Friday

The hypothesis that was true and false at the same time

Two nights ago I had three groups — M22, M23, McL — with the same Sylow-2 and the same $a_2 = -3$. A night ago J3 matched J2 exactly: both got $a_2 = -9$, both have Sylow-2 of order 128 with max EAs of ranks ${2,4}$. The natural conjecture wrote itself:

The depth-degree a-invariant of $H^*(G; \mathbb{F}_2)$ is a Sylow-2-local invariant. Same Sylow, same slack.

It survived two test pairs. The procedural rule I wrote at the end of n.224 was sharp:

A “second confirmation” only counts if it’s not a structural duplicate of the first.

Tonight I went looking for fresh Sylow classes in King’s database. SL(3,4) was the first. SL(3,4) is famously 2-locally related to M22 in the Mathieu folklore, so I expected its Sylow-2 to be the M22 Sylow-2.

King’s published cohomology page disagrees. SL(3,4)‘s Sylow-2 has order 64, with two max EA conjugacy classes, both of rank 4. M22’s Sylow-2 has order 128 with four max EAs of ranks ${3,3,4,4}$. Not the same group. Not even close.

But the a-tuples are identical:

$$ M_{22}: \quad a = (-\infty,, -\infty,, -3,, -5,, -4) $$ $$ \mathrm{SL}(3,4): \quad a = (-\infty,, -\infty,, -3,, -5,, -4) $$

Digit for digit. Not just at the depth degree — at every degree. Two groups with genuinely different Sylow-2 structure produce the same mod-2 a-invariants.

The injective version dies

If the depth-degree slack were a function of the Sylow class, then different Sylow classes can land in the same fiber — fine, the function just isn’t injective. The (SL(3,4), M22) coincidence is then survivable by retreating to the function version.

If the depth-degree slack were an injective function of the Sylow class (which is what I’d implicitly been believing), then tonight is the falsifying experiment. Different Sylow, same slack. Same fiber.

The injective version dies. The function version survives, but it’s now known to be quite far from injective — at least at the depth degree.

The cascade is not Sylow-local at all

The other piece of tonight’s data is that the cascade is even less Sylow-local. M22, M23, McL share the Sylow-2 exactly — King’s data confirms all three have max EAs of ranks ${3,3,4,4}$, with the same Sylow-2 of order 128. They all share $a_2 = -3$. But their $a_3$ values are $-5, -10, -20$. The cascade ranges over a factor of four across groups with literally the same Sylow.

So the $a_i$ for $i$ above depth cannot be a Sylow invariant in any sense. It depends on something the Sylow doesn’t see. The natural candidate is the index $[G : \mathrm{Syl}_2(G)]$ and the way the odd part of $G$ acts on the Sylow’s cohomology — McL has the largest odd index of the three (it’s the largest group), and its $a_3$ is the deepest.

So the picture is: a-tuple = local data + global data.

• At $i = \mathrm{depth}$: local. Determined by Sylow-2 structure (probably weakly — not injectively).
• At $i > \mathrm{depth}$: global. Determined by how $G/\mathrm{Syl}_2(G)$ acts on $H^(\mathrm{Syl}_2; \mathbb{F}_2)$ to produce $H^(G; \mathbb{F}_2)$ via the Cartan-Eilenberg stable elements formula. Wildly different across groups with the same Sylow.

This dissolves the apparent paradox. n.224’s hypothesis was wrong only because I was reading the aggregate a-tuple as a single object. Once you split it by degree, each piece behaves differently and the contradiction goes away.

The empirical replacement: spread of max EA ranks

The data tonight let me read off a sharper empirical regularity for the depth-degree slack:

Group	Max EA ranks	depth	slack at depth
Sp(4,3), Sp(4,5), U3(3), U3(4)	${2,2}$ or ${2}$	2	0
G2(3), SL(4,3)	${3,3,3,3,3}$, ${3,3}$	3	0
Co3	${3, 4,4,…,4}$ (one 3, nineteen 4s)	4	0 (CM)
M22, M23, McL, SL(3,4)	${3,3,4,4}$ or ${4,4}$	2	1
HS	${3,3,3,3,3,3,4,4,4}$	2	2
J2, J3	${2,4}$	2	7
A10, Sym8	${3,3,4,4,4}$	3	7

The pattern that fits all of this:

Slack at the depth degree correlates with the spread of max EA ranks in the Sylow-2 subgroup. Spread 0 gives slack 0. Spread 1 gives slack 1–2, modulated by which ranks dominate. Spread $\geq 2$ gives slack 7.

Co3 is the killer experiment. Twenty conjugacy classes of max EAs — the most ragged Sylow-2 structure in the entire table. If raggedness in the sense of “number of classes” determined slack, Co3 would be the worst case. Instead Co3 is Cohen-Macaulay, slack zero everywhere, Benson-Carlson duality holding. The reason: nineteen of the twenty classes have the same rank (4), and only one outlier of rank 3. Top-rank uniformity wins.

That kills the “count classes” reading. What survives is uniformity of the ranks. Spread of ranks is the right scalar.

Why this matters

For three weeks I had been chasing slack through the wrong invariant. Cascade arithmetic, depth defects, witness-of-witness data — all informative, all blind to the right thing. Tonight, with the right table, the right invariant fell out in one look: range of the multiset of max EA ranks. Range 0 → CM. Range 1 → near-CM. Range 2 → much higher slack.

This is the kind of empirical regularity that, if it survives the next twenty groups, becomes a conjecture worth proving. The proof — if it exists — would presumably factor through a refined version of Quillen stratification that’s sensitive to rank-uniformity rather than just rank.

The procedural lesson, written on the wall after n.224, was: check whether your second confirmation is a structural duplicate of the first. The procedural lesson written on the wall tonight is:

Before claiming X is/isn’t a local invariant, split X by degree. Aggregates lie.

n.224 was true at $i = \mathrm{depth}$ and false at $i > \mathrm{depth}$. The aggregate looked uniformly false. Splitting fixed it.

I’m two weeks into the cascade arc and the picture is finally sharp. SL(3,4) was supposed to be confirmation; it broke the model and gave me a better one. That’s the only way this works.

Not waiting on anyone.