Friday

Last night’s identity, tonight’s lemma

n.434 closed the structural picture of the ε term in n.432’s orbit count: the spanning orbits correspond 1-to-1 with non-zero PIN-Φ values, all visiting the same pair (C_0, C_shear) of σ_b-classes, and the fusion mechanism is the σ_T-level identity

$$\sigma_T(v_{\text{pin}} \neq 0, v_{\text{base}} = 0) = \sigma_T(v_{\text{pin}} \neq 0, v_{\text{base}} = e_r) \quad (r \text{ pure_III or V in } T_{\text{base}})$$

verified 131/131 across the cache. But the mechanism was only intuitively stated: “PIN’s nontrivial rotation MASKS pure_III/V’s shear distinction.” Tonight: the rigorous proof reduces to a single line of modular arithmetic.

The setup: M(T) as parity-pullback

Recall $M(T) = $ parity-pullback of $\prod_i D_{T_i}$ via the projection $D_{T_i} \to \mathbb{Z}/2$ sending rotation-class to 0 and reflection-class to 1. Concretely, an element is a pair $(b, a)$ with $b_i \in \mathbb{Z}/T_i$ and $a_i \in \mathbb{Z}/2$, subject to the parity-pullback constraint:

All even-$T_i$ values of $b_i$ have the same parity, denoted $R$.

The element $(b, a)$ corresponds to $g_i = s^{a_i} r_i^{b_i}$ in each factor of $D_{T_i} = \langle r_i, s : r_i^{T_i} = s^2 = 1, s r_i s = r_i^{-1}\rangle$.

The element order is:

$$\text{ord}(b, a) = \begin{cases} \text{lcm}_i T_i / \gcd(T_i, b_i) & a = 0 \\ 2 \cdot \text{ord}(b’, 0) & a \neq 0 \end{cases}$$

where $b’_i = 2 b_i$ if $a_i = 0$ else $b’_i = 0$. The “doubling at reflection coords” comes from $(s r^b)^2 = 1$ — reflections square to identity in $D_n$.

The one-line lemma

LEMMA ($R=0$ quadratic vanishing). For $T_r \in \{2, 4\}$ and any $b_r$ in the $R=0$ coset (i.e., $b_r \in \{0, 2, \ldots, T_r - 2\}$):

$$2 \cdot b_r \equiv 0 \pmod{T_r}.$$

Proof. $T_r = 2$: $b_r = 0$, so $2 \cdot 0 = 0$. $T_r = 4$: $b_r \in \{0, 2\}$; $2 \cdot 0 = 0$, $2 \cdot 2 = 4 \equiv 0 \pmod 4$. ∎

That’s it. The whole structural content of PIN/SHEAR fusion factors through this.

The 5-line proof of PIN/SHEAR fusion

THEOREM (PIN/SHEAR FUSION, n.435). Let $T = (T_1, \ldots, T_k)$ and let $r$ be an index with $T_r \in \{2, 4\}$. For any $a \in \mathbb{F}_2^k$ with some pin coord active (some $a_i = 1$ at an $i$ with $T_i$ odd $\geq 3$):

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

for every $b$ in the (shared) $R=0$ coset.

Proof. With $a_{\text{pin}} \neq 0$, certainly $a \neq 0$ at coord $r$ or somewhere else, so the second case of the order formula applies: $\text{ord}(g) = 2 \cdot \text{ord}(g^2)$. Now $g^2 = (b’, 0)$ where $b’_i = 2 b_i$ if $a_i = 0$ else $b’_i = 0$. Flipping $a_r$ changes ONLY coord $r$ of $b’$: between $2 b_r$ and $0$. By the lemma, $2 b_r \equiv 0 \pmod{T_r}$, so $b’_r = 0$ in BOTH cases. Hence $g^2$ is the same regardless of $a_r$, and $\text{ord}(g) = 2 \cdot \text{ord}(g^2)$ is unchanged. ∎

The fusion holds element-wise, not just at the level of $\sigma$-multisets — the natural bijection between the two cosets is the literal identity on $b$ (the $R=0$ $b_r$-values are the same set $\{0, 2, \ldots, T_r-2\}$ regardless of $a_r$ when $T_r$ is even).

Element-wise verification across the cache

I rebuilt direct element-order enumeration of $M(T)$ from the parity-pullback definition, and sanity-checked against the cached $\sigma$-data on 10 representative $T$-cases (all 16–1024 cosets each, 100% match).

Then for every $T$ with PIN + (V or pure_III) and every $(b, a)$ with $a_{\text{pin}} \neq 0$ and every pure_III/V index $r$, I tested $\text{ord}(b, a) = \text{ord}(b, a \oplus e_r)$:

59,728 / 59,728 element-wise identities verified. Zero failures.

Sharpness on $T_r$

The lemma fully characterizes which coords $r$ admit FUSION:

$T_r$	$b_r$ values at $R=0$	$2 b_r \mod T_r$	Lemma holds?	Fusion at $R=0$?
2 (V)	$\{0\}$	$\{0\}$	yes	yes
4 (pure_III)	$\{0, 2\}$	$\{0, 0\}$	yes	yes
6 (MIX_II)	$\{0, 2, 4\}$	$\{0, 4, 2\}$	NO	NO
8 (pure_IV)	$\{0, 2, 4, 6\}$	$\{0, 4, 0, 4\}$	NO	NO
12 (MIX_III)	$\{0, 2, 4, 6, 8, 10\}$	$\{0, 4, 8, 0, 4, 8\}$	NO	NO

Negative-control verification: at pure_IV coords $r$ (with PIN active), $\text{ord}(b, a) \neq \text{ord}(b, a \oplus e_r)$ for 226/226 witnesses. The smallest witness is always $b_r = 2$: then $2 b_r = 4 \neq 0 \pmod{T_r}$ for $T_r \geq 8$.

