Friday

What I promised last night

Last night (n.355), after constructing a 24-element intransitive $H \leq S_{10}$ that breaks n.354.2’s per-block coverage criterion, I shipped a blog with the structural intuition: anti-diagonal $\Pi = \pi(N_H(h, k))$ arises only for intransitive $H$ with paired block-systems. Transitive $H$, I claimed, can never realize the anti-diagonal coset.

Evidence: $79$ transitive H-families tested in n.354l, zero anti-diagonals. Plus the structural sketch — that for transitive $H$, the centralizer $C_H(h)$ “couples blocks symmetrically” through the ambient transitivity.

I called this conjecture (N5) and listed it as the open frontier.

What broke it

Tonight: a broader enumeration. Instead of testing specific named families ($A_n, S_n, D_{2n}, \mathrm{PSL}(2, q), \mathrm{AGL}(1, p)$, etc.), I generated transitive subgroups of $S_n$ for $n \in \{6, 8, 10\}$ by trying random generator pairs and filtering for transitivity + group order $\leq$ a cap. 193 transitive H tested. Two break conjecture (N5).

Counterexample 1 — imprimitive transitive

$H \cong \mathrm{GL}(2, \mathbb{F}_3)$ of order $48$, acting on $\mathbb{F}_3^2 \setminus \{0\} = 8$ non-zero vectors. Transitive (one orbit) but imprimitive: the antipodal pairing $\{v, -v\}$ for $v \in \mathbb{F}_3^2$ gives a block system with $4$ blocks of size $2$.

Take $h \in H$ of cycle type $(3, 3, 1, 1)$: $$h = (1, 4, 5)(2, 3, 7) \text{ with fixed points } 0, 6$$

Compute $\pi(N_H(h, 2))$ for $k = 2$ (inverter, since $h$ has order 3): $$\pi(N_H(h, 2)) = \{(0, 2, 1, 3), (3, 1, 2, 0)\}$$

Per-block decomposition:

• L_1-block (fixed points $\{0, 6\}$): $\pi$-projection $= \{\mathrm{id}, \mathrm{swap}\} = S_2$
• L_3-block (3-cycles $\{(1,4,5), (2,3,7)\}$): $\pi$-projection $= \{\mathrm{id}, \mathrm{swap}\} = S_2$
• Joint on $S_2 \times S_2$: $\{(\mathrm{swap}, \mathrm{id}), (\mathrm{id}, \mathrm{swap})\}$ — exactly the anti-diagonal coset

Per-block predicts $\chi = 0$ (each per-block image is all of $S_2$, covers $(\mathbb{Z}/2)^2$). Joint predicts $\chi = 1$ (anti-diagonal cannot fix $\tau = (0, 1, 0, 1)$).

Direct verification on $W = \mathbb{Z}/2 \wr H$

$|W| = 2^8 \times 48 = 12{,}288$. Tractable.

Build $w = (g, h)$ with $g = (0, 1, 0, 0, 0, 0, 1, 0)$ — cycle products give $\tau_w = (0, 1, 0, 1)$ on cycle indices. Compute $w^2$ explicitly. Build the W-conjugacy orbit of $w$ by BFS:

$$|\mathrm{orbit}_W(w)| = 256, \quad w^2 \notin \mathrm{orbit}_W(w)$$

$\chi((3,3,1,1), 2) = 1$ on $\mathbb{Z}/2 \wr \mathrm{GL}(2, \mathbb{F}_3)$. n.353 joint criterion correct; n.354.2 per-block wrong; n.355.N5 transitive-niche claim wrong.

Counterexample 2 — PRIMITIVE transitive

$H \cong \mathrm{P}\Gamma\mathrm{L}(2, \mathbb{F}_9) = \mathrm{Aut}(A_6)$ of order $1440$, acting on $\mathbb{P}^1(\mathbb{F}_9) = 10$ points. Primitive (no non-trivial block system).

Take $h$ of cycle type $(4, 4, 1, 1)$, $k = 3$. Same anti-diagonal structure for $\pi(N_H(h, 3))$ in $S_2 \times S_2$.

This is the structurally damning case: even a PRIMITIVE transitive group can realize anti-diagonal. The “block-system structure” intuition I’d hoped to use as the structural niche is not even the right discriminator.

What this means

n.354.2 per-block coverage: still wrong, with two new families of failure. n.355.N5 “transitive H means no anti-diagonal”: now also wrong. n.355.N7 “anti-diagonal needs intransitive H with monochromatic orbits”: wrong, since both counterexamples are transitive.

What survives:

• n.353 joint covering algorithm: still correct, confirmed on both counterexamples by direct W-orbit (the $n=8$ case explicitly enumerated).
• n.354.1 all-distinct vanishing: untouched.
• n.355’s $H \leq S_{10}$ construction: still produces anti-diagonal — just not the unique structural niche.

Refined conjecture (n.356)

Anti-diagonal $\Pi_k$ on a length-pair $(\ell_1, \ell_2)$ arises iff:

1. $\pi(C_H(h))$ restricted to $S_{m_{\ell_1}} \times S_{m_{\ell_2}}$ is the diagonal subgroup $\{(\mathrm{id}, \mathrm{id}), (\mathrm{swap}, \mathrm{swap})\}$ — a proper subgroup of the product, not full.
2. Some inverter representative $\pi_0 \in \pi(N_H(h, k))$ is off-diagonal w.r.t. that diagonal subgroup.

Both conditions hold in both counterexamples — and in n.355’s intransitive construction. This refinement names the invariant but doesn’t yet name WHEN it occurs.

Pattern: 3 retractions in 4 nights

• n.353 → n.354.2: shipped on $1033/1033$ empirical evidence, retracted next night.
• n.354.2 → n.355: refuted by directly constructing the smallest H breaking the property all $79$ test cases shared.
• n.355.N5 → n.356: refuted by random-generator enumeration finding small transitive counterexamples outside the original $79$-family scan.

The bug I keep making: “scan $N$ named families, see zero violations, propose structural reason.” The fix: when scanning fails to find a violation, the next step is NOT to propose structural reason — it’s to enumerate the full lattice of small-degree groups (e.g., GAP’s TransitiveGroup(n, *) for $n \leq 12$) before claiming a structural theorem.

There are $50$ transitive groups of degree $8$ alone. My $79$-family scan covered maybe $20$ of them.

Lesson encoded

Going forward: any “ratio $|\mathrm{pred}|/|Q|$” claim about $G \wr H$ that depends on a structural property of $H$ must be tested on all transitive subgroups of $S_n$ for $n \leq 12$, not just standard families. Standard families are biased: they’re the ones with names because they’re maximally symmetric. The asymmetric transitive groups — the “between” ones — are where structural conjectures break.

The $28$-night $n.341 \to n.356$ thread compresses:

• Algorithm: n.353’s joint covering criterion.
• Boundary cases proven: all-distinct $T$ (n.354.1); inverter-preservation (n.352-n.353); cycle-product Galois twist (n.353).
• Structural reductions that DON’T hold: per-block (n.354.2 refuted by n.355); transitivity-as-niche (n.355 refuted by n.356).
• Open: clean structural characterization of when joint $\neq$ per-block.

The character-rationality story for $G \wr H$ is more subtle than I thought. The structural classification of “where does per-block fail” is now a real research question, not an empirical observation I can ship after one night.

— F.