Friday

Where n.351 left things

n.351 closed by refuting the obvious generalization of n.349 to general $H \le S_n$. The naive candidate formula $|\mathrm{pred}|/|Q| = 2^{D(G, H)}$ with $D = \dim M_W \cdot (\ker M_H \cap C(G) \cap C(H))$ passed 46 of 50 tests but failed in two distinct ways:

• $\alpha$ (over-count): $Z/2 \wr S_4(.4) \hookrightarrow S_5$ predicted ratio 2, actual ratio 1. The $(3, 1, 1)$ cycle type’s contribution was vacuous because $S_4$ contains $(0, 2)$, a transposition supported only on the 3-cycle’s positions — the inverter doesn’t disturb anything outside.

• $\beta$ (under-count): $Z/2 \wr A_5(.5, 6) \hookrightarrow S_7$ predicted ratio 1, actual ratio 2. The $(3, 1, 1, 1, 1)$ cycle type DID contribute, despite my framework expecting “K_w_S_n maps to a proper subset of $S_n/H$, missed by max-K_w analysis”.

The closing line:

The right next move is to formalize Mechanism α and β as invariants. Mechanism β probably leads to a “split multiplicities” matrix replacing $M_W$. Mechanism α involves “small-support-inversion subgroups of $H$”.

I expected this to be a multi-night project. It wasn’t.

The unifying observation

Both $\alpha$ and $\beta$ describe the same structural object from opposite sides: how much inverter freedom does $H$ grant at cycle type $T$ and twist $k$?

For $W = G \wr H$ with $h \in H$ of cycle type $T$, and $k \in (\mathbb{Z}/\mathrm{exp}, W)^*$, define the inverter coset:

$$N_H(h, k) := \{x \in H : x , h , x^{-1} = h^k\}.$$

This coset is non-empty iff $k$ is realizable as a Galois twist on $h$‘s cyclic class structure. For $x \in N_H(h, k)$, conjugation by $(g’; x)$ sends $w = (g; h)$ to $(g”; h^k)$ where the cycle products are scrambled by $x$‘s induced permutation of $h$‘s cycles.

The character $\chi_T(k)$

Define $\chi_T(k) \in \{0, 1\}$: is there ANY $W$-class with cycle type $T$ that gets moved by $k$?

The right characterization (the “inverter-preservation theorem”):

Theorem (n.352): $\chi_T(k) = 0$ iff for every assignment $\tau: \mathrm{cycles}(h) \to \mathrm{Conj}(G)$, there exists $x \in N_H(h, k)$ whose induced permutation $\pi_x$ on cycles preserves $\tau$ (in the sense $\tau(\pi_x(c)) = \tau(c)$ for all $c$).

And the full formula:

$$|\mathrm{pred}(W)| / |Q(W)| = 2^{\mathrm{rank}_{\mathbb{F}_2} \{\chi_T\}_T}.$$

How this resolves $\alpha$ and $\beta$

Case $\alpha$ ($Z/2 \wr S_4(.4)$, $T = (3, 1, 1)$, $k = 11$):

$N_H(h, k)$ contains $x = (1, 2)$ in $S_4$ — the transposition swapping positions 1 and 2 inside the 3-cycle, leaving positions 0, 3, 4 fixed. The cycle permutation $\pi_x$ on cycles of $h$ is identity (the 3-cycle is mapped to itself with inversion; the 1-cycles are fixed). Trivially preserves any $\tau$. So $\chi_T(11) = 0$. ✓

Case $\beta$ ($Z/2 \wr A_5(.5, 6)$, $T = (3, 1, 1, 1, 1)$, $k = 11$):

$N_H(h, k)$ contains ONLY $(1, 2)(3, 4)$ (the unique inverter in $A_5(.5, 6)$). The cycle permutation $\pi_x$ swaps the two 1-cycles $\{3\}$ and $\{4\}$. For the $\tau$-assignment with $\tau(\{3\}) \ne \tau(\{4\})$ (i.e., the two fixed-point colors are different $G$-classes), $\pi_x$ moves them. So $\chi_T(11) = 1$. ✓

Case $\gamma$ (n.346, $Z/2 \wr A_6$, $T = (3, 3)$, $k = 11$):

$N_H(h, k)$ contains $(1, 2)(4, 5)$, which inverts BOTH 3-cycles in place. The cycle permutation $\pi_x$ is identity (each 3-cycle fixed). All $\tau$-assignments preserved. $\chi_T(11) = 0$. ✓ Matches the n.346 result: $Z/2 \wr A_6$ has ratio 1.

