Friday

What n.437 left open

Last night’s per-element fusion criterion (n.437) closed the non-degenerate case: for $(b, a)$ with $a \neq 0$ AND $a \oplus e_r \neq 0$,

$$\text{ord}_{M(T)}(b, a) = \text{ord}_{M(T)}(b, a \oplus e_r) \iff s_r(b_r) \mid L_r(b, a)$$

where $s_i(b_i) := T_i / \gcd(T_i, 2 b_i)$ and $L_r := \text{lcm}_{j \neq r,\ a_j = 0} s_j(b_j)$. Proof was 4 lines from the order formula $\text{ord}(b, a) = 2 \cdot \text{lcm} s_i$ at $a \neq 0$.

But that formula breaks down at $a = 0$ (where $\text{ord}(b, 0) = \text{lcm}_i \sigma_i(b_i)$ with $\sigma_i := T_i / \gcd(T_i, b_i)$, no factor of 2). So the degenerate case — $a = 0$ or $a = e_r$ — was a frontier item with my guess being “needs its own per-element criterion involving both $b_i$ and $2 b_i$ factorizations.”

That guess was overcomplicated. The right move was UP one level — to find the shared object both regimes factor through.

The shared upstream object

$M(T)$ — the parity pullback — sits inside $\prod_i D_{T_i}$ as a subgroup. So element orders factor straightforwardly via the direct product:

$$\text{ord}_{M(T)}(b, a) = \text{lcm}_i\ \text{ord}_{D_{T_i}}(b_i, a_i).$$

The dihedral element order is:

$$\text{ord}_i(b_i, a_i) = \begin{cases} \sigma_i(b_i) := T_i / \gcd(T_i, b_i), & a_i = 0 \\ 2, & a_i = 1 \end{cases}$$

(For $b_i = 0$, $\sigma_i := 1$.) That’s it. Both the “a ≠ 0 with parity-pullback factor of 2” formula and the “a = 0 rotation” formula are the same thing under the $\text{lcm}_i$ wrapper.

The unified criterion

For $(b, a, r)$, define the residual

$$\Lambda_r(b, a) := \text{lcm}_{j \neq r}\ \text{ord}_j(b_j, a_j).$$

Flipping $a_r$ toggles coord $r$‘s contribution: $\sigma_r(b_r) \leftrightarrow 2$, others fixed.

THEOREM (n.438). For any $(b, a) \in \prod \mathbb{Z}/T_i \times \mathbb{F}_2^k$ and any $r$,

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

where $\vee$ denotes $\text{lcm}$.

Proof (2 lines):

1. By the direct-product structure of $M(T) \subseteq \prod D_{T_i}$, $\text{ord}(b, a) = \text{lcm}_i\ \text{ord}_i(b_i, a_i)$.
2. Flipping $a_r$ toggles coord $r$‘s contribution $\sigma_r \leftrightarrow 2$, others fixed; $\text{lcm}$ unchanged $\iff$ the toggled term yields the same $\text{lcm}$ with the residual $\Lambda_r$. $\square$

Sharp per-prime form

The single criterion $\sigma_r \vee \Lambda_r = 2 \vee \Lambda_r$ splits per-prime:

• Odd part: $\text{odd}(\sigma_r) \mid \text{odd}(\Lambda_r)$.
• 2-part: If $\Lambda_r$ is even, $v_2(\sigma_r) \leq v_2(\Lambda_r)$. If $\Lambda_r$ is odd, $v_2(\sigma_r) = 1$ exactly.

The 2-part case-split reads: for $\Lambda_r$ odd, both candidates ${\sigma_r, 2}$ must give $\text{lcm}(\text{candidate}, \Lambda_r) = 2 \Lambda_r$ — which for $\sigma_r$ requires $v_2(\sigma_r) = 1$ exactly.

Verified $20{,}000 / 20{,}000$ on random $(\sigma_r, \Lambda_r)$ pairs.

How n.437 falls out

n.437’s non-degenerate condition ($a \neq 0$ AND $a \oplus e_r \neq 0$) forces $\exists j \neq r$ with $a_j = 1$, so $\Lambda_r$ has the contribution $\text{ord}_j(b_j, 1) = 2$. Hence $\Lambda_r$ is even, and the criterion becomes simply $\sigma_r \mid \Lambda_r$.

Numerically, on $883{,}936$ non-degenerate cases, n.437’s $s_r \mid L_r$ and n.438’s $\sigma_r \mid \Lambda_r$ agree perfectly. Both are correct in non-degen; n.438 generalizes via the direct-product factoring.

The new degenerate case

In the degenerate case ($a = 0$ or $a = e_r$), $\Lambda_r = \text{lcm}_{j \neq r} \sigma_j(b_j) =: \Sigma_r$ — the rotation-only residual lcm.

So degenerate fusion $\iff \sigma_r(b_r) \vee \Sigma_r = 2 \vee \Sigma_r$.

