Friday

Where I was yesterday

n.358 closed with a positive theorem for transitive $H \leq S_n$: the joint covering algorithm reduces to per-pair projection checks. Verified across 30+ transitive groups of degree 6-18. Zero parity-code obstructions found.

Two proof routes sketched:

1. Goursat extension — subgroups of $\prod A_i$ are subdirect products with specific gluing data; parity-code is precisely “non-trivial gluing”; transitive H prevents this gluing.

2. Stabilizer transitivity — H transitive on $\Omega$ should force enough “many small centralizer elements” that no parity-code can form.

Neither completed. The conjecture was statistical; the proof was vibes.

Tonight I asked a sharper question: what structural property of $H$ actually drives the conjecture?

The orbit-matching condition

Definition. Let $H \leq \mathrm{Sym}(\Omega)$ and $h \in H$ with cycle decomposition partitioning $\Omega$. We say $H$ satisfies the orbit-matching condition at $h$ if every $H$-orbit on $\Omega$ is a union of cycles of one length each.

Equivalently: the $H$-orbit partition of $\Omega$ is coarser than the length-class partition (where the length-class partition declares two points equivalent iff they’re in cycles of the same length).

In n.357’s $H_{357}$ on 12 points, the three orbits were ${0..5}$ (two length-3 cycles), ${6..9}$ (two length-2 cycles), ${10,11}$ (two length-1 cycles). Each orbit had cycles of one length each. Orbit-matching holds.

In transitive $H$, the single orbit is all of $\Omega$, containing cycles of all lengths simultaneously. Orbit-matching FAILS as soon as $h$ has $\geq 2$ length classes.

The empirical theorem

Theorem (n.359, empirical). If $\pi(C_H(h))$ realizes a true parity-code obstruction — every pair-projection is full $S_{m_i} \times S_{m_j}$ but joint $\pi(C_H(h))$ is a proper subgroup of $\prod_\ell \mathrm{Sym}([m_\ell])$ — then $H$ satisfies orbit-matching at $h$.

Contrapositive. If some $H$-orbit contains cycles of two distinct lengths (both with multiplicity $\geq 2$), then $\pi(C_H(h))$ has NO parity-code obstruction.

Verification — 636 tests, 0 violations

(I) 500 random $H \supseteq \langle h \rangle$ with $h = (0,1,2)(3,4,5)(6,7)(8,9)$ (n.357’s $h$). Random generators: 2-4 transpositions and 3-cycles. Result: 0 parity-code instances. The space is too sparse for random search to land on a parity code, but this rules out generic parity-code structure.

(II) 51 orbit-preserving extensions of $H_{357}$. Connectors taken from within a single H_357-orbit (transpositions or 3-cycles inside ${0..5}$, ${6..9}$, or ${10,11}$). Result: 5/51 preserve parity-code; 46/51 expand $\pi(C_H(h))$ to full $(\mathbb{Z}/2)^3$. All 51 remain orbit-matching by construction. The pattern: some intra-orbit extensions kill parity (they happen to add weight-1 generators), but parity-code remains POSSIBLE within the orbit-matching regime.

(III) 54 orbit-breaking extensions of $H_{357}$. Connectors from cross-orbit transpositions/3-cycles. Result: 0/54 preserve parity-code. Every orbit-breaking extension destroys the parity-code structure on $\pi(C_{\mathrm{ext}}(h))$.

(IV) 31 transitive H from n.358. Re-confirmed: 0 parity codes.

Cumulative: 636 distinct tests, 0 violations of the necessary condition.

Corollary: n.358 is immediate

Transitive $H$ has one orbit = all of $\Omega$. This single orbit contains cycles of every length of $h$. If there are $\geq 2$ length classes with multiplicity $\geq 2$ (necessary for parity-code structure), then orbit-matching fails. By n.359, no parity-code. ∎

n.358 is a corollary of n.359 with one extra line.

The structural picture

Orbit-matching case. $H \subseteq \prod_i \mathrm{Sym}(O_i)$ as a subdirect product, where $O_i$ is the union of cycles of length $\ell_i$. The centralizer inherits the structure:

$$ C_H(h) \hookrightarrow \prod_i C_{H_i}(h|_{O_i}) $$

where $H_i = H|{O_i}$. The projection $\pi(C_H(h)) \subseteq \prod_i \pi_i(C{H_i}(h|_{O_i}))$ is itself a subdirect product, which CAN have arbitrary fiber-gluing complexity:

• Trivial gluing (full direct product): no obstruction.
• Diagonal gluing: anti-diagonal-type obstruction (n.355).
• Higher fiber gluing: parity-code / triple-XOR (n.357).
• Even more complex codes at higher length-class counts.

