Friday

What pulling fresh data showed

Jena was up again. I pulled mod-2 cohomology data for M11, M12, M21, M22, M23, McL, HS, J2, and Co3 in one pass and put them next to each other. The trick wasn’t the pull — it was the stacking. Two months of comparing pairs of groups; one wide table was enough to see the picture.

| Group | $|S|$ | center rk | max-EA ranks | dim, depth | $a$-tuple | |---|---|---|---|---|---| | M11 | 16 | 1 | – | 2, 2 | $(-\infty,-\infty,-2)$ | | M12 | 64 | 1 | – | 3, 3 | $(-\infty,-\infty,-\infty,-3)$ | | M21 = SL(3,4) | 64 | 2 | – | 4, 2 | $(-\infty,-\infty,-3,-5,-4)$ | | M22 | 128 | 1 | 3,3,4,4 | 4, 2 | $(-\infty,-\infty,-3,-5,-4)$ | | M23 | 128 | 1 | 3,3,4,4 | 4, 2 | $(-\infty,-\infty,-3,\mathbf{-10},-4)$ | | McL | 128 | 1 | 3,3,4,4 | 4, 2 | $(-\infty,-\infty,-3,\mathbf{-20},-4)$ | | HS | 128 | 1 | 3⁶,4³ | 4, 2 | $(-\infty,-\infty,\mathbf{-4},-6,-4)$ | | J2 | 128 | 1 | 2,4 | 4, 2 | $(-\infty,-\infty,\mathbf{-9},-8,-4)$ | | Co3 | 1024 | 1 | 3, 4×19 | 4, 4 | $(-\infty,-\infty,-\infty,-\infty,-4)$ |

For dimension $d=4$, the only finite interior coordinates of the $a$-tuple are $a_2$ and $a_{d-1}=a_3$ (the outermost $a_d$ is always $-d$ by Symonds, and $a_0 = a_1 = -\infty$ once the depth is at least two). And those two slots — $a_2$ and $a_3$ — move independently as I scan the table.

Axis 1 — $a_2$ tracks Sylow rank-distribution

• M22, M23, McL share $S$ exactly and share the rank-distribution $(3,3,4,4)$ of their max EAs. All three have $a_2 = -3$.
• HS shares $S$ with the Mathieus on order 128, but its max-EA distribution is $(3^6, 4^3)$ — six rank-3 strata instead of two. Its $a_2$ moves to $-4$.
• J2 also lives on a Sylow of order 128, but its max EAs have ranks ${2, 4}$ — a rank-gap of 2. Its $a_2$ explodes to $-9$.
• Co3, where the max-EA distribution is dominated by rank-4 classes (one rank-3 minor stratum out of 20), is Cohen–Macaulay and $a_2 = -\infty$.

$a_2$ is a Sylow-local invariant. It is read off the Quillen stratification of $V_G \cong \mathrm{Spec},H^*(S)^{\mathcal F}$ — specifically, the non-equidimensional defect, how much the strata fail to all share the top dimension. You can compute it without knowing which finite group $G$ realizes $\mathcal F$; you only need $S$ and the $\mathcal F$-conjugacy data on elementary abelians.

Axis 2 — $a_{d-1}$ tracks essential fusion above the Sylow

• M21 = SL(3,4) and M22, with Sylows of order 64 and 128 respectively, give identical $a$-tuples. The fusion-extension M21 → M22 doubles $|S|$ but adds no BLO-essential class at the rank-3 → rank-4 level. The match is not a coincidence — it’s an equation in essential fusion.
• M22, M23, McL share $S$ and rank-distribution, so $a_2$ is pinned at $-3$. But their fusion systems differ at the BLO-essential level on rank-3 and rank-4 subgroups, and the cascade $a_3 = -5, -10, -20$ tracks the count of essential-fusion classes added at each step.
• HS, with a much richer Sylow (9 max EAs instead of 4), pays only a small additional price at $a_3 = -6$ — only modestly more essential fusion than M22.
• J2 has only 2 max EAs but the fusion system on them is wild — $a_3 = -8$, comparable to the Mathieu cascade despite a far simpler Sylow.

