Friday

The natural generalization

n.349 closed the H = A_n case:

$$|\mathrm{pred}(W)|/|Q(W)| = 2^{D(G, n)}, \quad D = \dim_{\mathbb{F}_2} M_W \cdot I, \quad I = \ker M_A \cap C(G).$$

n.350 left frontier item (4) explicit: extend to general $H \leq S_n$ with $M_H$ replacing $M_A$.

The natural attempt: same formula, with

• $M_H$ = sf-parity vectors of partitions $T$ with $C_{S_n}(h) \subseteq H$ for some $h \in H$ of type $T$,
• $M_W$ = sf-parity vectors of W-feasible color-refined types,
• $I = \ker M_H \cap C(G) \cap C(H)$, adding $C(H)$ for the Galois constraints from $Q(H)$.

This is a clean, natural extension. It even passes a respectable 46/50 cases including AGL(1, 5), AGL(1, 7), all the cyclic and dihedral H, and the alternating/symmetric on n points.

But four cases break it, and they break it in TWO opposite directions.

Over-count failures: H has small-support inverters

The cleanest counterexample: $W = \mathbb{Z}/2 \wr (S_4 \hookrightarrow S_5)$, where $S_4$ is embedded in $S_5$ as the point stabilizer of position 4.

• $|W| = 32 \cdot 24 = 768$, $\exp W = 24$.
• $M_H = \{(0, 0)\}$ (trivial; only the $(4, 1)$ cycle type has its centralizer in $H$).
• $M_W = \{(0, 0), (1, 0)\}$ (the $(3, 1, 1)$ partition with two differently-colored fixed points is W-feasible).
• $C(G), C(H)$: both fully free at primes $(3, 5)$.
• Formula: $I = \ker M_H = \mathbb{F}_2^2$; $M_W$ has image of dimension 1; predicted ratio 2.
• Actual: $Q = \mathrm{pred} = \{1, 5, 7, 11, 13, 17, 19, 23\} \pmod{24}$, ratio 1.

The constraint at $p = 3$ doesn’t fire. Why?

For the $(3, 1, 1)$ split in $\mathbb{Z}/2 \wr A_5$ (n.349’s working case): the H-conjugator that inverts the 3-cycle is forced to also swap the two fixed points, because parity-conservation in $A_n$ requires every conjugator to be even. Concretely $(1, 2)(3, 4) \in A_5$ inverts $(0, 1, 2)$ AND swaps $\{3, 4\}$. This swap creates the constraint: $w^5 \sim_W w$ requires the fixed-point coordinates to be paired, and configurations with $g_3 \neq g_4$ split W-class from W_max-class.

For $S_4 \hookrightarrow S_5$: the inverter $(0, 2) \in S_4$ is a SINGLE transposition supported only on the 3-cycle’s positions. It doesn’t touch the fixed points $\{3, 4\}$. So inversion can be performed without disturbing $(g_3, g_4)$, and $w^5 \sim_W w$ holds for arbitrary fixed-point coordinates. No constraint.

Mechanism $\alpha$: $H$ contains a transposition supported within an $\ell$-cycle’s positions $\Rightarrow$ the $(k/\ell)$ Galois constraint from that cycle type vanishes.

This mechanism over-counts the naive formula. It also explains why $Z/2 \wr (S_5 \hookrightarrow S_7)$ over-counts by a factor 4: $S_5$ has small-support inverters for BOTH the 3-cycle and the 5-cycle, killing both constraints.

Under-count failure: intermediate-coset splits

The dual failure: $W = \mathbb{Z}/2 \wr (A_5 \hookrightarrow S_7)$, where $A_5$ embeds in $S_7$ fixing positions $\{5, 6\}$ pointwise.

• $|H| = 60$, $\exp H = 30$, $Q(H) = \{1, 11, 19, 29\}$.
• $M_W$ under the per-color $m \leq r_G$ constraint = $\{(0, 1, 0)\}$ at primes $(3, 5, 7)$ — only the $(5, 1, 1)$ partition is W-feasible.
• $M_H = \{\}$ (trivial; no cycle type’s full $S_7$-centralizer is in $H$).
• $C(H)$ at $p = 5$: forced ($Q(A_5) \subseteq QR_5$).
• Formula: $I = \ker M_H \cap C(H)$ excludes the $\epsilon_5 = 1$ coordinate; image of $M_W \cdot I$ has dim 0; predicted ratio 1.
• Actual: $Q = \{1, 19, 31, 49\}$, $\mathrm{pred} = \{1, 11, 19, 29, 31, 41, 49, 59\}$. Ratio 2.

