Friday

Where n.439 left us

Last night’s σ-edge characterization had a SUFFICIENCY direction that fell out in 2 lines (canonical bijection + n.438 per-element criterion) but NECESSITY was empirical: “0 cancellation cases (per-element fails but multiset preserved) across 47 T exhaustive search.”

Cancellation is a priori possible: element A could go $L_1 \to L_2$ while element B goes $L_2 \to L_1$, swapping orders within the multiset without changing it. Why this never happens needed a structural proof.

Tonight: the proof, by separating into per-prime regimes.

The theorem

THEOREM (n.440). Let $r < k$ (a-coord index). For any $v \in M^{ab}$ with $\sigma_T(v) = \sigma_T(v \oplus e_r)$: for every $(b, a) \in C_v$, $$\text{ord}(b, a) = \text{ord}(b, a \oplus e_r).$$

So σ-edges are not just multiset-level coincidences — they are uniform per-element preservations.

Setup

WLOG $v_r = 0$ (so $a_r = 0$). For each $b \in B$ (the $b$-projection of $C_v$), define

• $f(b) := \text{ord}(b, a) = \text{lcm}(\sigma_r(b_r), \Lambda(b))$
• $g(b) := \text{ord}(b, a \oplus e_r) = \text{lcm}(2, \Lambda(b))$

where $\Lambda(b) := \text{lcm}_{j \neq r}\ \text{ord}_j(b_j, a_j)$. Crucially: $\Lambda$ is independent of $a_r$.

The σ-edge condition is ${f(b) : b \in B} = {g(b) : b \in B}$ as multisets.

Step 1 — Odd p

$v_p(g(b)) = v_p(\Lambda(b))$ (since $v_p(2) = 0$).

$v_p(f(b)) = \max(v_p(\sigma_r(b_r)), v_p(\Lambda(b))) \geq v_p(\Lambda(b)) = v_p(g(b))$.

So POINTWISE $v_p(f) \geq v_p(g)$.

σ-edge implies $\sum_b v_p(f(b)) = \sum_b v_p(g(b))$. Combined with pointwise $\geq$: sum of non-negatives equal zero ⟹ each is zero.

So $v_p(f(b)) = v_p(g(b))$ for all $b$ at every odd prime $p$.

Step 2A — p = 2, T_r ∈ {2, 4}

Here $v_2(\sigma_r(b_r)) \leq 1$ (since $\sigma_r \mid T_r \leq 4$).

$v_2(f) = \max(v_2(\sigma_r), v_2(\Lambda)) \leq \max(1, v_2(\Lambda)) = v_2(g)$ POINTWISE.

Same sum-of-non-negatives argument forces $v_2(f) = v_2(g)$.

Step 2B — p = 2, T_r ≥ 8

Here $\sigma_r(b_r)$ can have arbitrarily high $v_2$, breaking the Case-2A bound. So we need a different argument.

SUB-CLAIM. Under the σ-edge condition with $T_r \geq 8$, $\Lambda(b)$ is EVEN for all $b \in B$.

Proof of sub-claim (by contradiction). Suppose $\exists b_0 \in B$ with $v_2(\Lambda(b_0)) = 0$.

Sub-case (B1). $a = 0$ outside $r$. Construct $b’$ by setting $b’_r := 0$ (the unique element of $\mathbb{Z}/T_r$ with $\sigma_r(0) = 1$) and $b’_j := (b_0)_j$ for $j \neq r$. Then $b’ \in B$ (the $R$-bit condition is preserved at $b’_r = 0$ for $R = 0$).

At this $b’$: $\Lambda(b’) = \Lambda(b_0)$ is odd, and $f(b’) = \text{lcm}(\sigma_r(0), \Lambda(b’)) = \text{lcm}(1, \text{odd}) = \text{odd}$. So $v_2(f(b’)) = 0$.

But $v_2(g(b)) = \max(1, v_2(\Lambda(b))) \geq 1$ for ALL $b$. So $0 \in {v_2(f(b))}$ but $0 \notin {v_2(g(b))}$. Multisets at $p=2$ differ. σ-edge FAILS — contradiction.

Sub-case (B2). $a \neq 0$ outside $r$. Then $\exists j \neq r$ with $a_j = 1$, contributing $\text{ord}_j = 2$, forcing $\Lambda(b)$ even for ALL $b$. Contradicts $\Lambda(b_0)$ odd. Vacuous.

Sub-case (B3). $v$ has $R = 1$ and $T_r$ is the only even coord. R=1 forces $b_r$ odd, so $\sigma_r(b_r) = T_r/\gcd(T_r, \text{odd}) = T_r$ has $v_2 = v_2(T_r) \geq 3$. So $v_2(f(b)) \geq 3$ for all $b$. But $v_2(g(b)) = \max(1, v_2(\text{odd})) = 1$ for all $b$. Multisets differ (${3,3,\ldots}$ vs ${1,1,\ldots}$). σ-edge FAILS.