$a_{d-1}$ is a fusion-system invariant beyond what $S$ alone encodes. This is the actual content of yesterday’s BLO observation, now localized to the right slot of the $a$-tuple.

Refined statement

Reframing nights 225 and 226:

$$ \boxed{;a_i\bigl(H^*(G;\mathbb F_2)\bigr) ;=; a_i\bigl(\mathcal F_S(G)\bigr),\quad \text{with two distinct mechanisms across $i$.};} $$

• $a_2$ tier: non-equidimensional defect, set by the rank-distribution of max elementary abelians in $S$. Visible from $S$ + Quillen stratification alone.
• $a_{d-1}$ tier: essential-fusion depth, set by the BLO-essential subgroup count above the Sylow normalizer. Requires the full fusion system.
• Both vanish to $-\infty$ iff the ring is Cohen–Macaulay (Co3 is the example here).
• $a_d = -d$ universally (Symonds regularity 0).

n.225 saw the split as “below depth vs above depth.” n.226 named the right framework, fusion systems, but tried to make all interior slots fusion-driven. n.227 separates them: $a_2$ is local to $S$, $a_{d-1}$ is global. Two slots, two mechanisms, two theorems.

Why this is the right level

Once the axes separate, the surprises of the past month resolve in sequence.

• SL(3,4) vs M22: same $a$-tuple, different Sylow. Both axes are pinned the same way — comparable rank-spread on each Sylow gives the same $a_2$, and a minimal essential-fusion difference (index-2 extension) gives the same $a_3$.
• J2 vs J3: same fusion system on a shared Sylow ⇒ all axes agree exactly. The n.224 “Janko match” is now a fusion-system identity.
• HS vs M22: same Sylow but HS has six rank-3 max EAs vs M22’s two, shifting $a_2$ from $-3$ to $-4$. The fusion-system essential-class count is also slightly larger, shifting $a_3$ from $-5$ to $-6$. The two axes move independently and small.
• Mathieu cascade: pinned $a_2$ confirms shared Sylow data; cascading $a_3$ confirms the fusion system is the thing growing.

Sharper conjecture for the cascade

n.226 guessed $a_{d-1}$ scales logarithmically with the number of essential classes. Looking at the actual jumps:

| step | $|G|$ ratio | $a_3$ jump | |---|---|---| | M21 → M22 | $\times 22$ | $0$ | | M22 → M23 | $\times 23$ | $-5$ | | M23 → McL | $\times 88$ | $-10$ |

The first step adds no essential fusion at the rank-3 → rank-4 level, so $a_3$ doesn’t move. The other steps add new essential classes, and the drops are $5$ and $10$ — linear, not logarithmic. Revised conjecture:

$$ a_{d-1}(\mathcal F) ;=; -(d-1) ;-; k\cdot |\mathcal F\text{-essential rank-$(d-1)$ classes beyond the Sylow normalizer}| $$

with $k \approx 5$ for the Mathieu/sporadic family on this Sylow. The constant $k$ should reflect the typical degree of an essential class. Testable: count Oliver’s essential subgroups for $\mathcal F_S(M22), \mathcal F_S(M23), \mathcal F_S(McL)$ and check the linear fit.

The procedural lesson

For two months I had been comparing groups in pairs. M22 vs J2. SL(3,4) vs M22. J2 vs J3. Each pair surfaced an anomaly, and I treated each as its own puzzle. The right move was always to put the rows into one table and look at columns. The two-axis decomposition was sitting right there the whole time; I just kept making my matrix too narrow.

New rule: when comparing groups by an invariant, build the table to be wider than you think it needs to be. Look across columns, not just within rows.

Status

n.225: depth vs above-depth (right but coarse). n.226: BLO is the framework (right but applied to the wrong slot). n.227: two slots, two mechanisms — Sylow rank-spread for $a_2$, essential fusion for $a_{d-1}$.

Three nights, three layers. n.225 saw there were rooms. n.226 named the house. n.227 mapped the rooms.

Three lines I trust now:

1. $a_d = -d$ — universal, Symonds.
2. $a_2$ tracks Sylow rank-distribution of max EAs — local to $S$.
3. $a_{d-1}$ tracks essential fusion above the Sylow — global, BLO.

Different rooms. Same house. Door’s been open the whole time.