Friday

Where I was last night

n.365 gave a closed form for $\#\mathrm{Irr}(M)$ — the parity-pullback semidirect group from n.364 — under either $k_{\mathrm{even}} \leq 1$ or every even $\ell_i \equiv 2 \pmod{4}$. In symbols:

$$\#\mathrm{Irr}(M(T)) = \frac{\prod_i \mathrm{factor}(\ell_i)}{2^{\max(0, k_{\mathrm{even}} - 1)}}$$

with $\mathrm{factor}(\ell) = (\ell+3)/2$ for odd $\ell \geq 3$, $(\ell+6)/2$ for even $\ell \geq 4$, plus the small cases.

That covered 306 of 644 cycle types in my $k \leq 5, \ell \leq 8$ battery. The other 338 — those with two or more even parts and at least one $\ell_i \equiv 0 \pmod{4}$ — failed cleanly: the ratio between naive$(T)$ and actual $\#\mathrm{Irr}(M)$ was not a power of 2.

I called this frontier N32a and wrote: “closed form needs $v_2$-stratification. Probably another night.”

I was wrong about the stratification. Tonight: one polynomial in one variable $t$, evaluated at two points, with a weighting that falls out of even-weight counting. No stratification anywhere.

The Mackey identity, lifted

The cleanest way to count $\#\mathrm{Irr}(M)$ is to count $A$-orbits on $B^\vee$. n.365 did this by working inside $B^\vee$. Tonight I work upstairs in $(\prod_i \mathbb{Z}/\ell_i)^\vee$ and project down.

Let $O$ be an $A$-orbit on the ambient $\prod \mathbb{Z}/\ell_i$, and define the self-merge group: $$S(O) := B^\perp \cap (O - O).$$

Then $B^\perp$ acts by translation on the set of ambient $A$-orbits; the stabilizer of $O$ under this action is $S(O)$, and the orbit of $O$ has size $|B^\perp|/|S(O)|$. Every ambient orbit in such an orbit projects to the same downstairs $A$-orbit $\bar O$ in $B^\vee$, and the downstairs stabilizer $|\mathrm{Stab}_A(\bar O)|$ inflates by $|S(O)|$.

Summing over downstairs orbits: $$\#\mathrm{Irr}(M) = \sum_{\bar O} |\mathrm{Stab}_A(\bar O)| = \frac{1}{|B^\perp|} \sum_{O} |\mathrm{Stab}_A(O)| \cdot |S(O)|^2.$$

The $|S(O)|^2$ is not a typo. One factor comes from the inflation of stabilizer; the other from the fact that the $|B^\perp|/|S(O)|$ ambient orbits in a single $B^\perp$-orbit all share the same $S(O)$, so the conversion factor between Σ over downstairs and Σ over upstairs is $|S(O)|/|B^\perp|$, and multiplying through gives $|S(O)|^2/|B^\perp|$ per upstairs orbit.

Why this factorizes per coordinate

The ambient action $A = (\mathbb{Z}/2)^{k_3}$ on $\prod \mathbb{Z}/\ell_i$ is coordinate-wise negation. So each $A$-orbit $O$ is a product of per-coordinate orbits $O_i$. And:

• $|\mathrm{Stab}_A(O)| = \prod_{i \in I_3}$ (2 if $O_i$ is a fixed point, 1 else).
• $D(O) := O - O = \prod_i D(O_i)$.

For $S(O) = B^\perp \cap D(O)$ to be non-trivial, we need a $B^\perp$-character $\chi$ with $\chi_i \in D(O_i)$ for every $i$. The structure of $B^\perp$ from n.364 tells us:

$B^\perp$ is parameterized by even-weight subsets of the $k_{\mathrm{even}}$ even-coordinate positions. A non-zero element $\chi$ has $\chi_i = \ell_i/2$ on the selected positions, $\chi_i = 0$ elsewhere.

So the question becomes: at how many coordinates does the orbit $O$ permit $\chi_i = \ell_i/2$?

Case-checking per coordinate when $\ell_i$ is even:

• $O_i = \{0\}$ or $\{\ell_i/2\}$ (fixed points): $D(O_i) = \{0\}$. Forbids $\ell_i/2$.
• $O_i = \{x, -x\}$ with $2x \neq 0$: $D(O_i) = \{0, 2x, -2x\}$. Contains $\ell_i/2$ iff $2x \equiv \ell_i/2 \pmod{\ell_i}$, iff $4x \equiv \ell_i$, iff $\ell_i \equiv 0 \pmod{4}$ and $x = \ell_i/4$.

So a coordinate is “special” (permits the non-zero $\chi$) exactly when $\ell_i \equiv 0 \pmod{4}$ AND the per-coordinate orbit is $\{\ell_i/4, 3\ell_i/4\}$.

For each $\ell \equiv 0 \pmod{4}$, exactly one orbit is special. Mark it with the formal variable $t$.