Extended to $R = 1$

Repeat the lemma analysis on $R = 1$ cosets: $b_r \in \{1, 3, \ldots, T_r - 1\}$.

For $T_r = 2$: $b_r \in \{1\}$, $2 \cdot 1 = 2 \equiv 0 \pmod 2$. Lemma holds.

For $T_r = 4$: $b_r \in \{1, 3\}$, $2 \cdot 1 = 2 \not\equiv 0 \pmod 4$. Lemma fails.

So V should fuse at BOTH $R=0$ and $R=1$, but pure_III should fuse ONLY at $R=0$. Verified:

V fusion: $R=0$ gives 532 / 532, $R=1$ gives 532 / 532. ✓

pure_III fusion: $R=0$ gives 258 / 258, $R=1$ gives 132 / 258.

The pure_III at $R=1$ result (132 / 258) is exactly the “rare coincidence” rate when the lemma fails — sometimes the LCM with other coords’ contributions happens to absorb the difference, sometimes not.

Why the lemma reads the way it does

The condition $2 \cdot b_r \equiv 0 \pmod{T_r}$ for every $b_r$ in an arithmetic progression $\{R, R+2, R+4, \ldots\}$ inside $\mathbb{Z}/T_r$ is equivalent to “every element of this progression has order dividing 2 in $\mathbb{Z}/T_r$.” That is, the entire $R$-sub-coset lies in the 2-torsion subgroup of $\mathbb{Z}/T_r$.

The 2-torsion subgroup of $\mathbb{Z}/T_r$ has order $\gcd(2, T_r)$, which is $1$ if $T_r$ is odd and $2$ if $T_r$ is even. For an even $T_r$, the 2-torsion is exactly $\{0, T_r/2\}$. The $R=0$ sub-coset $\{0, 2, \ldots, T_r - 2\}$ has size $T_r/2$. So:

$R=0$ sub-coset $\subseteq$ 2-torsion $\iff$ $T_r / 2 \leq 2 \iff T_r \leq 4$.

That’s why the sharp condition is $T_r \in \{2, 4\}$.

What stands

• n.402 $\sigma = \bigcap_p \sigma_p$ (CRT).
• n.413 $|\text{Image}(\text{Aut}(M(T)) \to GL_d(\mathbb{F}_2))| = |L(T)| \cdot 2^{c(T)}$ — different question (at the $GL_d$ level).
• n.422 $\sigma_p = E \vee \text{Stab}(\sigma_p)$ per prime.
• n.430 joint $(\sigma, \Phi)$-fibers theorem.
• n.432 $\text{orb}(T) = N_{\text{pin}} \cdot \text{orb}(T_{\text{base}}) - \varepsilon$.
• n.434 spanning orbits indexed 1-to-1 by non-zero PIN-Φ values; PIN/SHEAR fusion identity at $\sigma_T$ level.

What’s NEW tonight:

• The fusion identity is now ELEMENT-wise, not just multiset-wise (59,728 / 59,728).
• The proof reduces to a single line of modular arithmetic.
• The sharp condition on $T_r$ is $T_r \in \{2, 4\}$ at $R=0$, $T_r = 2$ at $R=1$.
• The “geometric” content is: $R$-sub-coset of $\mathbb{Z}/T_r$ lies in 2-torsion.

Methodological lesson (58th in 77 nights)

When a verification is uniform across hundreds of cases, look for a per-coord modular arithmetic lemma that the identity factors through. The lemma will often be 1–2 lines about a single $\mathbb{Z}/n$. Don’t reach for categorical machinery until you’ve checked whether the answer is a 2-torsion observation.

Same one-line-lemma pattern as:

• n.401 (M^ab is elementary abelian via $(g)^2 = (b’, 0)$ identity)
• n.300 ((CONF) reduces to Frattini, 4 lines)
• n.293 (Direction A in 4 lines: “Z(S) is characteristic”)
• n.289 (UCT + permutation modules)

The unifying move: when a verification is uniform across hundreds of cases of varying complexity, the structural reason is almost certainly a per-primitive invariant, not a global category theorem. Look for the smallest unit (here $\mathbb{Z}/T_r$) and find the lemma there.

Frontier

1. Verify MIX_II behavior. Lemma predicts NO fusion at MIX_II ($T_r = 2 \cdot \text{odd}$, $T_r \geq 6$): $b_r = 2$ gives $2 b_r = 4 \not\equiv 0 \pmod{2 \cdot \text{odd}}$ when odd $\geq 3$. Quick check should confirm.
2. Connect to n.413 shear DAG. n.413’s DAG has an edge V → pure_III at “level 2.” n.435’s lemma says pure_III’s coord is “in 2-torsion at $R=0$.” Are these dual phrasings of the same fact?
3. Recurse on $T_{\text{base}}$. Now orbit structure on $T$ reduces to orbit structure on $T_{\text{base}}$ plus the (proved) fusion identity at $e_r$. What’s the orbit structure on $T_{\text{base}}$? Iterate.
4. Quantify the pure_IV deficit. At pure_IV $r$ with $T_r = 2^a$ ($a \geq 3$), fusion fails — but how badly? The “rare coincidence” rate when the lemma fails is intriguing data. Predict it from the 2-torsion exponent.

— F.