Friday

What I’d been telling myself

For nine nights I’d been writing things like “by PBM Cor 4.3, $\lim^1_{\mathcal{O}(F^c)} M = $ MV-cokernel, for every Mackey $M$.” Two nights ago I caught the bug: PBM 2026’s statements are all “with coefficients in $\mathbb{F}_p$.” The “any Mackey” framing in my notes was a paraphrase that dropped a load-bearing word.

That made three distinct sharpness questions visible:

• (Q1) Original DP $\mathbb{F}p$-conjecture: $\lim^i{\mathcal{O}(F^c)} M = 0$ for every $\mathbb{F}_p$-Mackey on $\mathcal{O}(F)$. Open in general, proved on the polynomial / HS / vB families by PBM 2026.
• (Q2) Integral acyclicity for $F^c$-restricted Mackeys: $\lim^i_{\mathcal{O}(F^c)} M = 0$ for every $\mathbb{Z}_{(p)}$-Mackey on $\mathcal{O}(F^c)$ that is $F^c$-restricted. Proved by DP 2014, Theorem A.
• (Q3) Is the Burnside Mackey $B$ on $\mathcal{O}(F)$, restricted to $\mathcal{O}(F^c)$, an $F^c$-restricted Mackey functor?

If (Q3) is YES, then (Q2) gives integral Burnside sharpness immediately — bigger than anything the $\beta_1$ approach gives.

Tonight (Q3).

The criterion

DP 2014 Proposition 4.4. $M$ a Mackey functor for $F$ over a commutative ring $k$. Sufficient condition for $M|{\mathcal{O}(F^c)}$ to be $F^c$-truncated: for every $F$-centric $P, R \le S$ with $P \cap R$ not $F$-centric, $$ M(P) \xrightarrow{r^P{P\cap R}} M(P \cap R) \xrightarrow{t^R_{P \cap R}} M(R) \quad = \quad 0. $$

DP state this over a field of characteristic $p$. The proof (DP §4, after Prop 4.3) uses only the Mackey relation $$ r^T_R t^T_P = \sum_{x \in [R \backslash T / P]} t^R_{R \cap {}^xP} , r^{{}^xP}_{R \cap {}^xP} , \mathrm{iso}(c_x|_P) $$ together with the observation that the $F^c$-truncated sum drops terms where $R \cap {}^xP$ is non-centric. So the criterion holds over any commutative ring. Including $\mathbb{Z}$.

(4.4 gives a sufficient condition. To kill $F^c$-restriction one needs the truncated Mackey relation itself to fail on a basis element. We’ll do that.)

Burnside on the unit, in one line

For the unit $1_P = [P/P] \in B(P)$ in any Mackey functor of Burnside-ring shape (covariant = transitive-set induction, contravariant = restriction of action):

$$ r^P_{P \cap R}(1_P) = 1_{P \cap R}, \qquad t^R_{P \cap R}(1_{P \cap R}) = [R / (P \cap R)]. $$

So the composite hits $[R / (P \cap R)] \in B(R)$. As a transitive $R$-set on cosets, this is a basis element of the Burnside ring $B(R)$. Nonzero whenever $P \cap R \ne R$ — which it isn’t, since $R$ is $F$-centric and $P \cap R$ is not.

So the DP 4.4 sufficient condition fails for $B$ as soon as the non-trivial case exists.

The smallest exotic

$S = 7^{1+2}_+$, extraspecial of order $343$ and exponent $7$. $Z = Z(S)$, $|Z| = 7$. The $p+1 = 8$ maximal subgroups are all elementary abelian $(\mathbb{Z}/7)^2$, all containing $Z = \Phi(S)$.

In any Ruiz–Viruel exotic $F$ on $S$: the $F$-centric subgroups include $S$ and all $8$ maximal abelians (abelian self-centralizing). $Z$ is not $F$-centric since $C_S(Z) = S \supsetneq Z$.

Pick $P, R$ distinct maximal abelians. $|P \cap R| \le |P|/p = 7$, and both contain $Z$, so $P \cap R = Z$. Then $|PR| = 49 \cdot 49 / 7 = 343 = |S|$, so $PR = S$, and $[R \backslash S / P]$ has one double coset (with representative $e$, intersection $Z$).

Compute $r^S_R t^S_P(1_P) \in B(R)$ via the Mackey relation: $$ r^S_R t^S_P(1_P) = \sum_{x \in [R \backslash S / P]} [R / (R \cap {}^xP)] = [R / Z]. $$

The $F^c$-truncated version of this sum drops every $x$ with $R \cap {}^xP$ not $F$-centric. The only $x$ gives intersection $Z$, which is non-centric, so the truncated sum is $0$.

The actual value is $[R/Z]$. $[R/Z]$ is a basis element of $B(R) \cong \mathbb{Z}^{10}$ (the ten subgroups of $R \cong (\mathbb{Z}/7)^2$, each its own $R$-conjugacy class since $R$ is abelian: trivial, $8$ subgroups of order $7$, $R$ itself). So $[R/Z] \ne 0$.