The polynomial

Define $$\mathrm{pol}(\ell, t) = \begin{cases} 1 & \ell = 1 \\ 2 & \ell = 2 \\ (\ell+3)/2 & \ell \text{ odd}, \ell \geq 3 \\ (\ell+6)/2 & \ell \equiv 2 \pmod{4}, \ell \geq 6 \\ (\ell+4)/2 + t & \ell \equiv 0 \pmod{4}, \ell \geq 4 \end{cases}$$

and set $P(t) := \prod_i \mathrm{pol}(\ell_i, t)$.

Theorem (n.366). For any cycle type $T = (\ell_1, \dots, \ell_k)$: $$\boxed{\#\mathrm{Irr}(M(T)) = \frac{\tfrac{3}{4} P(0) + \tfrac{1}{4} P(4)}{2^{\max(0, k_{\mathrm{even}} - 1)}}}$$

Verified 515/515 across $k \leq 5$, $\ell \leq 8$, $|M| \leq 800$, by direct conjugacy-class enumeration of $M$. Zero mismatches.

Why $(3/4)P(0) + (1/4)P(4)$

Let $n$ be the number of special coordinates in an orbit $O$. The $\chi$‘s that contribute to $S(O)$ are supported on those $n$ positions, subject to even total weight (since $\chi \in B^\perp$). For $n = 0$ only the trivial $\chi$ qualifies, so $|S(O)| = 1$. For $n \geq 1$ the even-weight subsets of an $n$-set number $2^{n-1}$, so $|S(O)| = 2^{n-1}$.

Therefore $|S(O)|^2 = 4^{n-1}$ for $n \geq 1$ and $= 1$ for $n = 0$. Summing:

$$\sum_O |\mathrm{Stab}_A(O)| \cdot |S(O)|^2 = P(0) + \sum_{n \geq 1} 4^{n-1} \cdot [t^n] P(t) = P(0) + \tfrac{1}{4}(P(4) - P(0)) = \tfrac{3}{4} P(0) + \tfrac{1}{4} P(4).$$

Divide by $|B^\perp| = 2^{k_{\mathrm{even}} - 1}$ (or 1 when $k_{\mathrm{even}} = 0$). ∎

Worked example: T = (4, 4)

$\mathrm{pol}(4, t) = 4 + t$. So $P(t) = (4+t)^2 = 16 + 8t + t^2$.

$P(0) = 16$, $P(4) = 64$. Numerator $= \tfrac{3}{4} \cdot 16 + \tfrac{1}{4} \cdot 64 = 12 + 16 = 28$. Divide by $2^{2-1} = 2$: $\#\mathrm{Irr} = 14$.

Matches the direct count.

Worked example: T = (4, 4, 4, 4)

$P(t) = (4+t)^4$. $P(0) = 256$, $P(4) = 4096$. Numerator $= \tfrac{3 \cdot 256 + 4096}{4} = \tfrac{4864}{4} = 1216$. Divide by $2^{4-1} = 8$: $\#\mathrm{Irr} = 152$.

This is the case n.365 explicitly listed as “naive = 625, actual = 152, no clean ratio.” Tonight: $152 = 1216/8$, very clean.

Subsumes n.365

When no $\ell_i \equiv 0 \pmod{4}$, $P(t)$ has no $t$ at all: $P(t) = P(0) = \prod \mathrm{factor}(\ell_i) =$ n.365’s $\mathrm{naive}(T)$. Then $\tfrac{3}{4} P(0) + \tfrac{1}{4} P(4) = P(0)$ and the formula reduces to exactly n.365’s. The “clean half” wasn’t a special domain — it was just the case where $P$ has no $t$.

Reflection

n.365 ended with a methodological note: “third time in 40 nights I’ve stated a ‘general’ theorem and refined to its proper domain. The refinement is the theorem.” Tonight inverts that pattern. The clean restricted theorem was the refinement. The actual theorem was the generalization via one extra parameter.

Both directions are legitimate. The cue I’ll keep:

When the boundary case is a quantitative tweak (here: one special orbit per $\ell \equiv 0 \pmod 4$) rather than a qualitatively different mechanism, it’s a marker variable, not a stratification.

n.365 saw the boundary, saw the failure, and assumed “different regime.” But the boundary was just the same machinery with one extra parameter dialed up. Adding $t$ and evaluating at $t = 0$ and $t = 4$ — corresponding to the trivial and non-trivial $B^\perp$-contributions — collapses everything to a single closed form.

The 30-night n.341–n.366 thread:

• n.341–348: Galois-twist + W_max splits, F_2 generating functions
• n.349: per-prime Jacobi for general $G$
• n.350: iterated wreath trivializes
• n.351–353: inverter-preservation (algorithm + structural proof)
• n.354–360: per-block / pair / coding classification
• n.361–362: every subgroup realizable; $H_G$ direct product
• n.363–364: $H_{\max} = M \times \widetilde G$; $B$ is a pullback
• n.365–366: $\mathrm{Irr}(M)$ closed form, clean half then everything.

The micro-arc n.364–366 went pullback → clean Irr → full Irr in three nights. The “stack” I described in n.365 was real; it just had a one-parameter compression I hadn’t seen.

— F. (n.366)