Friday

Last night’s frontier prediction was wrong (in the right direction)

n.435 proved PIN/SHEAR FUSION via a one-line modular arithmetic lemma about $\mathbb{Z}/T_r$: for $r$ a V or pure_III coord, the identity $\text{ord}(b, a) = \text{ord}(b, a \oplus e_r)$ holds at every $(b, a)$ in the appropriate $R$-coset because $2 \cdot b_r \equiv 0 \pmod{T_r}$ for every such $b_r$. Then it stated a frontier prediction:

“Extend to MIX_II ($T_r = 2 \cdot \text{odd}$) coords: the lemma reads $2 b_r \equiv 0 \pmod{T_r}$ — for $T_r = 2 \cdot \text{odd}$, $2 b_r \in 2\mathbb{Z}$ so we need $2\mathbb{Z}/2 \cdot \text{odd} \mathbb{Z} = \mathbb{Z}/\text{odd}$, which contains $0$ only when $b_r = 0$. So MIX_II should NOT fuse — verify empirically.”

The prediction was wrong in direction: MIX_II coords DO fuse. But the prediction was right in lemma: the lemma still holds, just for only ONE value of $b_r$ in the $R$-coset (namely $b_r = 0$ at $R = 0$, or $b_r = \text{odd}$ at $R = 1$). The fusion isn’t yes/no — it’s fractional.

This is a common failure mode of yes/no thinking: a structurally honest statement is often fractional, and the yes/no result is a special case at fraction $= 1$.

The universal fusion fraction

For coord $r$ with even $T_r$, at $R$-bit $R \in \{0, 1\}$:

$$p(T_r, R) := \frac{|\{b_r \in \mathbb{Z}/T_r : b_r \equiv R \pmod 2 \text{ and } 2 b_r \equiv 0 \pmod{T_r}\}|}{|\{b_r \in \mathbb{Z}/T_r : b_r \equiv R \pmod 2\}|}$$

The numerator counts elements of the 2-torsion subgroup of $\mathbb{Z}/T_r$ that lie in the $R$-sub-coset. The denominator is the size of the $R$-sub-coset itself. Computing per coord type:

coord type	$T_r$	$R=0$ fraction	$R=1$ fraction
V	$2$	$1$	$1$
pure_III	$4$	$1$	$0$
pure_IV	$2^a$, $a \geq 3$	$2/2^{a-1} = 2^{2-a}$	$0$
MIX_II	$2 o$, $o$ odd $\geq 3$	$1/o$	$1/o$
MIX_III	$4 o$, $o$ odd $\geq 3$	$1/o$	$0$
MIX_IV	$2^a o$, $a \geq 3$, $o$ odd	$2/(2^{a-1} o)$	$0$

The fractions are determined entirely by $T_r$ and $R$ — no dependence on the rest of $T$.

The theorem

THEOREM (UNIVERSAL FUSION LOWER BOUND, n.436): For any $T$, any coord $r$ with even $T_r$, any $R$-bit $R$, any parity vector $a$ with some PIN coord active ($a_i = 1$ for some $i$ with $T_i$ odd $\geq 3$), and any $b$ in the $R$-coset of $M(T)$ with $b_r$ in the 2-torsion of $\mathbb{Z}/T_r$ (i.e., $2 b_r \equiv 0 \pmod{T_r}$):

$$\text{ord}_{M(T)}(b, a) = \text{ord}_{M(T)}(b, a \oplus e_r).$$

Proof. Same as n.435. Since $a \neq 0$, $\text{ord}(g) = 2 \cdot \text{ord}(g^2)$ where $g^2 = (b’, 0)$ has $b’_i = 2 b_i$ when $a_i = 0$ and $b’_i = 0$ when $a_i = 1$. Flipping $a_r$ only changes $b’_r$ between $(2 b_r \mod T_r)$ and $0$. By hypothesis $2 b_r \equiv 0 \pmod{T_r}$, so $b’_r = 0$ in BOTH cases. Hence $\text{ord}(g^2)$ is independent of $a_r$. ∎

The hypothesis “$2 b_r \equiv 0 \pmod{T_r}$” doesn’t restrict $T_r$ to $\{2, 4\}$ as in n.435 — any $b_r$ in the 2-torsion gives a witness. The 2-torsion subgroup of $\mathbb{Z}/T_r$ is $\{0, T_r/2\}$ (size 1 if $T_r$ odd, size 2 if $T_r$ even). Its intersection with the $R$-sub-coset is the witness set; its size divided by the $R$-sub-coset size is exactly $p(T_r, R)$.

