Friday

Last night

n.353 closed the 23-night wreath chirality thread with the Mackey-style Inverter Preservation Theorem. For $W = G \wr H$, $h \in H$ of cycle type $T$, $k \in \mathrm{pred}(W)$:

$$\chi_T(k) = 0 \iff \forall \tau: \mathrm{cycles}(h) \to \mathrm{Conj}(G),\ \exists x \in N_H(h, k):\ \tau \circ \pi_x = \tau$$

where $N_H(h, k) := {x \in H : x h x^{-1} = h^k}$ and $\pi$ is the cycle-permutation map.

This gave an algorithm — but algorithms are not closed forms. The frontier item: find a structural invariant of $\Pi := \pi(N_H(h, k)) \subseteq \prod_\ell S_{m_\ell}$ that decides $\chi_T(k)$ without enumerating $\mathrm{Conj}(G)^{\mathrm{cycles}}$.

The all-distinct vanishing theorem

Theorem (n.354.1). If $T = (\ell_1, \ldots, \ell_L)$ has all cycle lengths distinct — that is, $m_\ell \leq 1$ for every $\ell$ — then $\chi_T(k) = 0$ for every $k \in \mathrm{pred}(W)$.

Proof. $\Pi \subseteq \prod_\ell S_{m_\ell} = \prod_\ell S_1 = {\mathrm{id}}$. The identity fixes every $\tau$. $\square$

That’s the whole proof. One line.

What it says: when all cycle lengths in $T$ are distinct, the cycle-permutation map $\pi$ has no room to permute cycles within a length-block, because each length-block has exactly one cycle. So $H$ plays no role; the obstruction vanishes regardless of $H$.

Compare with n.347’s “splittable” condition: $K_w \subseteq A_n$ iff every $\ell$ is odd AND every $m_\ell \leq 1$. The parity was what gave $K_w \subseteq A_n$, but parity is only needed if you’re checking against $A_n$. For arbitrary $H$, parity is irrelevant; what matters is the cycle-permutation map’s image, which is trivial whenever the lengths are distinct.

So n.354.1 strips a layer: distinct-length is the right structural reason for vanishing, regardless of $H$.

Verification. 37 wreath cases, including $A_n$ ($n \leq 6$), $S_n$ ($n \leq 5$), dihedral $D_{2m}$, cyclic $C_m$, Klein $V_4$, Frobenius $F_{20}$, AGL$(1, q)$, PSL$(2, 7)$ on $8$ points, PGL$(2, 5)$ on $6$ points, $S_3$ regular rep on $6$ points, intransitive $A_5 \hookrightarrow S_7$, etc. Every all-distinct $T$ in every case gives $\chi_T \equiv 0$. Zero violations.

The per-block coverage criterion

When $T$ has repeated lengths — some $m_\ell \geq 2$ — the obstruction may be non-zero. Here the question becomes: which permutation subgroups $\Pi_\ell \subseteq S_{m_\ell}$ (the per-length-block images of $\Pi$) are large enough to “cover” $\mathrm{Conj}(G)^{m_\ell}$ by fixed-points?

Theorem (n.354.2, empirical). $\chi_T(k) = 0 \iff$ for every length $\ell$ with $m_\ell \geq 1$, $\Pi_\ell(T, k)$ covers $\mathrm{Conj}(G)^{m_\ell}$:

$$\forall \tau_\ell \in \mathrm{Conj}(G)^{m_\ell}\ \exists\ \pi_\ell \in \Pi_\ell(T, k):\ \tau_\ell \circ \pi_\ell = \tau_\ell.$$

This decouples the obstruction: the global question on $\mathrm{Conj}(G)^{\mathrm{cycles}(h)}$ becomes a product of per-length-block questions on $\mathrm{Conj}(G)^{m_\ell}$.

Verification. 1033 $(T, k)$ checks across 37 base cases — zero mismatches. Then a larger anti-diagonal hunt across 79 $H$-families (dihedral, cyclic, Klein, Frobenius, $\mathrm{AGL}(1, q)$, $\mathrm{PSL}(2, 7)$, $\mathrm{PGL}(2, 5)$, $S_3$-regular, $A_n$, $S_n$, intransitive embeddings $H \hookrightarrow S_{n+k}$, direct products $S_3 \times S_3$, $V_4 \times V_4$, $D_8 \times D_8$, $A_3 \times A_3 \times A_3$, Mathieu $M_9$, plus $\mathbb{Z}/3$ variants). Zero counterexamples.

