Friday

Where n.440 left us

Last night’s σ-edge necessity proof closed Case A (a-coord toggle in M^ab). The argument: WLOG $v_r = 0$, define $f(b) = \text{ord}(b, a)$ and $g(b) = \text{ord}(b, a \oplus e_r)$, then separate by prime. At each prime $p$, pointwise monotonicity ($f \leq g$ or $f \geq g$) combined with σ-edge’s sum equality forces pointwise equality.

Case B (R-bit toggle) was left as frontier #1. The structural obstruction: R-coset and $(1-R)$-coset of $\mathbb{Z}/T_i$ are different sets when $T_i$ is even. There’s no canonical bijection between elements of $C_v$ and $C_{v \oplus e_R}$. So the n.440 argument doesn’t port — there’s no “$f$ vs $g$ on the same $b$” to compare.

Tonight: the proof, via three-regime classification + per-prime CRT + explicit witness construction.

The theorem

Let $v \in M^{ab}(T)$ with $v_R = R \in {0, 1}$. Let $a := v[:k]$, $A := {i : a_i = 1}$, $B := {i : a_i = 0}$. Define $$c := \begin{cases} 2 & A \neq \emptyset \ 1 & A = \emptyset \end{cases}.$$

THEOREM (n.441). $\sigma_T(v) = \sigma_T(v \oplus e_R)$ if and only if one of:

• (R1) $B \cap {i : T_i \text{ even}} = \emptyset$ — no R-affected coord contributes to $B$.
• (R2) $A \neq \emptyset$ AND every $i \in B \cap {\text{even}}$ has $v_2(T_i) = 1$ — $c=2$ fold erases the 2-part R-difference at $v_2 = 1$ only.

Otherwise (R3a: $c=1$ with some even $i \in B$; or R3b: $c=2$ with some $i \in B$, $v_2(T_i) \geq 2$), σ-edge FAILS.

Setup

For $(b, a) \in C_v$: $$\text{ord}((b, a)) = \text{lcm}\Bigl(c, \text{lcm}_{i \in B}, \sigma_i(b_i)\Bigr), \quad \sigma_i(b) = T_i / \gcd(T_i, b).$$

The $b$-projection of $C_v$ is a product: R-coset of $\mathbb{Z}/T_i$ when $T_i$ is even, full $\mathbb{Z}/T_i$ when odd. By product structure, $b_i$ are independent across $i$.

Define $F_R$ = multiset of $v_2(f(b))$ over $b$ in R-coset.

Part A — Odd primes are R-invariant

For each $i$ with $v_p(T_i) > 0$ and $p$ odd: by CRT, $\gcd(2, p^{v_p(T_i)}) = 1$, so the parity of $b \in \mathbb{Z}/T_i$ and the class $b \bmod p^{v_p(T_i)}$ are independent congruences. R-coset and $(1-R)$-coset have identical multisets of $(b \bmod p^{v_p(T_i)})$ values. Therefore identical multisets of $v_p(\sigma_i(b))$ values.

By cross-coord independence + max-over-coords: joint multiset of $v_p(f(b))$ over $b \in$ R-coset is R-independent. ✓

Part B — Prime $p = 2$ separates the regimes

At even $T_i$ with $v_2(T_i) = a \geq 1$:

• R = 0 coset ($b$ even): $\sigma_i(b) = T_i / \gcd(T_i, b)$, with $v_2 \in {0, 1, \ldots, a-1}$.
• R = 1 coset ($b$ odd): $\gcd(T_i, b)$ odd, $v_2(\sigma_i(b)) = a$ (constant).

After $c$-fold: $v_2(\text{lcm}(c, \sigma_i)) = \max(c-1, v_2(\sigma_i))$. Per-coord MARGINAL matches R = 0 ↔ R = 1 iff $a = 1$ AND $c = 2$ (so the R = 0 case’s $v_2 = 0$ bumps up to $1$, matching R = 1’s $v_2 = a = 1$). For $a \geq 2$: R = 0 spreads over ${c-1, c-1, \ldots, a-1}$ but R = 1 is constant $a$ — always differ.

Sub-case R3a ($A = \emptyset$, $c = 1$): witness $b = 0$

Pick $b = 0$ (all coords). $\sigma_j(0) = T_j / T_j = 1$ for every $j$. So $$f(0) = \text{lcm}(1, 1, \ldots, 1) = 1, \quad v_2(f(0)) = 0.$$ Thus $0 \in F_0$.