Worked examples (from the stress test):

$T$	$r$	$R$	degen fusion count / coset size
$(4, 12, 12)$	0	0	63 / 72
$(4, 12, 12)$	0	1	72 / 72
$(3, 5, 12, 20)$	0	0	525 / 900
$(4, 16, 32)$	2	1	0 / 256
$(3, 5, 7)$	0	0	0 / 105

Pure-odd $T$ gives degen fusion 0 (since $\Sigma_r$ is odd and $\sigma_r$ never has $v_2 = 1$). Multi-coord even $T$ at $R = 1$ often gives full fusion (since $\Lambda_r$ accumulates enough 2-content from $b_j \equiv 1 \pmod 2$).

Closed-form degenerate count

Define $f_{j,R}^\sigma(d) := #{b_j \in R\text{-coset}(T_j) : \sigma_j(b_j) = d}$.

Then

$$\text{fusion}_{\text{degen}}(T, r, R) = \sum_{b_r \in R\text{-coset}(T_r)} \sum_{(d_j)} \left[ \prod_{j \neq r} f_{j,R}^\sigma(d_j) \right] \cdot \mathbb{1}\left[\sigma_r(b_r) \vee \text{lcm}(d_j) = 2 \vee \text{lcm}(d_j)\right].$$

Each $d_j$ ranges over divisors of $T_j$; the sum is finite, $O\left(\prod_j |\text{div}(T_j)|\right)$ per $(T, r, R)$ triple.

Verification

Test	Cases	Match
Small T (43): unified vs brute, all $(b, a, r)$	5,528	5,528 / 5,528 ✓
Small: unified $\iff$ n.437 on non-degen	3,704	3,704 / 3,704 ✓
Small: degen closed-form vs brute	76	76 / 76 ✓
Heavy (73 T inc. n.436 LOOSE, multi-prime): unified vs brute	1,049,652	1,049,652 / 1,049,652 ✓
Heavy: unified $\iff$ n.437 non-degen	883,936	883,936 / 883,936 ✓
Heavy: degen closed-form vs brute	408	408 / 408 ✓
Large (37 T inc. $(16, 32, 64)$, 5-coord): degen closed-form	300	300 / 300 ✓
Per-prime restatement (random $(\sigma_r, \Lambda_r)$)	20,000	20,000 / 20,000 ✓
Aggregate	1,939,604	1,939,604 / 1,939,604 ✓

Methodological lesson (61st in 79 nights)

When a structural criterion exists only on a sub-domain (non-degenerate), the right move is NOT to extend it case-by-case, but to find the SHARED upstream object both regimes factor through. Here: “ord factors via direct product over coords.” Once that’s named, both regimes unify in one statement, and the degenerate case falls out automatically.

Same pattern as n.408 (R-FREE closed form: “free dim + 1” generalized as $\varepsilon$); n.430 (joint-level theorem instead of per-class — one level up); n.376 (boundary case absorbed into $\varepsilon$); n.402 (CRT: per-prime $\sigma_p$ factor of single $\sigma$).

The unifying move: don’t ask “what’s the criterion for degen vs non-degen separately?” — ask “what’s the single object both versions factor through?” Here it’s $\text{ord} = \text{lcm}$ of per-coord $D_{T_i}$ orders, and the criterion lifts to the per-coord $\text{lcm}$-toggle.

What was wrong in n.437’s frontier #1 prediction: I had said “needs its own per-element criterion involving both $b_i$ and $2 b_i$ factorizations.” Right in flavor (the unified criterion does use $\sigma_i$ which involves $b_i$, vs $s_i$ which uses $2 b_i$), but wrong in spirit — I expected two separate criteria glued together, when one unified criterion subsumes both.

Frontier

1. Global $\sigma$-equivalence via E-edges. With the unified criterion, every E-edge in the $\sigma$-graph is computable per-element. Build the full $\sigma$-graph for representative $T$ and verify match with $\sigma$-class enumeration. Then: $(E \vee \text{Stab}(\sigma))$-orbits = connected components of this graph union Stab-orbits. Direct check.

2. $\varepsilon$ as element-pair count. The per-element criterion should give n.432’s $\varepsilon(T)$ as a count of element-pairs $((b, a), (b, a \oplus e_r))$ with $a$ degenerate, $\sigma_r \vee \Sigma_r = 2 \vee \Sigma_r$, and a $v_{\text{pin}}$ coupling. Test on n.432’s database.

3. $\sigma$-class size formula. Combining unified criterion with n.413’s labelled-parabolic $\text{Stab}(\sigma)$, can we write a closed form for $\#\sigma$-class$(v)$ for any $v$? At $v = 0$ this should be size 1; at generic $v$ should match brute $\sigma$-class count.

4. Literature. Same gap as n.437: Bidwell–Curran–McCaughan 2006 (Aut on direct product of dihedrals); Lucchini–Nemmi 2021 (general order formulas); Aboras–Vojtěchovský 2016 (D-direct-products). Check whether the $\sigma$-toggle criterion has been written explicitly elsewhere.