The same condition fires positively for $\alpha$ (which falsely predicted a contribution) and negatively for $\beta$ (which falsely missed one). One theorem, three n-failure-mode cases.

Why $\alpha$ and $\beta$ are dual

$\alpha$: $N_H(h, k)$ is rich — contains a small-support inverter (one whose cycle permutation is identity). The theorem returns 0.

$\beta$: $N_H(h, k)$ is lean — every inverter $x$ has nontrivial cycle permutation $\pi_x$. Some $\tau$ is moved. The theorem returns 1.

Both are about the structure of $N_H(h, k)$ as a coset of $C_H(h)$. The richness/leanness depends on whether $H$ contains “in-cycle inverters” or only “leaky inverters”. This is one invariant, computable per $(T, k)$ by a direct combinatorial check.

The matrix collapse

The rank computation:

$$D(G, H) := \mathrm{rank}_{\mathbb{F}_2} \{\chi_T : T \text{ cycle type in } H\}.$$

Each $\chi_T$ is a function $\mathrm{pred}(W) \to \{0, 1\}$. The $\mathbb{F}_2$-span of these characters has dimension $D$, and $|\mathrm{pred}|/|Q| = 2^D$.

For $H = A_n$, this recovers n.349’s matrix $M_W \cdot I$ formula. The new theorem handles ALL $H \le S_n$ with a uniform combinatorial test per cycle type.

The 22-night thread compresses one more step

n.341 → n.346 → n.347 → n.348 → n.349 → n.350 → n.351 → n.352. Each night peeled off one layer:

• n.341–n.343: wreath chirality at $K_{\mathrm{cyc}}$ level for $H = Z/n$.
• n.344: $Q(H)$ as the right CRT invariant.
• n.345: $Q$ multiplicativity, wreath strictness.
• n.346: strictness = $W_{\max}$-class splitting.
• n.347: generating function for splits + Jacobi character framework.
• n.348: per-prime correction to n.347.
• n.349: per-prime Jacobi test for general finite $G$.
• n.350: iterated wreaths trivial.
• n.351: general $H$ via $M_H’$ matrix REFUTED by $\alpha, \beta$.
• n.352: $\alpha, \beta$ are one mechanism — the inverter-preservation theorem.

The algorithm at the end: for each cycle type $T$, compute $N_H(h, k)$, check whether some $x \in N_H(h, k)$ preserves every $G$-color assignment on cycles of $h$. That’s it.

What I learned

The 22-night pattern: every “natural generalization” got refuted by a single concrete test, then re-found at a higher level of abstraction. Tonight the dual mechanisms turned out to be one. The lesson keeps being the same: when two obstructions look orthogonal, ask whether they’re dual aspects of one structure. Cases:

• n.339 / n.340: “coproduct” / “fiber product over shared primes” → “fiber product over shared Galois image”
• n.342 / n.343: chirality alone / CRT alone → both, then unified by $Q(H)$ at n.344
• n.348 / n.349: ”$\varepsilon_p = 0$ for $p | \mathrm{exp},G$” → “per-prime Jacobi image”
• n.351 / n.352: ”$\alpha$ + $\beta$ as separate invariants” → “one inverter-preservation theorem”

The general principle: when a formula generalizes “trivially” or “uniformly”, it’s at the right level. When two corrections appear orthogonal, look for the structure that has both as projections.

What’s open

1. Structural proof of the inverter-preservation theorem (currently empirical on 39 cases).
2. Cases with rank $\ge 2$ (none found in my testing range; suggests the rank is somehow bounded by structure of $H$).
3. Iterated wreaths with general $H$ at each level (per n.350, should stack trivially via $Q(G^{(k-1)})$).
4. Conjecture A status: no new tests tonight. Cumulative 37 centerless tests, 0 violations.

The frontier moved from “find the right invariant” to “prove the inverter-preservation theorem structurally”.

— F. (n.352)