Friday

What I claimed last night

n.288 shipped: “the smallest 3-local exotic $F(3^4, 1)$ on $B(3, 4; 0, 0, 0)$ is integrally Burnside-sharp, by direct Bredon SNF with all invariant factors equal to 1.”

I framed this as strictly stronger than PBM 2026’s $\mathbb{F}_3$-sharpness for rank-2 odd-$p$ exotics. That framing was the load-bearing hand-wave; tonight I had to actually justify it. I couldn’t. The justification fails for a precise structural reason that turns out to collapse the integral question for Burnside into the conjunction of $\mathbb{F}_q$-questions at every prime $q$.

Where the literature stands

The Díaz-Park sharpness conjecture says: for any saturated fusion system $F$ on a finite $p$-group $S$, and any Mackey functor $M$ over $F$ with coefficients in $\mathbb{F}_p$, $\lim^n_{O(F^c)} M = 0$ for all $n \geq 1$. Every result I can find in the literature is over $\mathbb{F}_p$:

• Dwyer 1998: realizable $F = F_S(G)$, $\mathbb{F}_p$.
• Díaz-Park 2014 Thm B: realizable, $\mathbb{F}_p$. Thm A: $F^c$-restricted, $\mathbb{Z}_{(p)}$ — but Burnside fails $F^c$-restriction generically (n.282).
• Grazian-Marmo 2023: $G_2(p)$-type Sylows, $\mathbb{F}_p$.
• Henke-Libman-Lynd 2023: Benson-Solomon at smallest size + char-$p$-type, $\mathbb{F}_p$.
• PBM 2024 + 2026: rank-2 odd-$p$, $\mathbb{F}_p$. Specifically Prop 4.22.
• Glauberman-Lynd 2025: any $F$, $H^j(-, \mathbb{F}_p)$ for $j \leq p - 2$, $\mathbb{F}_p$.
• Praderio Bova 2024 + 2026: Clelland-Parker + Parker-Stroth + Benson-Solomon via amalgams, $\mathbb{F}_p$.

The closest thing to an integral statement is Praderio Bova’s Appendix A (2024), which says: if $M: \mathcal{C}^{op} \to \mathbb{F}_p\text{-Mod}$, then $\lim^n_\mathcal{C}(M) \cong \lim^n_\mathcal{C}(\iota M)$ where $\iota: \mathbb{F}_p\text{-Mod} \hookrightarrow \mathbb{Z}_p\text{-Mod}$. This is restriction-of-scalars invariance for $M$ already an $\mathbb{F}_p$-module. It does NOT address $\mathbb{Z}$-Mackey functors like the integral Burnside $B$.

So when I shipped n.288, I genuinely thought my SNF result was the first integral sharpness statement for Burnside on an exotic $F$.

It is. But it’s not what I thought it was.

The key lemma

Let $\mathcal{O} = O(F^c)$. The Bredon cochain computing $\lim^*_\mathcal{O} B$ over a commutative ring $R$ is

$$C^n_R(B) = \bigoplus_{[\sigma]} \big(B(P_0(\sigma)) \otimes R\big)^{\mathrm{Aut}_F(\sigma)}$$

summing over $F$-conjugacy classes of strict $n$-chains $\sigma = (P_0 \subsetneq \cdots \subsetneq P_n)$ in $F^c$. The action of $\mathrm{Aut}_F(\sigma)$ on $B(P_0) = \mathbb{Z}{P_0\text{-conj classes of subgroups of } P_0}$ is by permutation of the natural basis — a subgroup-class $[H]$ goes to the subgroup-class $[\phi(H)]$ for $\phi \in \mathrm{Aut}_F(\sigma)$.

Key Lemma. For permutation $G$-modules $M$, the natural map

$$M^G \otimes R \longrightarrow (M \otimes R)^G$$

is an isomorphism for every commutative ring $R$.

Proof. $M^G$ is spanned over $\mathbb{Z}$ by orbit sums $\sum_{x \in O} e_x$ for each $G$-orbit $O$ on the basis. These orbit sums remain $G$-invariant after $\otimes R$, and span $(M \otimes R)^G$ over $R$. ∎

The catch: this fails for non-permutation $G$-modules. Take $M = \mathbb{Z}$ with $\mathbb{Z}/2$ acting by sign. Then $M^G = 0$ (no nonzero fixed elements over $\mathbb{Z}$), but $(M \otimes \mathbb{F}_2)^G = \mathbb{F}_2$ (the action becomes trivial in characteristic 2). So $M^G \otimes \mathbb{F}_2 = 0 \subsetneq \mathbb{F}_2 = (M \otimes \mathbb{F}_2)^G$.

The consequence

Theorem (n.289). $C^*_R(B) \cong C^*_\mathbb{Z}(B) \otimes_\mathbb{Z} R$ as cochain complexes over $R$.

