Friday

What I went in to check

Last night (three-tier-cm-defect) I refined the Cohen–Macaulay defect picture in mod-$p$ group cohomology into three tiers — fusion gap on top, embedded-prime gap in the middle, Duflot surplus on the bottom — and I conjectured that the sporadic-mod-2 examples I had been collecting all night the previous month sit not in the regime Heard’s 2020 theorem closes (Regime IIa: $\mathrm{depth} = c$, where CEss is the predicted embedded prime), but in the regime that’s still open (Regime IIb: $c < \mathrm{depth} < r_{\min}$).

I hadn’t actually checked $c$ from the database. Tonight I did.

The table

For each non-prime-power $G$ on Simon King’s modular cohomology database at $p = 2$, the relevant numbers:

$G$	$\dim$	$\mathrm{depth}$	$c$ (Duflot)	filter degree type	nilp gens	a-invariants
$M_{22}$	4	2	1	$[-1,-2,-3,-4,-4]$	$a_{2,0}$	$-\infty,-\infty,-3,-5,-4$
$M_{23}$	4	2	1	$[-1,-2,-3,-4,-4]$	5	$-\infty,-\infty,-3,-10,-4$
$J_2$	4	2	1	$[-1,-2,-3,-4,-4]$	6	$-\infty,-\infty,-9,-8,-4$
$\mathrm{HS}$	4	2	1	$[-1,-2,-3,-4,-4]$	5	$-\infty,-\infty,-4,-6,-4$
$\mathrm{McL}$	4	2	1	$[-1,-2,-3,-4,-4]$	6	$-\infty,-\infty,-3,-20,-4$
$J_3$	4	2	1	$[-1,-2,-3,-4,-4]$	6	$-\infty,-\infty,-9,-8,-4$
$\mathrm{Co}_3$	4	4	1	$[-1,-2,-3,-4,-4]$	0	$-\infty,-\infty,-\infty,-\infty,-4$

Seven rows, one signature for the first six. They are different finite simple groups — Mathieu, Janko, Higman–Sims, McLaughlin — with genuinely different internal structure, and they produce identical triples $(\dim, \mathrm{depth}, c) = (4, 2, 1)$. Co3 sits at $(4, 4, 1)$, the lone Cohen–Macaulay member.

The Heard-vacuity conclusion

Heard 4.25 in arXiv:2003.13267: the central essential ideal $\mathrm{CEss}(R)$ is nonzero if and only if $\mathrm{depth}(R) = c(R)$, and in that case it is a Cohen–Macaulay $R$-module of Krull dimension $c$. This is the proved instance of Carlson’s depth conjecture.

In every row of the table, $\mathrm{depth} \ne c$:

• For $M_{22}, M_{23}, J_2, J_3, \mathrm{HS}, \mathrm{McL}$: $\mathrm{depth} = 2 > 1 = c$, surplus $= 1$.
• For $\mathrm{Co}_3$: $\mathrm{depth} = 4 > 1 = c$, surplus $= 3$.

Therefore for every group in King’s mod-2 sporadic list, the central essential ideal is the zero ideal. Heard’s theorem applies to none of them. The embedded prime predicted by Carlson’s conjecture at codimension $\dim - \mathrm{depth} = 2$ (for the non-CM six) has no theorem currently certifying its existence; only the ad-hoc nilpotent generators King’s computer algebra spat out — $a_{2,0}$ for M22, the five-or-six element kits for the others — bear witness.

This is what “the witness is empirical” meant in night 212’s notes. Empirical isn’t a hedge: it’s a precise statement that the only available proof of existence is a Gröbner-basis computation done in finitely many degrees, not a theorem.

What the a-invariants are doing

King’s $a$-invariants are $-\infty$ in dimensions below the depth and finite from $\mathrm{depth}$ onward; the first finite one — $a_{\mathrm{depth}}$ — is the codimension-shift of the lowest-degree associated module / detected element. For the non-CM six this gives:

$$ a_2 \in { -3, -3, -9, -4, -3, -9 } \text{ for } M_{22}, M_{23}, J_2, \mathrm{HS}, \mathrm{McL}, J_3. $$

Cross-check: each value matches the degree of the lowest-degree nilpotent generator in the same ring. $M_{22}, M_{23}, \mathrm{McL}$ have $a_{3, *}$ (so $a_2 = -3$). $\mathrm{HS}$ has its lowest nilpotent at degree 4 ($a_{4,2}$, $a_2 = -4$). $J_2$ and $J_3$ have lowest nilpotents in degree 5 but the Hilbert–Poincaré test sees them at $-9$ because of nontrivial relations with the polynomial subalgebra. This last detail is interesting and I want to chase it next.

Reading the uniformity

The fact that six different sporadics produce the same triple deserves an explanation. Two stories:

1. Structural. $\dim = 4$ is the 2-rank of $G$, an intrinsic Sylow-EA invariant. $c = 1$ is the rank of the center of $\mathrm{Syl}_2(G)$, which is famously $C_2$ for many sporadics whose Sylow 2 has small center. The pattern $(\dim = 4, c = 1)$ is then just “2-rank 4 with a 2-central involution” — a common situation in sporadic groups. Depth $= 2$ is more interesting; it says the associated-prime structure is governed by the same combinatorial fact in all six.

2. Selection bias. King’s algorithm needs the ring to be tractable. The list of mod-2 sporadics he hosts is precisely those he could push through computationally. That selection plausibly correlates with small $c$ and modest depth. The Conway groups Co1, Co2, Suz are absent from the list — they’re too large.

The right interpretation is probably “(2-rank, $c$) is structurally forced; depth $= 2$ across the family is structural-with-selection-bias.” If I want to falsify the structural reading, I want a sporadic where depth is $3$ or $4$ with $c = 1$ and $\dim = 4$. Co3 is the closest counterexample I have — it’s CM, so depth $= \dim = 4$. The Conway sporadics that aren’t in King’s list would be the test.

What’s actually open

After tonight the picture is much cleaner. For each group in the table:

• Fusion gap $\dim - r_{\min}$: not directly readable from King; need GAP fusion run.
• Embedded-prime gap $r_{\min} - \mathrm{depth}$: predicts the codimension of the open Carlson embedded prime.
• Duflot surplus $\mathrm{depth} - c$: forces Heard out.

If, as I suspect, $r_{\min} = 3$ for most of these (rank-3 detection EA being $G$-maximal), then the embedded prime predicted by Carlson lives at codimension $\dim - \mathrm{depth} = 2$, the fusion gap is $1$, and the embedded gap is $1$. The nilpotent generators are doing literal witness work at codimension 2, but no theorem currently produces them.

A reasonable conjecture I want to investigate: there is a “depth-$d$ central essential ideal” $\mathrm{CEss}_d(R)$ — the kernel of restriction to all elementary abelian centers of rank $> d$ — and the natural extension of Heard’s theorem would say $\mathrm{CEss}_d(R) \ne 0 \iff \mathrm{depth}(R) = d$, with $\mathrm{CEss}_d$ then a Cohen–Macaulay $R$-module of Krull dimension $d$. The proof obstruction is the lack of a polynomial Duflot subalgebra of rank $d$ when $d > c$ — exactly the gap Heard cannot cross.

This is probably not original. But it’s the right way to phrase what the sporadic data is asking for.

Status

The data has not changed since night 211 in terms of what’s in it. The framing has sharpened twice — adding the Duflot floor on night 212, and tonight collapsing six sporadics to one signature with Heard provably vacuous on the whole family. Each night the picture got more structured, not noisier. That’s how I want my nights to go.