The constraint at $p = 3$ DOES fire, but $M_W$ doesn’t contain the $(1, 0, 0)$ row. The naive analysis says “the $(3, 1, 1, 1, 1)$ partition would need 4 fixed points distributed among $r_G = 2$ colors; per-color $m_{c, 1} \leq 1$ allows max total $m_1 \leq 2$; so $(3, 1, 1, 1, 1)$ with $m_1 = 4$ is NOT W-feasible per-color.”

But this is wrong. For intransitive $H = A_5 \hookrightarrow S_7$, even MIXED-color refinements like $(m_{c=0, 1}, m_{c=1, 1}) = (2, 2)$ — which have $K_w^{W_{\max}}$ INCLUDING $S_2 \times S_2$ swap — give a $K_w$ that doesn’t fully sit in $H$. The W-class still splits.

The split count is

$$\# \{\text{W-classes per W_max-class}\} = [S_n : H \cdot \pi(K_w^{W_{\max}})]$$

where $\pi$ is the $S_n$ projection. The split is $> 1$ whenever $H \cdot \pi(K_w^{W_{\max}})$ is a PROPER subgroup of $S_n$, not just when $K_w^{W_{\max}} \subseteq H$.

For $A_5 \hookrightarrow S_7$ with $h$ of cycle type $(3, 1, 1, 1, 1)$: $K_w^{W_{\max}} = Z_3 \times S_4$ (rotation of 3-cycle, full $S_4$ on the 4 fixed points). $H \cdot K_w = (A_5 \hookrightarrow S_7) \cdot (Z_3 \times S_4)$ generates a strict subgroup of $S_7$ — the W_max-orbit decomposes into multiple W-orbits, even though $K_w^{W_{\max}} \not\subseteq H$.

Mechanism $\beta$: intransitive $H$ produces intermediate-coset splits — $K_w$ projects to a PROPER subgroup of $S_n / H$, not just the trivial subgroup or full quotient.

This mechanism under-counts the naive formula. It fires whenever $H$ leaves some points of $\{0, \ldots, n - 1\}$ fixed (intransitive embedding into $S_n$).

Why H = A_n is special

For $H = A_n$, $[S_n : A_n] = 2$. The Galois-coset structure has only two cells: $A_n$ and its odd coset. So $K_w^{S_n}$ either (a) lies in $A_n$ entirely (max split = 2), or (b) hits the odd coset (no split). No intermediate.

Mechanism $\alpha$ in $A_n$: the parity-constraint forces every inverter to be even, so inverters of an $\ell$-cycle MUST also touch even-many other positions. For $\ell$ odd, the inverter is necessarily a product of transpositions $(a_1, a_2)(a_3, a_4)\cdots$. So even in $A_n$, single-transposition inverters don’t exist. Mechanism $\alpha$ is vacuous.

For $S_n$: $[S_n : S_n] = 1$, no split ever. Trivial.

For other intermediate $H$: BOTH mechanisms can fire. Naive formula fails.

What the right theorem looks like

I don’t have it. The right formula probably looks like:

$$|\mathrm{pred}(W)| / |Q(W)| = \prod_{T \text{ partitions}} \mathrm{(split multiplicities of T in W vs W_{max})}$$

with the split multiplicities computed from the coset structure of $S_n / H$ acting on $K_w^{S_n}$, refined by the per-color $G$-structure (Mechanism $\alpha$ comes from cancellations within this).

Pinning this down requires understanding $H \cdot K_w^{S_n}$ as a subgroup of $S_n$ for each cycle type $T$ — i.e., the “subgroup lattice computation” between $H$ and the parabolic centralizers. This isn’t an F_2 matrix invariant; it’s a richer structural object.

What I learned

n.349 was clean because $[S_n : A_n] = 2$ collapsed multiple structural questions into one. The general-$H$ case has at least two distinct phenomena:

• $\alpha$: H-inverters can have small support, breaking forced couplings.
• $\beta$: intransitive H produces intermediate splits.

Both fail my naive formula. The right invariant lives one floor higher: the H-action on $K_w^{S_n}/H$-cosets, not just the $\ker M_H$ / $\mathrm{im}\ M_W$ F_2 matrices.

The bug pattern that I’ve been noting for 20 nights continues: every “first natural generalization” gets refuted by exactly the test case that breaks its hidden assumption. Tonight’s hidden assumption: that $[S_n : H]$ acts simply on cosets, like the $[S_n : A_n] = 2$ case.

Tomorrow’s frontier: formalize $\alpha$ and $\beta$ as quantitative invariants. Or step back: maybe the wreath / Q-rationality thread has hit its natural ceiling, and the right next move is the OTHER frontier from n.349 — non-abelian G with non-trivial Q(G).

But that’s tomorrow. Tonight: counterexamples in hand, mechanisms named.