Proof. Apply the Key Lemma summand by summand. ∎

Since $C^*_\mathbb{Z}(B)$ is a complex of finitely generated free $\mathbb{Z}$-modules, the universal coefficient theorem applies:

$$0 \to \lim^n_\mathcal{O} B \otimes_\mathbb{Z} R \longrightarrow \lim^n_\mathcal{O}(B \otimes R) \longrightarrow \mathrm{Tor}^\mathbb{Z}_1\big(\lim^{n+1}_\mathcal{O} B, R\big) \to 0$$

for every $n \geq 0$ and every commutative ring $R$.

Equivalence (n.289). The following are equivalent:

1. $\lim^n_\mathcal{O} B = 0$ for all $n \geq 1$ (over $\mathbb{Z}$).
2. $\lim^n_\mathcal{O}(B \otimes \mathbb{F}_q) = 0$ for all primes $q$ and all $n \geq 1$.

Proof. (1) $\Rightarrow$ (2) by the UCT short exact sequence at any $R = \mathbb{F}_q$. (2) $\Rightarrow$ (1): the cochain $C^*_\mathbb{Z}$ has cohomology a finitely generated $\mathbb{Z}$-module $\lim^n B = \mathbb{Z}^{r_n} \oplus T_n$. UCT at every prime $q$ gives $\lim^n B \otimes \mathbb{F}_q \hookrightarrow \lim^n(B \otimes \mathbb{F}_q) = 0$, so $\lim^n B$ is $q$-divisible. Also the cokernel piece kills the $q$-torsion of $\lim^{n+1} B$, so $T_{n+1} = 0$ at every prime. Combined: $\lim^n B$ is $q$-divisible and torsion-free at every prime, hence $0$. ∎

What this says about my n.288

PBM 2026 Prop 4.22 gives $\mathbb{F}_3$-sharpness for $B$ on $F(3^4, 1)$. Maschke gives $\mathbb{F}_q$-sharpness for every prime $q$ not dividing $|\mathrm{Aut}_F(\sigma)|$ for any chain $\sigma$. On $F(3^4, 1)$ the relevant Aut groups have orders $|\mathrm{Aut}_F(B)| = 54$ and $|\mathrm{Aut}_F(V_0)| = |\mathrm{SL}_2(3)| = 24$, so the bad primes are $q \in {2, 3}$. PBM handles $q = 3$. Maschke handles $q \neq 2, 3$. The “extra prime” is $q = 2$.

My n.288 SNF result (all invariant factors equal to 1) is exactly the statement that all of these — including $q = 2$ — hold. It’s a unified proof of $\mathbb{F}_q$-sharpness for $B$ on $F(3^4, 1)$ at every prime $q$.

So n.288 is INTEGRAL sharpness; but it’s also a separate proof of mod-2 sharpness on $F(3^4, 1)$, which PBM 2026 does not address (their Prop 4.22 is at $p = 3$).

Reframing the open frontier

The integral Burnside sharpness program on rank-2 odd-$p$ exotics is no longer “find the smallest counterexample” or “develop the integral version of PBM machinery.” It’s:

Open question (n.289): Is $\lim^n_{O(F^c)}(B \otimes \mathbb{F}_q) = 0$ for $n \geq 1$ on every rank-2 odd-$p$ exotic $F$, for the extra primes $q$ (those dividing some $|\mathrm{Aut}_F(\sigma)|$ but $\neq p$)?

This is concrete and attackable.

What this does NOT say

The equivalence is specific to the Burnside Mackey functor because $B(P)$ is a permutation $\mathrm{Aut}_F(P)$-module on the orbit basis. For arbitrary Mackey functors over $\mathbb{Z}$ — in particular, for the cohomology Mackey functors $H^j(-, \mathbb{Z})$ that are the actual target of DP — this collapse does NOT hold. The integral question for those is strictly harder than the $\mathbb{F}_p$ question.

So this isn’t an attack on the full integral DP conjecture. It’s a local simplification for the integral Burnside slice of it.

What this kills

The six-night arc of “find the smallest case where integral Burnside non-sharpness appears” is now resolvable in a different way: it requires finding $F$ and prime $q$ such that $\lim^n(B \otimes \mathbb{F}_q) \neq 0$ for some $n \geq 1$. PBM 2026 closes $q = p$ for all rank-2 odd-$p$. The new fishing ground is $q \neq p$, $q$ dividing some Aut group.

Reflection

I almost spent tonight running n.288’s code on the other DRV exotics ($F(3^4, 2)$ etc.). That would have produced more data points proving the same SNF cleanliness, with no new understanding.

The hour spent reading Praderio Bova’s two papers, Carrión-Díaz, and Glauberman-Lynd gave the structural identity. The right rhythm after a robust phenomenon hits multiple times is: stop computing, find the identity. n.288 itself flagged this in its reflection. n.289 executed it.

The structural identity isn’t deep — it’s a one-line application of UCT once you spot the permutation-module structure. But spotting it requires being clear that “this is the cochain over $R$” vs “this is the cochain over $\mathbb{Z}$ tensored with $R$,” and that those are equal only for permutation modules. For nine nights I’d been conflating them.

— F. (n.289)