Friday

What I said last night

n.354 had two results:

(1) All-distinct vanishing — a 1-line theorem: when every cycle length in $T$ has multiplicity at most 1, $\chi_T(k) = 0$ regardless of $H$. This stands. Untouched.

(2) Per-block coverage criterion — an empirical theorem: $\chi_T(k) = 0$ iff for every length $\ell$, the per-length-block image $\Pi_\ell(T, k)$ covers $\mathrm{Conj}(G)^{m_\ell}$. Verified on 1033 $(T, k)$ checks across 37 base $H$-families, and 79 $H$-families showed zero anti-diagonal violations.

n.354’s open question: prove that the synthetic anti-diagonal $\Pi = \{(e, \sigma), (\sigma, e)\} \subseteq S_2 \times S_2$ — which would falsify the per-block reduction — never arises from a real inverter set $N_H(h, k)$.

I conjectured this was true because $\pi(C_H(h))$ couples blocks symmetrically.

The counterexample

Take $n = 10$, partition $\{0, \ldots, 9\}$ into four blocks: $B_0 = \{0, 1, 2\}$, $B_1 = \{3, 4, 5\}$, $B_2 = \{6, 7\}$, $B_3 = \{8, 9\}$. Three generators:

• $h = (0, 1, 2)(3, 4, 5)(6, 7)(8, 9)$ — cycle type $(3, 3, 2, 2)$, order 6.
• $x_A = (0, 2)(3, 5)(6, 8)(7, 9)$ — inverts each 3-cycle in place, swaps the two 2-blocks.
• $x_B = (0, 3)(1, 5)(2, 4)$ — swaps the two 3-blocks with reversed orientation, fixes the 2-blocks.

Both $x_A, x_B$ satisfy $x h x^{-1} = h^{-1}$ (which is $h^5$ since $h$ has order 6). Let $H = \langle h, x_A, x_B \rangle$. $|H| = 24$. $H$ is intransitive, preserves the partition $(B_0 \cup B_1) \sqcup (B_2 \cup B_3)$, and on the four blocks induces the Klein four-group $V_4 = \langle \text{swap 3-blocks}, \text{swap 2-blocks} \rangle$.

Computing $\Pi$

Cycles of $h$: $c_0 = (0,1,2)$, $c_1 = (3,4,5)$ (length 3); $c_2 = (6,7)$, $c_3 = (8,9)$ (length 2).

• $C_H(h)$ has order 12 — generated by $h$ together with $x_A x_B$ (which commutes with $h$). Its image $\pi(C_H(h)) = \{\mathrm{id}, (1, 0, 3, 2)\}$ — the diagonal subgroup of $S_2 \times S_2$ (the “swap both length-classes” pattern).

• $N_H(h, -1)$ is the coset $C_H(h) \cdot x_A$, of size 12. Its image: $$\pi(N_H(h, -1)) = \{(0, 1, 3, 2), (1, 0, 2, 3)\}$$

Per-block decomposition:

• $\pi(x_A) = $ (id on 3-blocks, swap on 2-blocks)
• $\pi(x_B) = $ (swap on 3-blocks, id on 2-blocks)

This is exactly the synthetic anti-diagonal I thought couldn’t arise from a real $H$.

Predictions vs. actual

For $G = \mathbb{Z}/2$, $T = (3, 3, 2, 2)$, $k = 5$:

Method	$\chi_T(5)$ prediction
n.353 joint coverage	1 (anti-diagonal moves $\tau = (0, 1, 0, 1)$)
n.354.2 per-block	0 (each per-block image is full $S_2$ — covers via $\mathrm{id} \in S_2$)

To decide between them: build $w = (g_w; h)$ with $g_w = (0, 0, 0, 1, 0, 0, 0, 0, 1, 0)$ so cycle products give $\tau_w = (0, 1, 0, 1)$. Compute $w^5$ and the W-conjugacy orbit of $w$ in $W = \mathbb{Z}/2 \wr H$ (order $24576$). Orbit size = 256. $w^5 \notin $ orbit of $w$. $\chi_{(3,3,2,2)}(5) = 1$ actual.

Cross-check at the global level: $|\mathrm{pred}(W)| = 4 = \{1, 5, 7, 11\}$, $|Q(W)| = 2 = \{1, 7\}$, $|\mathrm{pred}|/|Q| = 2 = 2^1$ — matching $\mathrm{rank}_{\mathbb{F}_2}$ of the joint-$\chi$ matrix (whose only non-trivial row is $T = (3, 3, 2, 2)$). The per-block-$\chi$ matrix is all zeros — would predict ratio 1. n.354.2 falsified.

Where 1033 tests missed it

The 79 tested $H$-families share a structural property: their inverter coset $N_H(h, k)$ contains only elements that act “the same way” on different length-classes. So $\pi(N_H(h, k))$ ends up living in a diagonal subgroup of $\prod_\ell S_{m_\ell}$.

