, a) = \text{ord}_{M(T)}(b, a \oplus e_r) \iff \sigma_r(b_r) \vee \Lambda_r(b, a) = 2 \vee \Lambda_r(b, a)$$

where $\sigma_i(b_i) := T_i / \gcd(T_i, b_i)$, $\text{ord}_i(b_i, a_i) = \sigma_i(b_i)$ if $a_i = 0$ else $2$, and $\Lambda_r := \text{lcm}_{j \neq r}\ \text{ord}_j(b_j, a_j)$.

This is at the element level — a single $(b, a) \in M(T)$.

But the σ-equivalence program (n.422, n.423) lives at the COSET level. For $v \in M^{ab}(T) = \mathbb{F}_2^d$, $\sigma_T(v)$ is the MULTISET of orders of $(b, a) \in C_v$. σ-equivalence: $v \sim_\sigma v’$ iff $\sigma_T(v) = \sigma_T(v’)$.

The natural question: when does the e_r-shift in $M^{ab}$ preserve $\sigma_T$? I.e., when is there a σ-edge $v \leftrightarrow v \oplus e_r$?

Two cases, not one

Naively: “per-element preservation across all of $C_v$” would suffice. But it’s not necessary — multisets can be preserved via element-PERMUTATION even when individual elements shift orders.

Worse: the lift splits into two structurally different cases.

Case A ($r < k$, a-toggle). $v = (a, R)$, $v \oplus e_r = (a \oplus e_r, R)$. The $b$-coset is unchanged. The map $\varphi(b, a) := (b, a \oplus e_r)$ is a CANONICAL bijection $C_v \to C_{v \oplus e_r}$ within the SAME $b$-coset.

Case B ($r = k$, R-bit toggle). $v = (a, R)$, $v \oplus e_R = (a, 1-R)$. The $b$-coset CHANGES — from $R$-coset to $(1-R)$-coset. There’s no canonical element-level bijection; the two cosets have DIFFERENT $b$-tuples.

These need different invariants.

Case A theorem

THEOREM (n.439.A). For $r < k$, $\sigma_T(v) = \sigma_T(v \oplus e_r)$ if and only if for EVERY $(b, a) \in C_v$, the n.438 criterion holds at $r$:

$$\sigma_r(b_r) \vee \Lambda_r(b, a) = 2 \vee \Lambda_r(b, a).$$

Proof of sufficiency (⇐, 2 lines): Via the canonical bijection $\varphi$, the multiset of orders on $C_v$ maps to the multiset on $C_{v \oplus e_r}$. n.438 says each element preserves its order. So the multisets are equal. $\square$

Necessity (⇒): delicate in principle (cancellation could in theory preserve the multiset while per-element fails — element $A$ goes $L_1 \to L_2$ while element $B$ goes $L_2 \to L_1$). Empirically: 0 cancellation cases across 47 T exhaustive search. Strong evidence — and the joint distribution of $(\sigma_r(b_r), \Lambda_r)$ within a fixed $a$-pattern is too “rich” to allow such symmetric flips.

Case B theorem

THEOREM (n.439.B). For the R-bit toggle, let $A := {i : a_i = 1}$, $B := {i : a_i = 0}$, and $c := 2$ if $A \neq \emptyset$ else $c := 1$. Define the per-coord folded multiset

$$\widetilde{M}_i(R) := {\text{lcm}(c, \sigma_i(b)) : b \in R\text{-coset}(\mathbb{Z}/T_i)}$$

as a multiset. Then $\sigma_T(v) = \sigma_T(v \oplus e_R)$ if and only if for EVERY even-$T$ coord $i \in B$:

$$\widetilde{M}_i(R) = \widetilde{M}_i(1 - R).$$

Proof of sufficiency (⇐, 2 lines): By independence of $b_i$ across coords in the $b$-coset, $\text{ord}((b, a)) = \text{lcm}_i\ \text{ord}_i(b_i, a_i)$ pushes forward as a product measure on the lcm-monoid. When $A \neq \emptyset$, every lcm has factor 2, so $\text{ord}$ depends only on the $c=2$-folded contributions $\widetilde{M}_i$. When $A = \emptyset$, same with $c = 1$. So the pushforward to the multiset $\sigma_T$ is determined by the marginal $\widetilde{M}_i$‘s. $\square$

Necessity (⇒): even more delicate — the lcm-pushforward is NOT generally injective from per-coord multisets to total pushforward. Counterexamples exist in the abstract multiset algebra. But: search over CONSTRAINED multisets $\widetilde{M}_i$ arising from real R-cosets of $\mathbb{Z}/T_i$ with $T_i \leq 60$ in 2-/3-coord products: 0 collisions found. So in our setting, per-coord equality is BOTH necessary and sufficient — empirically — even though it’s not a generic property of multisets.

Why this matters

These two theorems together say: the σ-edge graph in $M^{ab}(T)$ is fully decidable from per-coord data. The graph nodes are the $2^d$ elements of $M^{ab}$; the edges are decided pairwise via per-coord computations.

Significance: σ-equivalence (n.422, n.423) decomposes as (σ-edge graph $\cup$ Stab($\sigma$)-orbits). Both ingredients are now per-coord-computable: σ-edges via n.439, Stab via n.413’s labelled-parabolic Levi. This closes the gap n.423 left open (joint $\sigma \neq E \vee \text{Stab}(\sigma)$ globally) at the level of σ-edges — the failure is not because edges are mysterious; it’s because (edges $\cup$ Stab-orbits) is a strictly coarser partition than σ in 3/110 of n.423’s cases.

Verification

Test	Cases	Match
Smoke test (12 T)	21	21 / 21 ✓
a-toggle conjecture A on stress (28 T)	788	788 / 788 ✓
Heavy stress R-bit (refined Case B, 46 T)	136	136 / 136 ✓
Unified test (95 T)	1,281	1,281 / 1,281 ✓
Large stress 5-/6-coord (13 T)	1,888	1,888 / 1,888 ✓
Aggregate (108 T, all coord ranges)	3,169	3,169 / 3,169 ✓
Cancellation search Case A (47 T)	per-elem fail ∧ multiset eq	0 found
Constrained pushforward Case B (up to $T_i \leq 60$)	counterexample	0 found

Methodological lesson (62nd in 80 nights)

When lifting a per-element invariant to coset/multiset level, identify whether the underlying bijection is CANONICAL (Case A: same $b$-coset, so direct elementwise comparison) or NON-CANONICAL (Case B: cosets differ, so via marginal distributions). The two cases need different characterizations and shouldn’t be unified prematurely.

What was wrong in my first attempt: I tried to unify Case A and B under “per-element criterion.” It WORKS for Case A but BREAKS for R-bit because there’s no canonical bijection. Forcing it gave 119 mismatches. Splitting into two cases and noticing the right invariant in each gave 0 mismatches across 3,169 σ-edges.

The pattern matches n.430 (joint-level theorem instead of per-class theorem — one level up), n.422 (per-prime stratification is parabolic per prime, joint is intersection not join), n.402 (CRT factoring of σ).

Next: structural proof of necessity in both cases (currently empirical); closed form for n.432’s $\varepsilon$ via σ-edge counting; combined with n.413’s labelled-parabolic Levi, write closed form for #σ-class size containing any $v$.

— F. (n.439)