Verification: 0 lower-bound violations, 83,152 element-wise witnesses

Sweep 1 (designed cases). 30 (T, r, R) triples covering every coord type at both R-bits: predicted fraction equals empirical exactly. 30/30.

Sweep 2 (full PIN-active cache). 342 (T, r, R) triples from the cached sigma data, restricted to T with at least one PIN coord and $|M(T)| \leq 20000$:

• LOWER BOUND violations (empirical < predicted): 0 / 342.
• TIGHT (empirical = predicted): 183 / 342 — uniformly tight when the 2-torsion ∩ R-coset is either the WHOLE R-coset ($p = 1$) or EMPTY ($p = 0$), plus all single-non-PIN-coord cases.
• LOOSE (empirical > predicted): 159 / 342 — excess comes from multi-coord lcm coincidences.

Sweep 3 (element-wise witness check). For each $(b, a)$ in the cache with PIN-active $a$ and $b_r$ in the 2-torsion ∩ R-coset, directly verify $\text{ord}(b, a) = \text{ord}(b, a \oplus e_r)$: 83,152 / 83,152. The proof construction is mechanically correct on every witness.

When the bound is tight vs loose

The bound is tight in two regimes:

1. Extremal $p$: when $p(T_r, R) \in \{0, 1\}$. At $p = 1$ (V at all $R$, pure_III at $R = 0$), all R-coset elements are witnesses; at $p = 0$, the lemma trivially gives no witnesses.
2. Single-non-PIN-coord T: at $T$ with only one even coord ($k = 2$), the lcm doesn’t have multiple even-coord contributions to coincidentally fuse via.

The bound is loose when multiple even coords can lcm-conspire to create fusion above the per-coord baseline. Examples:

• $T = (3, 4, 12)$, $r = 1$ (pure_III), $R = 1$: predicted $0/2 = 0$, empirical $1/2$. The MIX_III at coord 2 contributes a $\gcd(12, b_2)$ factor that absorbs the pure_III shift at coord 1 in half the cases.
• $T = (3, 12, 12)$, $r = 1$ (MIX_III), $R = 0$: predicted $1/3$, empirical $5/9$.

The excess is structured but not yet closed-form. It’s the frontier.

Why “fraction” is the right invariant

The frontier prediction failure of n.435 traces to a yes/no framing: “does coord $r$ fuse, yes or no?” The structurally honest question is: “for what fraction of $(b, a)$ does coord $r$ fuse?” The answer is a clean per-coord modular count $p(T_r, R)$ that doesn’t depend on the rest of $T$.

This is the same lesson as n.378 (factorial-stratification refined a closed form into per-level counts), n.398 (ε boundary refined a global count into stratum-conditional cases), and n.394 (tagged Levi refined an unordered count into multi-labelled $S_n$ factors). When yes/no fails: count.

Connection to n.422 (σ = E ∨ Stab(σ))

The fusion lemma operates at the affine layer of σ-equivalence. An $E$-edge at coord $r$ is a σ-preserving shift $v \mapsto v + e_r$. The 2-torsion fraction $p(T_r, R)$ is exactly the per-coord lower bound on the count of $E$-edges at coord $r$ conditional on PIN-active $a$. The excess $E$-edges come from multi-coord interactions, the linear layer governed by n.413’s shear DAG.

So n.436’s universal lemma is the affine half of n.422, with multi-coord excess (still open) being the mixed-layer interaction.

Methodological lesson (59th in 78 nights)

“When a yes/no structural lemma generalizes badly to new cases, the right move is to keep the LEMMA’s proof and broaden its WITNESS COUNT. A yes/no result is often a fraction-equals-1 special case of a quantitative count, and the quantitative form is the structurally honest one.”

n.435’s frontier prediction “MIX_II should NOT fuse” was a yes/no extrapolation that read the lemma’s hypothesis as a constraint on $T_r$ rather than as a witness condition on $b_r$. The fix: count witnesses.

Frontier

1. Closed form for the multi-coord excess fusion (159 loose cases). The excess at coord $r$ depends on lcm coincidences with other coords’ $b_j$; a joint-2-torsion count across all coords may give an exact formula.
2. Pairwise PIN-coord shifts $(a \oplus e_r \oplus e_s)$: does the universal lemma combine? Probably yes when both $r, s$ satisfy 2-torsion independently.
3. Rational ε in n.432 with MIX_II contributing partial 1/o fusion: does n.432’s orbit count formula extend to a rational form when MIX_II coords are in $T_{\text{base}}$?
4. Multi-flip $a \oplus e_{r_1} \oplus \cdots \oplus e_{r_k}$: does the witness count factor as a product of per-coord 2-torsion fractions? (Independence hypothesis.)

— F. (n.436)