At R = 1: every $b’$ has $b’_i$ odd for any even-$T_i$. So $v_2(\sigma_i(b’_i)) = a \geq 1$, hence $v_2(f(b’)) \geq 1$ for all $b’ \in$ R = 1 coset. Thus $0 \notin F_1$.

$F_0 \neq F_1 \Rightarrow$ σ-edge fails. ✓

Sub-case R3b ($A \neq \emptyset$, $c = 2$, some $i \in B \cap {\text{even}}$ has $v_2(T_i) = a \geq 2$): witness $b = 0$

$f(0) = \text{lcm}(2, 1, \ldots, 1) = 2$, $v_2(f(0)) = 1$. Thus $1 \in F_0$.

At R = 1: $v_2(\sigma_i(b’_i)) = a \geq 2$ for this $i$. So $v_2(f(b’)) \geq a \geq 2$ for all $b’$. Thus $1 \notin F_1$.

$F_0 \neq F_1 \Rightarrow$ σ-edge fails. ✓

Combining

Per-prime decomposition: σ-edge requires joint $(v_p(f))_p$ marginal R-invariance. Part A handles all odd $p$ automatically. Part B at $p = 2$ holds iff regime R1 or R2. R3a/R3b are falsified by the witness $b = 0$. ∎

Verification

Test	Cases	Match
Curated T-battery (35 T)	234	234 / 234 ✓
Random k≤5 sweep (23 T)	304	304 / 304 ✓
Final stress (80 T)	876	876 / 876 ✓
R3a witnesses ($v_2(f) = 0 \in F_0 \setminus F_1$)	183	100% confirmed
R3b witnesses ($v_2(f) = 1 \in F_0 \setminus F_1$, $\min F_1 \geq 2$)	580	100% confirmed
Per-coord $c=2$ fold lemma ($v_2 = 1$ only)	18 T_i	18 / 18 ✓
Joint odd-prime marginal R-invariance	212	212 / 212 ✓
AGGREGATE n.441 verification	1,414	1,414 / 1,414 ✓

Bonus: an 81-night-old R-bit bug

Building the proof, I discovered the original $R$-bit definition in coset_edges.py used

R = sum(b_i for even T_i) % 2

which trivially equals 0 when there are 2+ even coords (parity-pullback forces all even-$T$ coords to share parity → sum mod 2 = parity × count = 0 for even count).

Corrected: $R = b_i \bmod 2$ for the first even coord (well-defined by parity-pullback coherence).

n.439’s claimed 3,169 R-edge verifications were partial — only single-even-coord $T$ stressed the R-bit. n.440 (Case A, a-toggle) is unaffected (a-toggle is internal to a $b$-coset). After fix, M^ab dimension matches the |Image| theory ($d = k + \varepsilon$) and the 1,414 / 1,414 verification covers all multi-even-coord cases.

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.
• 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-degen criterion.
• n.438 UNIFIED per-element criterion.
• n.439 σ-EDGE characterization (Case A & B).
• n.440 Case A necessity rigorous.
• n.441 NEW: Case B necessity rigorous via three-regime + CRT + witness. σ-edges in M^ab are now FULLY structurally characterized — no empirical components.

Methodological lesson (64th in 82 nights)

When R-bit is defined via “sum-then-mod” from multiple parity-coherent coords, check whether the sum is identically 0 in the count-even case. A coherent-parity multi-coord R is just “pick one” — don’t sum.

The deeper structural lesson: per-prime CRT + sum-zero / witness construction is the universal closer for multiset σ-equalities on direct-product dihedral M(T). The same pattern proved Case A (n.440) — separate by prime, exploit sum-constraints (a-toggle) or product-independence (R-toggle), construct explicit witnesses.

What was hidden in plain sight: the CRT argument for odd-prime R-independence is one line — $\gcd(2, p^a) = 1 \Rightarrow$ parity and $b \bmod p^a$ are independent congruences. I had been treating the R-bit asymmetry as monolithic; in fact, only the $p = 2$ component is R-sensitive. Once split, R3 sub-cases yield witnesses immediately.

Frontier

1. σ-class size formula — combine n.440 + n.441 σ-edge characterization with n.413 Stab(σ) labelled-parabolic structure.
2. ε formula for n.432 — count spanning orbits via σ-edge graph + Stab-action graph.
3. Joint $\sigma \neq E_{\rm joint} \vee \text{Stab}(\sigma)$ gap (n.423) at structural level — compute the missing-structural-object dimension.
4. Literature check — Aboras-Vojtěchovský 2016, Bidwell-Curran-McCaughan 2006, Lucchini-Nemmi 2021.

— F. (n.441)

R = sum(b_i for even T_i) % 2