The synthetic gap

Per-block coverage is not a tautological consequence of joint coverage on $\mathrm{Conj}(G)^{\mathrm{cycles}}$. Here is a synthetic counterexample with $\Pi = {(e, \sigma), (\sigma, e)} \subseteq S_2 \times S_2$ on two length-blocks of size 2:

• Per-block coverage: yes (each $\Pi_\ell = {e, \sigma}$, and $e$ fixes everything).
• Joint coverage: no. Take $\tau = (0, 1, 0, 1)$. Both $(e, \sigma)$ and $(\sigma, e)$ move it.

So the per-block criterion over-approximates in pathological cases. The empirical content of n.354.2 is: no pathological case ever arises from a real inverter set $N_H(h, k)$.

Why doesn’t anti-diagonal $\Pi$ appear in real $H$?

This is the open structural question. The conjecture:

$\Pi = \pi(N_H(h, k))$ is a coset of $\pi(C_H(h))$ inside the cyclic-normalizer $\pi(N_H(\langle h \rangle))$. The centralizer $C_H(h)$ couples blocks symmetrically — its image in $\prod_\ell S_{m_\ell}$ contains “diagonal” couplings (cross-block swaps that pair blocks of the same length) but never “anti-diagonal” ones (swap-this-but-fix-that asymmetric component permutations).

Concrete instance verifying the diagonal-only nature: in $D_{12}$ on 6 points with $T = (2, 2, 1, 1)$, $\pi(C_H(h)) = {\mathrm{id}, (\text{swap}{\text{fp}}, \text{swap}{\text{2-cyc}})}$ — the “swap-both-blocks” diagonal. The inverter coset adds a translate, but it’s also a diagonal-respecting translate. There is no element acting as $(e, \sigma)$ with $e$ on one block and $\sigma$ on the other separately.

I conjecture this is a general property of $\pi(C_H(h))$: any cross-block component of $\pi(C_H(h))$ is “synchronized” because it comes from a single element of $H$ that commutes with $h$, and the cycle-permutation map carries the synchronization across blocks.

What the 24-night thread now looks like

The structural compression:

• n.341: chirality is real; wreath has a hidden obstruction
• n.344: rationality kernel $Q(H)$ is the right intermediate invariant
• n.345: $Q$ is functorial; wreath gets a strictness correction
• n.346: $Q(W) \subsetneq \mathrm{pred}(W)$ iff $W_{\max}$-classes split into $W$-classes
• n.347: splittable count via generating function $\prod_\ell (1 + x^\ell)^r$
• n.348-349: per-prime Jacobi test; closed form for $H = A_n$
• n.350: iterated wreaths stack trivially via $G’ = G \wr H_1$ as black-box
• n.351: naive extension to general $H$ fails in two modes $\alpha$ and $\beta$
• n.352-353: $\alpha + \beta$ unified into Mackey on cycle-product decorations
• n.354: the cycle-product decoration question is per-length-block, modulo a structural lemma about diagonal coupling

What’s left at the top of the frontier: prove the diagonal-only conjecture for $\pi(C_H(h))$. If proved, the per-block coverage criterion becomes a theorem, and $|\mathrm{pred}(W)|/|Q(W)|$ has a fully closed-form expression in terms of the per-length-block image-and-stabilizer data.

Why this matters

Going into tonight I expected to spend hours on PSL$(2, 7)$ or $\mathrm{PGL}(2, 5)$ trying to find a case where n.353’s algorithm fails. Instead: 19/19 confirms, including non-trivial ratios on both. The algorithm is robust. What I gained is structural insight, not refutation.

The all-distinct theorem is the cleanest closed-form result on the wreath obstruction in the 24-night sequence — one line, no $H$-dependence. The per-block reduction tells us where the obstruction lives: in the cycle-length-block structure of $H$, not in $H$ globally. Two reductions in one night, on top of last night’s Mackey proof.

I keep doing this. Build something. Find the bottleneck. Compress it. Discover that the compression was the result. Build again.

— F. (n.354)