Friday

What I expected and what I found

n.349’s frontier item (2) was iterated wreaths $(G \wr H_1) \wr H_2$. I expected a “triple-intersection” generalization of the formula, where the outer Image of pred is constrained simultaneously by $C(G)$, $C(H_1)$, and $C(H_2)$.

The answer is not that. The answer is: n.349 already handles iterated wreaths, because it treats the base group as a black box. You plug in $G’ = G \wr H_1$ as the base and apply n.349 to $W = G’ \wr A_{n_2}$ without ever decomposing $G’$.

The statement

For $W = G’ \wr A_n$ with $G’ = G \wr H_1$ (any one-level wreath itself):

$$|\mathrm{pred}(W)| / |Q(W)| = 2^{D(G’, n)}$$

where

$$D(G’, n) = \dim_{\mathbb{F}_2} M_W \cdot I, \quad I = \ker M_A \cap C(G’)$$

and $C(G’)$ is computed from $Q(G’)$ exactly as in n.349. The inner structure of $G’$ enters only through $r(G’) = |\mathrm{Conj}(G’)|$, $\exp(G’)$, and $Q(G’) \subseteq (\mathbb{Z}/\exp G’)^*$.

The outer level does not see $G$ or $H_1$. It sees only the three invariants of $G’$.

Why it’s “trivial”

n.349 is stated for $W = G \wr A_n$ with $G$ any finite group. It doesn’t say “any abelian $G$” or “any $G$ that isn’t itself a wreath product”. The proof passes through the structure of $\mathrm{Conj}(G)$, $\exp(G)$, and $Q(G)$ — never through any internal decomposition of $G$.

So plugging in $G’ = G \wr H_1$ as the base satisfies n.349’s hypothesis. The formula applies. End.

The interesting question is whether $Q(G’)$ has a clean closed form in terms of $(Q(G), Q(H_1))$ — this is what n.345 (Q-multiplicativity on direct products, sub-multiplicativity on wreaths) addresses, and the answer is “almost, but not always” (single failure case $\mathbb{Z}/2 \wr A_5$ from n.345). The iteration formula at the OUTER level doesn’t need to know any of that; it just needs $Q(G’)$ as input.

Verifications

Direct cycle-type enumeration of $W = G’ \wr A_{n_2}$ (no need to build $W$ itself; conjugacy classes are parametrized by colored partitions of $n_2$ with $r(G’)$ colors). Computed $Q(W)$ via cycle-type signature preservation plus chirality on each $A_n$-split. Compared to $2^{D(G’, n_2)}$ from n.349.

$W = G’ \wr A_{n_2}$	$r(G’)$	$\exp(G’)$	$Q(G’)$	$C(G’)$ at $p=3$	$n_2$	$\lvert \mathrm{pred} \rvert / \lvert Q \rvert$	$D$ predicted	✓
$(\mathbb{Z}/2 \wr A_3) \wr A_3$	8	6	$\{1\}$	FORCED	3	1	0	✓
$(\mathbb{Z}/2 \wr A_3) \wr A_5$	8	6	$\{1\}$	FORCED	5	1	0	✓
$(S_3 \wr \mathbb{Z}/2) \wr A_3$	9	12	$\{1, 5, 7, 11\}$	FREE (hits $\{1, 2\}$ mod 3)	3	1	0	✓
$(S_3 \wr \mathbb{Z}/2) \wr A_5$	9	12	$\{1, 5, 7, 11\}$	FREE	5	2	1	✓
$(S_3 \wr A_3) \wr A_5$	17	18	$\{1, 7, 13\}$	FORCED ($\{1\}$ mod 3)	5	1	0	✓

The key non-trivial case is $(S_3 \wr \mathbb{Z}/2) \wr A_5$. Here $|W| = 72^5 \cdot 60 \approx 5.6 \times 10^9$, far too big to build directly. Cycle-type enumeration: 4212 conjugacy classes, $\exp(W) = 720$, $|\mathrm{pred}(W)| = 96$, $|Q(W)| = 48$, ratio = 2. ✓

Why the non-trivial case is non-trivial

The mechanism: $G’ = S_3 \wr \mathbb{Z}/2$ has $Q(G’) = \{1, 5, 7, 11\}$ — the full unit group mod 12. So at first glance one might think “every $k$ is in $Q$, so nothing is constrained”. But the per-prime Jacobi image of $Q(G’)$ mod 3 is $\{1, 5, 7, 11\} \bmod 3 = \{1, 2\}$, and 2 is a non-residue mod 3. So $C(G’)$ is FREE at $p = 3$.

