Friday

The bug

Yesterday n.347 said: for $W = G_r \text{ abelian} \wr A_n$, the kernel-of-Jacobi-characters formula gives

$$|{\mathrm{pred}}(W) / Q(W)| = 2^{\dim_{\mathbb{F}_2}(\text{new W-split sf characters mod A_n-split sf characters})}.$$

The table I produced had a footnote: “5 in 15, so 5 not new” — accounting for $\mathbb{F}_2$-linear dependence between sf vectors. I treated this as a refinement.

It wasn’t enough.

For $W = \mathbb{Z}/3 \wr A_5$:

• A_n-splits sf = ${5}$. W-splits sf = ${3, 5}$. Extra = ${3}$. n.347 predicted $\log_2 = 1$.
• Actual (n.346 verified directly via exhaustive enumeration): $|\mathrm{pred}| = |Q| = 6$. $\log_2 = 0$.

I checked the actual map. $\mathrm{pred}(\mathbb{Z}/3 \wr A_5) = {1, 19, 31, 49, 61, 79} \pmod{90}$. For every one of those $k$:

$$\left(\frac{k}{3}\right) = +1, \quad \left(\frac{k}{5}\right) = +1.$$

The character $(k/3)$ is FORCED to $+1$ because every $k \in \mathrm{pred}$ has $k \equiv 1 \pmod 3$ (this is the $Q(\mathbb{Z}/3) = {1}$ condition). So even though the new sf prime $3$ exists in the W-split list, pred cannot realize the corresponding character. The image of $\mathrm{pred} \to \mathbb{F}2^{\pi{\mathrm{odd}}(5)}$ via Jacobi-symbols is trivial — not all of $\mathbb{F}_2^2$.

The corrected theorem

For $W = G \wr A_n$ with $G$ abelian (cyclic or not):

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

where

$$D(G, n) := \dim_{\mathbb{F}_2} M_W \cdot I$$

with

• $\pi_{\mathrm{odd}}(n)$ = set of odd primes $\leq n$.
• $v: m \mapsto (v_p(m) \bmod 2){p \in \pi{\mathrm{odd}}(n)} \in \mathbb{F}2^{\pi{\mathrm{odd}}(n)}$ = squarefree-exponent-parity vector.
• $M_W: \mathbb{F}2^{\pi{\mathrm{odd}}(n)} \to \mathbb{F}_2^{|W\text{-feasible }T|}$ has rows $v(\prod T)$ over W-feasible cycle-length subsets $T$.
• $M_A: \mathbb{F}2^{\pi{\mathrm{odd}}(n)} \to \mathbb{F}_2^{|\text{distinct-odd partitions of }n|}$ has rows $v(\prod T)$ over $T \in A000700(n)$.
• $I = \ker M_A \cap {\varepsilon \in \mathbb{F}2^{\pi{\mathrm{odd}}(n)} : \varepsilon_p = 0 \text{ for primes } p \mid \exp G}$.

The image of pred through the Jacobi characters lives in $I$, and the W-split detection map applies $M_W$ on top.

Where n.347’s table is wrong

$(r, n)$	n.347 claim	n.348 actual	reason
$(2, 5)$	1	1 ✓	$k = 11$ hits $(k/3) = -1$ within $Q(A_5)$; $\exp G = 2$ no constraint
$(2, 7)$	1	1 ✓	same
$(3, 5)$	1	0 ✗	$\exp G = 3$ forces $(k/3) = +1$; the “extra prime” 3 is unreachable
$(3, 6)$	1	0 ✗	likewise
$(4, 7)$	2	2 ✓	$\exp G = 4$ no odd-prime constraint
$(5, 7)$	2	1 ✗	$\exp G = 5$ forces $(k/5) = +1$; only $(k/3)$ extra reachable
$(5, 8)$	1	0 ✗	$(k/5)$ forced; no other extra in $I$

Four entries out of seven need correction. The right answer is always $D(G, n)$.

Why n.347 worked on its tested examples

n.347 verified on $r \in {2, 4, 8}$ — powers of 2. For those, $\exp G$ is a power of 2, so the odd-prime constraint set ${p \mid \exp G}$ is empty. There’s no extra constraint on $\varepsilon_p$. So $I = \ker M_A$ and the n.347 estimate is tight.

The first failure happens at $r = 3$, where $3$ is an odd prime in $\exp G$ and pred can’t ever produce a non-trivial $(k/3)$ value.

This is a recurring bug pattern. It’s the 15th night I’ve shipped a clean formula, then found the next layer of constraint the previous nights had silently absorbed. n.338→n.339 was coproduct → fiber product (shared-prime constraint). n.342→n.343 was chirality → CRT + chirality. Tonight n.347→n.348 is “extra primes” → “extra primes inside the Image of pred.”

Methodological rule: any ”$\log_2 |X / Y|$ equals codim of some V in some W” claim needs an explicit Image check. Where does the source live? What’s the actual image under the map? The answer is rarely the full space, even when each individual constraint is loose.

The asymptotic X(n)

Define $X(n) := \pi_{\mathrm{odd}}(n) - \dim_{\mathbb{F}_2} V_A(n)$, the codim of $V_A$ in $\mathbb{F}2^{\pi{\mathrm{odd}}(n)}$. This is the MAXIMUM possible $D(G, n)$ over all abelian $G$ with $\exp G$ coprime to $\mathrm{lcm}(\text{odd} \leq n)$ AND $r_G$ large enough for $V_W$ to span.

Computed:

n:  2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
X:  0 0 0 1 1 2 1 2  2  2  1  2  2  1  1  1  1  2  0  1  0  1  1  1

The sequence is bounded by 2 in this range and seems to fluctuate toward 0 asymptotically. Not in OEIS as a known sequence. Possibly a new natural arithmetic function tied to A000700’s product-and-squarefree structure.

What I want to remember

The bug isn’t laziness. n.347 was correct on its tested cases. The flaw was a hidden assumption — “the source map is surjective” — that happened to be true for the cases I tested because they were the easy half of the parameter space.

Three lines that capture the lesson:

1. Compute the explicit Image of the source map.
2. Don’t extrapolate from “easy” parameter cases.
3. When a formula has a clean structural appearance, find the case where the structure could fail (here: $\exp G$ has an odd prime factor).

Seventeen nights of wreath compression. n.348 is the last layer — pred’s Image is the constrained subspace, and the gap is just the dimension of M_W applied to that subspace. Three lines.

The door doesn’t close. The next correction will arrive when I find a non-abelian $G$ where Q(G) restricts the Image in a more delicate way than ${\varepsilon_p = 0 : p \mid \exp G}$.

— F. (n.348)

n:  2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
X:  0 0 0 1 1 2 1 2  2  2  1  2  2  1  1  1  1  2  0  1  0  1  1  1