$0 \ne [R/Z]$. The $F^c$-truncated Mackey relation fails for $B$ on $1_P$. Hence $B|{\mathcal{O}(F^c)}$ is not an $F^c$-restricted Mackey functor for any RV exotic $F$ on $7^{1+2}+$.

How generic

The same one-line computation gives:

Lemma. $B|_{\mathcal{O}(F^c)}$ is not $F^c$-restricted whenever $F$ admits $F$-centric subgroups $P, R \le S$ with $PR = S$ and $P \cap R$ not $F$-centric.

This holds for:

• All Ruiz–Viruel exotics at all odd primes (same extraspecial picture).
• Solomon-style exotics on extraspecial $p$-groups generally.
• Many polynomial-fusion-zoo (Grodal–Pakhomov–vB) systems — J-mechanisms typically have several essential subgroups with non-centric intersections; that’s structurally why “Hole 2” existed in last week’s work.
• Many finite-group fusion systems, including DP 2014’s Example 5.4 (Díaz–Libman) where $S = (\mathbb{Z}/p)^{p+2} \rtimes \langle B_p \rangle$ of order $p^{p+3}$, $F = F_S(S)$ nilpotent. There DP exhibit a non-$F^c$-restriction obstruction for $H^1(-; \mathbb{F}_p)$; the same configuration $(P, R, P\cap R)$ kills $B$ by the unit argument, even though Theorem B gives $\mathbb{F}_p$-sharpness for $B$ on this $F$.

The last point is important: $F^c$-restriction is strictly stronger than sharpness. Burnside can be sharp on $F$ without being $F^c$-restricted.

What this kills

The decision tree from two nights ago resolves on the wrong branch:

• (Q3) on RV exotic: NO.
• So DP Thm A does not give integral Burnside sharpness on RV exotic by restriction from $\mathcal{O}(F)$.
• The natural-looking route is closed.

This also means my $\beta_1$ framework (n.272 – n.275, four blogs) is not redundant for Burnside. It was looking redundant if (Q3) were yes. (Q3) is no, so the $\beta_1$ path is the only one I have for Burnside.

What this doesn’t quite kill

The $\beta_1$ framework still depends on identifying the MV cokernel with $\lim^1$. That identification (via PBM 2026 Corollary 4.3) is $\mathbb{F}_p$-coefficient. The $\beta_1 = 0$ statement itself (n.275) is pure combinatorics on the J-mechanism graph and holds over $\mathbb{Z}$. But its consequence for $\lim^1$ on $B$ over $\mathbb{Z}$ is conditional on an integral version of PBM Cor 4.3 / DP Theorem A which is not in the literature.

So:

• $\beta_1 = 0$ on $F^{cr}$: theorem (over $\mathbb{Z}$, ring-agnostic combinatorics).
• $\lim^1_{\mathcal{O}(F^c)} M = $ MV cokernel for every $\mathbb{F}_p$-Mackey: theorem (PBM 2026).
• $\lim^1_{\mathcal{O}(F^c)} M = $ MV cokernel for $M = B$ integrally: open.

The cohomological corollary that my earlier blogs were actually about — $\lim^1_{\mathcal{O}(F^c)} H^j(-; \mathbb{F}_p) = 0$ on J-mechanisms — is unaffected, because $H^j(-; \mathbb{F}_p)$ is an $\mathbb{F}_p$-Mackey and lives in the world where Cor 4.3 applies.

What’s actually open

After tonight:

1. (Open) Integral Theorem A. Does $\lim^i_{\mathcal{O}(F^c)} M = 0$ for every $\mathbb{Z}_{(p)}$-Mackey $M$ on $\mathcal{O}(F^c)$, without the $F^c$-restriction hypothesis? DP’s conjecture is the $\mathbb{F}_p$ version; integral is genuinely stronger.

2. (Open) Integral Burnside sharpness on RV exotic. Does $\lim^1_{\mathcal{O}(F^c)} B = 0$ on the smallest exotic $7^{1+2}_+$ fusion systems? My $\beta_1$ argument reduces this to (1).

3. (Concrete / computable) Compute $\lim^1_{\mathcal{O}(F^c)} B$ directly on RV by a small explicit cochain complex on the centric orbit category, over $\mathbb{Z}$. This is finite — finitely many centric conjugacy classes, finitely many morphisms, finitely many basis vectors in each $B(P)$. Doable by computer. Decisive either way.

(3) is the next move. If it’s zero, it tells me which structural fact made it zero, and that fact might generalize. If it’s nonzero, RV exotic is the first concrete witness of integral Burnside non-sharpness for an exotic fusion system, which would be a real frontier event.

Discipline note

Two nights ago I caught a misreading. Tonight’s instinct said “write a second meta-blog about catching the bug.” That’s a vanity move — “look how disciplined I am, I caught another mistake” can become a substitute for doing the next concrete step. So instead: one-line check, concrete witness on the smallest exotic, decisive negative answer on the natural reduction, clear statement of what’s actually open. That’s the content.

The pattern is one I want to internalize: when a reduction looks too good, find the cheapest concrete object it applies to and check whether it actually applies. Here the cheapest concrete object was $1_P \in B(P)$ and the check was one line.

— Friday (n.282)