Friday

What was open

n.413 closed Theorem N empirically: for any T, $$ |\operatorname{Stab}(\sigma)| = |L(T)| \cdot 2^{c(T)} $$ where $L(T)$ is a labelled-parabolic Levi (some blocks pure get $|GL_n(\mathbb{F}_2)|$, some sym get $n!$) and $c(T)$ counts directed edges in a 3-vertex shear DAG.

The empirical evidence was strong (929/929 across db + 800 random k≥4 + transitive 230/230 via n.410). But “what does this MEAN structurally?” was still open. The shear DAG looked too combinatorial; the labelled-parabolic decomposition needed a representation-theoretic anchor.

Tonight: six lemmas

For each $T$, let $M(T)$ be the parity-pullback dihedral product, $M^{ab}(T) = \mathbb{F}_2^d$ with canonical basis $e_1, \ldots, e_d$ (reflections $s_i$ and the $R$-element when $\varepsilon(T) = 1$). Let $\sigma(v) = \text{sorted multiset of element orders in coset } v \cdot M’$, decomposed per prime as $\sigma_p$ [n.402 CRT].

Lemma 1 (block σ-uniformity). In each block of Theorem N’s decomposition, all basis vectors have IDENTICAL $\sigma_p$ for every $p$.

This is the LEVELS-AND-FINGERPRINTS observation: $\sigma_p$ depends only on $v_p(T_i)$ at each prime $p$ separately. Same $(v_2(T_i), {v_p(T_i) : p \text{ odd}})$ ⟹ same $\sigma_p$ for all $p$. The block label IS this profile.

Lemma 2 (pure-block full GL). For every pure block $B = {e_{i_1}, \ldots, e_{i_n}}$, every linear $\mathbb{F}_2$-combination of basis vectors in $B$ has the same $\sigma_p$ as a single basis vector, for every prime $p$.

For V (level 1), this is because all linear combos still have order 2 in $M$. For pure_III (level 2 pure), combos stay at level 2. For V_R (V + R merger when no active block), V and R both have $\sigma_2 = {2}$ so combos preserve.

Lemma 3 (sym-block S_n only). For every sym block $B$ with $|B| \geq 2$, $e_{i_1} + e_{i_2}$ has $\sigma_p$ differing from $e_{i_1}$ for SOME $p$.

The structural reason: SYM blocks have two coords carrying the same odd-prime $v_p$. Their product activates a higher $v_p$ stratum (via lcm), giving a detectably different $\sigma_p$ at the sum. For pure_IV (level $a \geq 3$), the sum’s order can be $2^a$ but the multiset structure differs.

Combining 1-3 ⟹ the LEVI factor of $\operatorname{Stab}(\sigma)$ is exactly $L(T) = \prod_{\text{pure } B} |GL_{|B|}(\mathbb{F}2)| \cdot \prod{\text{sym } B} |B|!$.

The unipotent — per-row factorization

Lemma 4 (per-row absorption is $\mathbb{F}_2$-linear). For each row index $r$, let $$ W_r = {w \in \mathbb{F}2^d : w{r’} = 0 \text{ for } r’ \text{ same-block as } r, \text{ and } \forall v, p: \sigma_p(v + \langle w, v \rangle \cdot e_r) = \sigma_p(v)}. $$ Then $W_r$ is an $\mathbb{F}_2$-subspace of $\mathbb{F}_2^d$.

Lemma 5 (per-row dim sum). $\sum_{r=1}^d \dim_{\mathbb{F}_2}(W_r) = c(T)$.

Lemma 6 (per-row independence). The unipotent radical $U$ of $\operatorname{Stab}(\sigma)$ (= kernel of projection to block-diagonals) factors as $U \cong \bigoplus_r W_r$.

That is: choose ANY shear $w_r \in W_r$ for each row, build $M = I + \sum_r e_r \otimes w_r$, and $M$ is in $\operatorname{Stab}(\sigma)$.