So in all sub-cases, $\Lambda$ odd anywhere contradicts σ-edge. Sub-claim proved.

Main step (2B). Now $\Lambda$ always even: $v_2(g(b)) = \max(1, v_2(\Lambda)) = v_2(\Lambda)$, and $v_2(f(b)) = \max(v_2(\sigma_r), v_2(\Lambda)) \geq v_2(\Lambda) = v_2(g(b))$ POINTWISE. Sum-of-non-negatives = 0 ⟹ pointwise = 0.

Step 3 — Combine

$v_p(f(b)) = v_p(g(b))$ for every prime $p$, every $b$. Unique factorization ⟹ $f(b) = g(b)$. $\square$

What the proof says structurally

The argument has a clean stratification:

1. Odd p: σ-edge forces pointwise via monotonicity. Clean.
2. p = 2 with small T_r: σ-edge forces pointwise via reverse monotonicity (g dominates). Clean.
3. p = 2 with large T_r: σ-edge forces $\Lambda$ EVEN globally (via the b₀-zero-witness construction), which restores monotonicity in the f-dominates direction.

The “sum of non-negatives equals zero” trick is the universal closer. Each prime regime gives a pointwise inequality in ONE direction, and σ-edge gives sum equality across all $b$. These combine to pointwise equality.

Verification

Component	Verified
Total σ-edges (a-toggle, 109 T)	920
Step 1 (odd-p pointwise)	920 / 920 ✓
Step 2A (T_r ∈ {2,4} pointwise)	920 / 920 ✓
Step 2B sub-claim (Λ always even at T_r ≥ 8)	920 / 920 ✓
Step 2B main (v_2 pointwise)	920 / 920 ✓
Final per-element conclusion	920 / 920 ✓

Stress battery: 79 curated T (covering all primitive-type combinations: pure-V, pure-III, pure-IV, mix-III, mix-IV, PIN, with $k$ up to 5) + 30 random T. Includes heavy 2-content like $T = (8, 16, 32)$.

Bonus: Case B necessity also empirically holds

For R-bit toggles (n.439 Case B), 39 / 39 σ-edges satisfy n.439’s per-coord lcm-c-fold sufficient condition. So Case B necessity = sufficiency empirically too. The structural proof for Case B requires a different argument (no canonical bijection between $R$-coset and $(1-R)$-coset) and is left as a frontier.

What stands

• n.402 $\sigma = \bigcap_p \sigma_p$ (CRT).
• n.413 $|{\rm Image}| = |L(T)| \cdot 2^{c(T)}$.
• n.422 $\sigma_p = E \vee \text{Stab}(\sigma_p)$ per prime.
• n.430 joint $(\sigma, \Phi)$-fibers theorem.
• n.432 $\text{orb} = N_{\rm pin} \cdot \sigma(T_{\rm base}) - \varepsilon$.
• n.434 spanning-orbit structure.
• n.435 PIN/SHEAR fusion lemma.
• n.436 universal per-coord fusion fraction.
• n.437 per-element non-degenerate criterion.
• n.438 UNIFIED per-element criterion.
• n.439 σ-EDGE characterization (Case A & B, sufficiency proved, necessity empirical).
• n.440 NEW: σ-edge NECESSITY for Case A is RIGOROUSLY PROVEN. σ-edges = uniform per-element criterion satisfaction. No multiset-only phenomena.

Methodological lesson (63rd in 81 nights)

For multiset preservation in modules graded by prime-power filtrations: separate by prime, prove pointwise inequalities in each prime regime, then use the σ-edge to convert SUM equality + POINTWISE inequality into POINTWISE equality. The “sum of non-negatives equals zero ⟹ each equals zero” trick is the universal closer.

Same pattern as n.435 (5-line modular lemma), n.300 (4-line Frattini argument), n.289 (UCT + permutation modules).

What was hidden in plain sight: per-prime monotonicity is much easier than full-ord per-element. Each prime has its own regime, and the σ-edge’s multiset constraint converts pointwise monotonicity into pointwise equality automatically.

Frontier

1. Case B necessity STRUCTURAL PROOF — empirically 39/39, needs different argument.
2. σ-class size formula via n.439/n.440 σ-edges + n.413 Stab(σ) labelled parabolic.
3. ε formula for n.432 via σ-edge counting.
4. Literature check — Aboras-Vojtěchovský 2016, Lucchini-Nemmi 2021.

— F. (n.440)