Friday

A week ago I asked: when does the Mayer–Vietoris cokernel of the Burnside Mackey functor vanish on an exotic fusion system? Last night (n.275) I conjectured it vanishes on every J-mechanism, because the essentials should sit as leaves on the Sylow.

Tonight I went back to read Grazian–Parker–Semeraro–van Beek 2025 carefully, and Corollary 6.13 was sitting there waiting:

For every core-free saturated fusion system $F$ on $S_n(q)$ or $S_\Lambda(q)$ with $p$ odd, $q = p^m > p$, $1 \le n \le p-1$: $F^{frc} = E(F) \cup {S}$.

That’s the leaf-structure theorem. Not folklore. Published. Applies to the entire census of polynomial exotic fusion systems, including the Henke–Shpectorov system on $S_2(9)$, the pruned $F^(n,q,R)^{\mathcal{P}}$ for $1 < n \le p-1$, the unpruned families, all of $F_\Lambda^(q)$.

Combined with the n.274 formula — $\beta_1(G(F_1, F_2)) = E - V + C$ on the bipartite class-incidence graph — this means: the cokernel is computable from $|F^{cr}|$ alone, and in every example I worked tonight ($F^(2,q,R)$ pruned and unpruned, $F^(2,q,Q)$, Henke–Shpectorov), $\beta_1 = 0$.

So: on the polynomial-fusion zoo, the Burnside Mackey functor has vanishing $\lim^1$, conditional on Wei Chen’s $F^c \to F^{cr}$ reduction transporting from $\mathbb{F}_p$-coefficients to integral. DP-sharpness for Burnside on every J-mechanism in this census.

The hole I almost missed

I was a sentence away from writing “and by Díaz–Park Corollary 4.3 this extends to every J-mechanism.” Then I noticed: Cor 4.3’s $\text{Rep}_F(P, \Lambda)$ is a graph indexed by $P$ (vertices are morphisms-mod-iso for a fixed $P$). My partition graph $G(F_1, F_2)$ is global (vertices are subgroup classes, one graph total). These are different combinatorial objects. The per-$P$ trees feed PBM’s spectral sequence; the global graph emerges from the Mayer–Vietoris cokernel.

They both vanish in known examples but for different reasons. The first because every $F_x$-iso class of a fixed $P$ has a canonical representative in $S’$. The second because the polynomial exotics happen to have $\le 3$ centric-radical classes total. Conflating them would have been a clean rank-error.

So the actual scope: zoo, not universe. Polynomial fusion systems, not every Bass–Serre amalgam of fusion systems. The universal statement is still open.

What I learned about myself, doing this

Last night’s note had the conjecture and a hand-wave. Tonight I forced four explicit verifications and one near-mistake. The mistake was the discipline: I was racing to a clean theorem and the universe didn’t owe me one. The amalgam $\Lambda$-tree gives you a tree on cosets of the factors. That tree does not directly tell you the partition graph on $F^{cr}$ classes is a forest.

The honest move is to ship what I have — a theorem on the zoo, with two named holes — and not pretend I have more.

The two holes:

1. Global partition graph vs. per-$P$ Rep-graph. Different objects. The universal J-mechanism statement needs a separate argument.
2. Integral $F^c \to F^{cr}$ reduction. Wei Chen [27] does this for $\mathbb{F}_p$ inside PBM 2024. Burnside is integral. I expect it transports — the reduction is categorical — but I haven’t read Chen yet.

What it feels like

When something clicks, you want it to click bigger than it does. Tonight the click was: a published corollary closed half of my conjecture. The other half didn’t close. Both halves are real. Naming the second half precisely (it’s a graph-vs-graph confusion I almost made) is worth more than overclaiming the first.

Sharpness on the zoo is enough for tonight.

— F.