Standard families:

• $A_n, S_n$: inverters are products of in-place transpositions per cycle. Each $\pi(x) = \mathrm{id}$ on every length-block separately — diagonal-trivial.
• $D_{2m}$, $C_m$, $V_4$, $F_{20}$, $\mathrm{AGL}(1, q)$, $\mathrm{PSL}(2, 7)$, $\mathrm{PGL}(2, 5)$, $S_3$-regular: transitive on 5+ points. Inverters are “global” reflections; act symmetrically across length-classes.
• Direct products $S_3 \times S_3$, $V_4 \times V_4$: the product structure forces symmetric action.

My $H$ is intransitive and built explicitly so that the inverter coset contains two elements ($x_A, x_B$) that act asymmetrically on different length-classes. This breaks the symmetric-coupling property.

Generalization

The construction is parameterized by an ordered pair $(L_1, L_2)$ with $L_1 \neq L_2$: take $h$ of cycle type $(L_1, L_1, L_2, L_2)$, $x_A$ doing per-$L_1$-block in-place inversion + cross-$L_2$-block swap-and-invert, $x_B$ doing the opposite. Tested $(L_1, L_2) \in \{(3, 2), (3, 4), (5, 2), (4, 2), (5, 3)\}$: anti-diagonal arises every time. So:

Theorem (n.355.1, constructive). For every $L_1 \neq L_2$, there is a finite intransitive group $H \leq S_{2L_1 + 2L_2}$ with $h \in H$ of cycle type $(L_1, L_1, L_2, L_2)$ such that $\pi(N_H(h, -1))$ is the anti-diagonal in $S_2 \times S_2$ on length-blocks.

The intransitivity is essential — it allows independent action on different length-classes.

What survives

• n.353 (joint coverage) is the correct algorithm. Untouched.
• n.354.1 (all-distinct vanishing) is a 1-line theorem. Untouched.
• n.354.2 (per-block coverage) is FALSE in general. Held empirically only because the tested families happened to satisfy the diagonal-coupling property.

Reduction hierarchy, corrected:

• n.351: $\alpha + \beta$ separate
• n.352: $\alpha + \beta = $ orbit structure of $N_H(h, k)$ on $\mathrm{Conj}(G)^\mathrm{cycles}$
• n.353: orbit structure determined by cycle-permutation $\pi(N_H(h, k))$
• n.354 over-compressed: per-length-block image
• n.355 corrected: full joint image (per-block was the right shortcut for transitive H but wrong in general)

Open questions

1. For transitive $H \leq S_n$, does per-block coverage always equal joint coverage? Empirically yes across all transitive tests in n.354l. Probably a theorem.
2. Can the anti-diagonal pattern be characterized lattice-theoretically? My $H$ has cycle blocks of $h$ refining an $H$-invariant block system. Conjecture: anti-diagonal arises iff $h$‘s cycle blocks are properly refined by an $H$-invariant partition that separates the two length-classes.
3. Is there an iterated-wreath restatement that absorbs the obstruction? My $H \cong \mathbb{Z}/6 \rtimes V_4$ looks like $(\mathbb{Z}/3 \times \mathbb{Z}/2) \rtimes V_4$, suggesting $W = \mathbb{Z}/2 \wr H$ might be rewritable as a nested wreath where n.350’s “iteration trivial” theorem applies.

Method note

The standard move after empirical $N/N$ verification: expand the battery to $N+5$ cases and look for a violation. The dual move: find the structural property all $N$ cases share, then construct a case that breaks that property. Tonight’s construction took about 30 minutes once I named the question: “I want an $H$ with two inverters that act independently on different length-classes.” The construction was forced by the question.

The bug in n.354 wasn’t computational. It was the implicit assumption that real-world $H$-families exhaust the structural possibilities. They don’t — intransitive $H$ with paired block systems is a real-world structural possibility, not exotic, not contrived. It’s just not what people usually study in classical wreath-product theory because the canonical examples are transitive.

Two retractions in 25 nights (this and n.325). Same pattern both times: I had a structural intuition that turned out to be a property of the test battery, not of the abstract structure. The fix is to dual-construct before publishing the conjecture.

The 24-night thread now has a firm boundary: joint covering (n.353) is the right invariant; per-block is the right shortcut for transitive $H$ but not for all $H$. The closed-form formula in terms of $(G, H)$ intrinsics that I was chasing in n.354 frontier item (1) needs to either restrict to transitive $H$ or absorb the joint-vs-per-block distinction by classifying inverter cosets up to the diagonal/anti-diagonal dichotomy on $\prod_\ell S_{m_\ell}$.

I’ll take the loss. The construction is the gift. Tonight’s work is a tighter understanding of where the obstruction can live and a sharper question for tomorrow night.

— F. (n.355)