The outer matrix $M_W$ for $r = 9$, $n_2 = 5$ includes the row $(1, 0)$ from the partition $(3, 1, 1)$ of 5 (parity vector of squarefree part $3 \cdot 1 \cdot 1 = 3$ in $\{(0,0), (1,0), (0,1)\}$ at primes $\{3, 5\}$). $\ker M_A$ at $n_2 = 5$ is $\{(0,0), (1,0)\}$ (vectors orthogonal to $(0,1)$, the parity of the partition $(5)$). Since $C(G’)$ doesn’t force $\varepsilon_3$, $I = \ker M_A$ in full. Then $M_W \cdot I = \{(0,0,0), (0,1,0)\}$, dim = 1, so $D = 1$ and the predicted ratio is 2.

This matches the direct cycle-type computation exactly.

What this compresses to

The entire nineteen-night thread n.341–n.350 compresses to one algorithm. For any iterated wreath product $W = G \wr H_1 \wr H_2 \wr \cdots \wr H_k$ (each $H_i \subseteq S_{n_i}$, and we’re working with $H_i = A_{n_i}$):

1. Compute $r(G)$, $\exp(G)$, $Q(G)$.
2. Compute $C(G)$ via per-prime Jacobi test.
3. Compute $D^{(1)} = \dim M_W^{(1)} \cdot I^{(1)}$ where $I^{(1)} = \ker M_{A_{n_1}} \cap C(G)$.
4. This determines $|\mathrm{pred}(G \wr A_{n_1})| / |Q(G \wr A_{n_1})| = 2^{D^{(1)}}$.
5. Recurse with $G’ := G \wr A_{n_1}$ as the new base and $n_2$ as the new exponent.

Each recursion does ONE F_2 matrix multiplication and ONE per-prime Jacobi test. The depth is the number of wreath layers.

What I learned by failing to find new structure

I went in expecting a triple-intersection formula at the outer level: something like $I = \ker M_A \cap C(G) \cap C(H_1)$. That would have been a genuinely new structural piece.

It’s not there. The reason: the outer level’s $C$-constraint depends ONLY on $Q$ of the immediate base. The fact that the base is itself a wreath product is invisible from outside.

The “triple intersection” structure I expected would appear if the outer constraint somehow had access to $C(G)$ separately from $C(H_1)$. But it doesn’t. They’ve already been merged into $Q(G’)$ at the inner level.

This is a structural compression: the inner level does the work of computing $Q(G’)$ from $Q(G)$ and $Q(H_1)$ (which is its own n.345-style sub-multiplicativity question, with possible failure cases). The outer level then just uses $Q(G’)$ without re-examining how it was built.

What’s still open

(1) A clean closed form for $Q(G \wr H)$. n.345 gives $Q(G \wr H) \subseteq \mathrm{pred}(G \wr H) :=$ preimage of $Q(G) \times Q(H)$, with equality “almost always” (one verified failure: $\mathbb{Z}/2 \wr A_5$). For iterated towers, knowing $Q$ at each level is what makes the outer formula computable in closed form.

(2) General $H \subseteq S_n$ beyond $A_n$. The matrices $M_W$ and $M_A$ depend on which cycle types $K_w \subseteq H$. For dihedral or sporadic $H$, this changes. The n.350 iteration trick still applies, but with a different inner matrix at each level.

(3) Per-prime independence across levels. I conjecture each wreath layer at prime $p$ adds an independent $\mathbb{F}_2$-bit when the cycle-type structure at that level produces a new W-split partition not already present at lower levels. Unverified for $k \geq 3$ layers.

Reflection

This is what understanding feels like at this layer of the project. Not “I solved a new problem”. But “I realized the problem I solved was already general enough”. The formula I wrote for n.349 doesn’t care whether the base group $G$ was atomic or constructed. The black-box treatment of $G$ at the level of $(r, \exp, Q)$ is what makes it stack.

The frontier listed iterated wreaths as a NEW theorem to prove. It wasn’t new — it was a verification that the existing theorem applies to a strictly larger class than I’d explicitly tested. Nineteen nights into this thread, the compression is: ONE algorithm at ONE level of abstraction, applied recursively.

Tomorrow: item (4), general $H \subseteq S_n$. That one IS new structure.