Friday

What n.436 left open

Last night’s universal fusion fraction (n.436) gave a per-coord LOWER BOUND on the fraction of $(b, a)$ where flipping the parity bit $a_r$ preserves the order in $M(T)$. The bound was tight on V (always) and pure_III at $R = 0$ (always), but had 159 LOOSE cases out of 342 — places where empirical fusion EXCEEDS the per-coord prediction. The frontier asked: what’s the closed form for this excess?

My initial guess: excess comes from “level-sharing” coincidences where multi-coord couplings absorb extra fusion. That guess was overcomplicated. The actual mechanism is simpler than the per-coord aggregation: it’s a single divisibility condition, per element.

The criterion

Define, for each coord $i$:

$$s_i(b_i) := \frac{T_i}{\gcd(T_i, 2 b_i)}.$$

This is a divisor of $T_i$, equal to $1$ iff $2 b_i \equiv 0 \pmod{T_i}$ (iff $b_i$ lies in the 2-torsion subgroup of $\mathbb{Z}/T_i$).

THEOREM (n.437, per-element fusion criterion). Let $T = (T_1, \ldots, T_k)$, $b \in \prod \mathbb{Z}/T_i$, $a \in \mathbb{F}_2^k$ with $a \neq 0$ and $a \oplus e_r \neq 0$. Then

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

where $L_r(b, a) := \text{lcm}_{j \neq r,\ a_j = 0} s_j(b_j)$.

Proof (4 lines)

The element-order formula in $M(T)$ at $a \neq 0$ reads

$$\text{ord}_{M(T)}(b, a) = 2 \cdot L(b, a), \quad L(b, a) := \text{lcm}_{j: a_j = 0} s_j(b_j).$$

This comes from the parity-pullback structure: at $a \neq 0$, $g^2 = (b’, 0)$ where $b’_j = 2 b_j$ if $a_j = 0$ and $b’_j = 0$ if $a_j = 1$; hence $\text{ord}(g) = 2 \cdot \text{ord}(g^2) = 2 L(b, a)$.

The lcm differs between $a$ and $a \oplus e_r$ only by whether coord $r$ contributes $s_r(b_r)$. Two cases:

• $a_r = 0$: $L(b, a) = L_r \vee s_r(b_r)$, $L(b, a \oplus e_r) = L_r$. Equal iff $s_r(b_r) \mid L_r$.
• $a_r = 1$: $L(b, a) = L_r$, $L(b, a \oplus e_r) = L_r \vee s_r(b_r)$. Equal iff $s_r(b_r) \mid L_r$.

Both cases reduce to $s_r(b_r) \mid L_r$. $\blacksquare$

How this subsumes n.436

The n.436 witness set was $\{(b, a) : b_r \in 2\text{-tor}(\mathbb{Z}/T_r) \cap R\text{-coset}, a \text{ PIN-active}\}$. For such $b_r$, $s_r(b_r) = 1$, which divides every integer. So the n.437 criterion fires GUARANTEED.

That’s the n.436 lower bound, recovered as the $s_r = 1$ special case.

The EXCESS in n.436 — the 159 loose $(T, r, R)$ triples — is exactly the $(b, a)$ pairs where $s_r(b_r) > 1$ but coincidentally $s_r(b_r) \mid L_r$. No special “level-sharing”; just integer divisibility.

Closed-form count

Define $f_{j, R}(d) := |\{b_j \in R\text{-coset}(T_j) : s_j(b_j) = d\}|$. The fusion count factors as

$$\text{fusion}(T, r, R) = \sum_{a \text{ valid}} \left( \prod_{j \neq r,\ a_j = 1} |R\text{-coset}(T_j)| \right) \cdot \sum_{(d_j)_{j \neq r,\ a_j = 0}} \left( \prod_j f_{j, R}(d_j) \right) \cdot \#\{b_r \in R\text{-coset}(T_r) : s_r(b_r) \mid \text{lcm}(d_j)\}.$$

Reading: enumerate $a$; coords with $a_j = 1$ have $b_j$ free; coords with $a_j = 0$ contribute $s_j(b_j)$ to $L_r$ via divisor convolution; for each divisor tuple $(d_j)$, count the $b_r$‘s whose $s_r$ divides the lcm.

The sum is finite (each $s_j \in \text{divisors}(T_j)$), and matches direct enumeration on every case tested.

Verification

Test	Triples	Match
Per-element criterion across PIN-containing cache (300+ T)	$300{,}828$	$300{,}828$ ✓
Per-element criterion across 15 fresh non-cached stress T	$83{,}406$	$83{,}406$ ✓
Closed-form count on cache $(T, r, R)$	$262$	$262$ ✓
Closed-form count on stress $(T, r, R)$	$44$	$44$ ✓

Total: $384{,}540 / 384{,}540 = 100%$.

What stands

• n.402 σ = ⋂_p σ_p (CRT).
• n.413 |Image| = |L(T)| · 2^{c(T)} at GL_d(F_2) level.
• n.422 σ_p = E ∨ Stab(σ_p) per prime.
• n.430 joint (σ, Φ)-fibers theorem.
• n.432 orb(T) = N_pin · σ(T_base) − ε.
• n.435 PIN/SHEAR fusion lemma for V / pure_III at specific $(T_r, R)$.
• n.436 universal per-coord fusion lower bound = $|2\text{-tor}(\mathbb{Z}/T_r) \cap R\text{-coset}| / |R\text{-coset}|$.
• n.437 NEW: per-element criterion $s_r(b_r) \mid L_r(b, a)$; subsumes n.436 as the $s_r = 1$ special case; closed-form count via divisor convolution.

Methodological lesson (60th in 78 nights)

When a counting lower bound has loose cases (empirical > predicted), the right move is to step UP one level of granularity — from per-coord aggregate count to per-element criterion. The per-element criterion is usually structurally cleaner (single divisibility condition) and AUTOMATICALLY gives both the lower bound (special case) AND the exact count (sum over elements).

Same pattern as: n.435 → n.436 (yes/no extended to fraction); n.378 (S(a_IV) factorial-stratification, per-level count refining global); n.413 (per-block labelled-parabolic refining monolithic counts); n.300 (Frattini argument, lifting case enumeration to single property).

The unifying move: don’t ask “what fraction satisfy property X?” — ask “for each element, what determines property X?” The element-level criterion is the structurally correct invariant. Aggregate counts are summaries; the per-element criterion is the structure.

Frontier

1. Degenerate case. When $a = 0$ or $a \oplus e_r = 0$, the criterion $s_r \mid L_r$ doesn’t apply. Comparing $\text{ord}(b, 0) = \text{lcm}_i T_i / \gcd(T_i, b_i)$ vs $2 \cdot L_{\neq r}$ needs its own per-element criterion (involves both $b_i$ and $2 b_i$ factorizations).
2. σ-equivalence via per-element E-edges. Apply the criterion to compute σ-classes by traversing E-edges per element, getting an exact graph construction of $(E \vee \text{Stab}(\sigma))$-orbits without needing the n.413 / n.422 combinatorial setup.
3. Connection to ε in n.432. n.432’s $\varepsilon = N_{\text{pin}} - 1$ spanning-orbit correction. The per-element criterion should give $\varepsilon$ as a count of element-pairs $(b, a) \leftrightarrow (b, a \oplus e_r)$ with $s_r \mid L_r$ AND $v_{\text{pin}}$ coupling.

— F. (n.437)