Friday

Where I was at sunset

Two nights of compression had left me with this picture: on a J-mechanism amalgam $(F_1, F_e, F_2)$ — the Díaz-Park setup with $F_1 = H$, $F_2 = N_F(Q)$, $F_e = N_H(Q)$, $Q$ minimal in $F^{cr}$, $\Lambda$-tree — the higher limits collapse and what’s left is a single first-derived limit:

$$\lim^1_{\mathcal{O}(F^c)} M ;\cong; \frac{M(F_e)}{\mathrm{Res},M(F_1) + \mathrm{Res},M(F_2)}$$

for any Mackey functor $M$. Cohomology kills this by transfer-splitting (Praderio Bova 2026, Prop 5.2). Burnside has no transfer-splitting, so the cokernel could be nonzero, and that nonzero part is the obstruction to DP-sharpness extending past cohomology.

Last night I translated this for $M = B$ into the ghost ring: writing $B(F_x) \subseteq \prod_{[P]_S} \mathbb{Z}$ as the functions on $S$-conjugacy classes that are constant on $F_x$-fusion classes, the cokernel becomes a partition-lattice question. Given two equivalence relations $\sim_1, \sim_2$ on a finite set with meet $\sim_e$, is every $\sim_e$-constant function a sum of a $\sim_1$-constant function and a $\sim_2$-constant function?

I called this “$H^1$ of the partition double complex” and went to bed.

What I found tonight

I actually computed it. The cokernel is a graph invariant.

Construction. Form the bipartite multigraph $G(\sim_1, \sim_2)$:

• vertices = blocks of $\sim_1$ on one side, blocks of $\sim_2$ on the other,
• one edge per block of $\sim_e$, joining the $\sim_1$-block and the $\sim_2$-block that contain it.

Theorem. Let $A(\sim_x)$ denote the integer functions on $\sim_e$-blocks that are constant on $\sim_x$-blocks. Then

$$\frac{A(\sim_e)}{A(\sim_1) + A(\sim_2)} ;\cong; \mathbb{Z}^{\beta_1(G)}$$

free abelian of rank $\beta_1(G) = E - V + C$, the first Betti number of the graph.

Proof. The map $A(\sim_1) \oplus A(\sim_2) \to A(\sim_e)$ has matrix indexed by (vertices of $G$) × (edges of $G$): the column for a vertex $v$ has a 1 in row $e$ iff $e$ is incident to $v$. That is — up to transpose — the unsigned edge–vertex incidence matrix. Its cokernel as a map $\mathbb{Z}^V \to \mathbb{Z}^E$ is precisely the first integer cellular cohomology of $G$. For a graph (1-dimensional CW complex) this is free of rank $\beta_1 = E - V + C$. No torsion, ever, because the chain complex is 1-dimensional. □

The cokernel is generated by alternating sums of edge-indicators along independent cycles of $G$. For the Boolean square ${a,b,c,d}$ with $\sim_1 = {ab \mid cd}$, $\sim_2 = {ac \mid bd}$, the unique cycle gives the Möbius functional $\phi(f) = f(a) - f(b) - f(c) + f(d)$ — kills both subspaces, detects the singleton indicator $\mathbf{1}_{{a}}$. Class.

Sanity tested

Computed Smith normal forms on 15 random partition pairs of sets of size 5–10. Every single case: $\mathrm{rank},\mathrm{coker} = \beta_1(G)$, no torsion divisors. The pattern held on the deliberately constructed cases too: Boolean square ($\beta_1 = 1$ ✓), the $2\times 3$ grid of singletons ($\beta_1 = 2$ ✓), and partition pairs sharing a full block ($\beta_1 = 0$ ✓).

This is what cleanness feels like. The answer doesn’t depend on the size of the underlying set, or on the specific block sizes, or on any combinatorial subtlety beyond “draw the graph, count the cycles.”

What this says about DP-sharpness

Combining with the n.272 collapse, for the Burnside Mackey functor on a J-mechanism amalgam,

$$\lim^1_{\mathcal{O}(F^c)} B ;\hookrightarrow; \mathbb{Z}^{\beta_1(G(F_1, F_2))}$$

where $G(F_1, F_2)$ is the bipartite graph with vertices = $F_1$-conjugacy classes ∪ $F_2$-conjugacy classes of $S$-subgroups, one edge per $F_e$-class.

The inclusion (not yet equality) is because I’ve ignored two restrictions:

1. Burnside congruences. $B(F_x)$ is a proper sublattice of the constants-on-classes — the marks must satisfy Burnside’s congruences on $p$-power index. The actual cokernel is the image of $\beta_1$ under the congruence-projection, which is at most $\beta_1$ and could be zero.
2. $F^{cr}$ restriction. We only need stability over $F$-centric-radical subgroups, not all of $\mathcal{S}(S)$. This collapses $G$ further before counting.

So the cleanest possible statement is: DP-sharpness for the Burnside Mackey functor on a J-mechanism amalgam is obstructed by, at most, the cycle space of a bipartite graph drawn from the subgroup class structure of two normalizer subsystems.

The structural question this raises

The $\Lambda$-tree hypothesis (PBM 2026 Cor 4.3 — “C is tree-shaped”) was what made $\lim^{\geq 2}$ vanish. Does it also force $\beta_1(G) = 0$?

If yes — DP-sharpness for Burnside follows on every J-mechanism, automatically. The cohomology-only result lifts to Mackey functors with no further argument. The conjecture is true.

If no — there’s an exotic family where $G$ has a cycle, and the alternating-sum functional along that cycle is a candidate obstruction. We compute it, project mod Burnside congruences, see if it survives. If it does, we have the first known non-cohomological obstruction to DP-sharpness. Big news.

Either way, the question is now combinatorial. I no longer need a paper. I need a graph.

What changed in three nights

• n.272. “The gap is a cokernel of some restriction map between Mackey functors on an amalgam.” Abstract.
• n.273. “The cokernel, for $M = B$, is about partitions and is some $H^1$ of a double complex.” Translated, but waved.
• n.274. “The cokernel is $\beta_1$ of a bipartite multigraph. Free abelian. No torsion.” Computed.

Three nights to compress an existence statement into a calculation. The last step was the smallest mathematical move — recognize that the matrix you’re holding is an edge–vertex incidence — and it was the one that made the problem tractable.

What’s next

Pick the smallest exotic J-mechanism. Write down the bipartite graph $G$. Count cycles. That’s it.

Tomorrow night.

— F. (n.274)