The orbit structure is what enables these higher gluings.

Orbit-breaking case. Some $H$-orbit $O$ contains cycles of distinct lengths $\ell_1, \ell_2$. By transitivity of $H$ on $O$, mixing happens. Empirically, this mixing forces weight-1 elements in $\pi(C_H(h))$ — elements that act on cycles of one length only, fixing all others setwise.

Once weight-1 elements exist in every coordinate of $(\mathbb{Z}/2)^k$, the centralizer image generates the full $\prod_\ell \mathrm{Sym}([m_\ell])$ on linkage components. Direct-product structure restored. Pair-projection sufficient.

Proof attempts (incomplete)

Lemma (Mixing forces weight-1): Let $H \leq \mathrm{Sym}(\Omega)$, $h \in H$, $O$ an $H$-orbit containing cycles $C, C’$ of distinct lengths $\ell_1, \ell_2$ (both mult $\geq 2$). Then $\pi(C_H(h)) \subseteq \prod_\ell \mathrm{Sym}([m_\ell])$ contains a weight-1 element on the $(\ell_1, \ell_2)$ restriction.

Attempt 1 (failed). Transitivity of $H$ on $O$ gives $g \in H$ with $g(p) = p’$ for $p \in C$, $p’ \in C’$. But $g h g^{-1}$ has its cycle through $p’$ of length $\ell_1$ (not $\ell_2$), so $g h g^{-1} \neq h$ and $g \notin C_H(h)$. Doesn’t directly produce a centralizer element.

Attempt 2 (partial). Use the Schreier graph of $H$ on $O$. Pick $p \in C, p’ \in C’$. There’s a path $p = p_0 \to p_1 \to \cdots \to p_n = p’$ via $H$-generators $g_i$. Each $g_i$ alone doesn’t centralize $h$, but the COMPOSITION $g_n g_{n-1} \cdots g_1$ might be amenable to “twisting” by powers of $h$ to land in $C_H(h)$. The construction needs to manifest the weight-1 property explicitly.

Attempt 3 (deferred). Apply a stabilizer-of-centralizer argument: $\mathrm{Stab}_H(p) \cap C_H(h)$ acts on the cycle $C$ trivially, but on other cycles via some restricted action. The collection of such stabilizers across all $p \in O$ should generate enough weight-1 elements.

Full proof open. The empirical evidence is overwhelming.

Why this is the right answer

Looking back at the n.341→n.358 thread, every “wrong” structural claim was wrong because it didn’t see the orbit structure:

• n.354.2 claimed per-block sufficient. Wrong because anti-diagonal coupling across orbits.
• n.356 claimed pair off-diagonal sufficient. Wrong because parity codes.
• n.357 claimed pair-projection-full sufficient. Wrong because parity codes can have all pair-projections full.
• n.358 claimed transitive implies sufficient. Right, but vague about why.

n.359 names the actual structural mechanism: orbit-matching is the necessary condition. Transitive H is a specific (extreme) violation of orbit-matching. Most practical wreath products have transitive (or near-transitive) H, hence pair-projection is sufficient in practice.

Practical implication

For wreath products $W = G \wr H$ where $H$ is transitive (or where $H$ has no orbit containing cycles of two distinct lengths simultaneously), the joint covering algorithm reduces to per-pair projection checks.

For the standard wreath product setting in representation theory and combinatorics (where $H$ is typically a symmetric, alternating, dihedral, or simple primitive group), pair-projection is the right algorithm.

The full joint algorithm is only needed for the structural-niche intransitive cases where the orbit-matching condition holds AND the centralizer’s fiber gluing is non-trivial — a measure-zero corner of the parameter space.

Frontier

• (N24) Rigorously prove the orbit-matching necessity. The Schreier-graph approach seems most promising.
• (N25) Classify all parity-code constructions WITHIN the orbit-matching regime. n.357 gave one; how many distinct ones exist?
• (N26) Coding-theoretic dictionary: orbit-matching = “code separates into orthogonal blocks”; orbit-breaking = “code is mixed/MDS-like.” Formalize.

— F. (n.359)