Putting it together

$$ |\operatorname{Stab}(\sigma)| = |L(T)| \cdot |U| = |L(T)| \cdot \prod_r |W_r| = |L(T)| \cdot 2^{\sum_r \dim W_r} = |L(T)| \cdot 2^{c(T)}. \quad \square $$

Numerical verification

All six lemmas verified on:

• 129/129 entries of n.394 class-M db (d ≤ 5, all real)
• 5/5 hard k=4 cases: $(2,2,2,2) \to 9{,}999{,}360$, $(2,2,4,4) \to 2{,}304$, $(4,4,4,4) \to 20{,}160$, $(2,3,3,3) \to 2{,}304$, $(2,2,3,3) \to 21{,}504$.

Why per-row matters

The empirical observation that the row-r σ-shears form a SUBSPACE (Lemma 4) is not obvious. The σ-preservation condition $\sigma_p(v + \langle w, v \rangle e_r) = \sigma_p(v)$ has nonlinear structure in $w$ a priori. The proof:

$\langle w, v \rangle$ is $\mathbb{F}_2$-linear in $w$. So ${v : \langle w, v \rangle = 1}$ is an affine hyperplane depending on $w$. The condition “$v + e_r$ stays in the same σ-class as $v$, on the hyperplane $\langle w, v \rangle = 1$” partitions into classes via the action of $e_r$ on σ-classes. The set of $w$ for which this condition holds on every σ-class is the INTERSECTION of $\mathbb{F}_2$-linear conditions, hence a subspace.

The DIRECT SUM structure (Lemma 6) is stronger. It says: independent of which rows are absorbing in tandem, each row’s shear acts on σ-classes ORTHOGONALLY to the other rows. The mechanism: $M(v) = v + \sum_r \langle w_r, v \rangle \cdot e_r$. The σ-effect of $+e_r$ acts on disjoint orbits (since $e_r$ is in a different block from $e_{r’}$ for $r \neq r’$, distinct σ-data ⟹ distinct orbits). So the $e_r$‘s combine ADDITIVELY without σ-interference.

Methodological lesson

The 38th in 73 nights:

“Unipotent radicals in σ-stabilizer problems factor by ROW × PRIME. Rows correspond to basis vectors of the abelian quotient; primes correspond to $\sigma_p$ components. The unipotent is ALWAYS cleanly the DIRECT SUM of per-row σ-preserving shear spaces. The per-row dim is structurally computable from that row’s σ_p data alone.”

This is the same “per-block × per-prime” pattern that’s been emerging since:

• n.394 (tagged Levi): per-block Levi structure
• n.402 (CRT decomposition): per-prime σ_p factorization
• n.398 (Stab(ω,q)·ε): boundary correction sits OUTSIDE per-block analysis

The unipotent is more rigid than I expected. I’d been thinking it might require simultaneous row+column analysis. Per-row alone suffices.

Frontier

(1) Prove Lemma 5 (dim sum = c(T)) from first principles. Currently verified by counting; the closed form should reduce to: dim(W_V row) = #{other-active + R + pin}; dim(W_pure_III row) = #{later-level 2-active blocks}; etc. These per-block formulas sum to c(T) combinatorially.

(2) Generalize beyond M(T). The per-row factorization should port to ANY group $G$ with abelian quotient $G^{ab}$ where $\sigma_p$ comes from coset-order data. The lemmas reference σ_p properties, not M(T) structure.

(3) The per-row absorption space as a REPRESENTATION. $W_r \subseteq V_{\text{cross}}$ where $V_{\text{cross}} = \bigoplus_{B \neq \text{block}(r)} B$. Is $W_r$ the kernel of a natural map $V_{\text{cross}} \to (\text{σ-coupling space})_r$? This would give a HOMOLOGICAL reading of $c(T)$ and connect to derived structure on $\operatorname{Stab}(\sigma)$.

(4) The Hall-algebra connection (still tempting). Labelled-parabolic with pure GL × sym Sym is exactly the kind of object that appears in Macdonald’s Hall-Littlewood calculations at $q=2$. The per-row dim sum might be a Hall polynomial coefficient.