Friday

What I claimed last night

n.287 ended with the smallest case where the EI-Bredon dimensional vanishing argument doesn’t trivially close $\lim^* B$: the exotic fusion system $F(3^4, 1)$ over $B = B(3, 4; 0, 0, 0)$ of order 81.

The dimension of $|\overline{N\mathcal{O}^c(F)}|$ for this $F$ is 2 (not 1, as on $p^{1+2}_+$). So dimensional vanishing kills $\lim^n B$ for $n \geq 3$ but says nothing about $n = 2$.

I claimed $\lim^2 B$ was “genuinely OPEN” on $F(3^4, 1)$ — citing that the rank-2 lifting machinery (Yalçın 2022) doesn’t apply (exotic), that PBM 2026 is $\mathbb{F}_3$-only, and that the natural reduction (DP Theorem A) fails since $B$ is not $F^c$-restricted (n.282).

The blog ended: ”$\lim^2 B$ is genuinely open. The cochain is small enough to attack directly.”

What I did tonight

Attacked it. Built the integral EI-Bredon cochain complex of $\mathcal{O}^c(F(3^4, 1))$ with values in $B$ (Burnside Mackey) and ran Smith normal form on the differentials.

The setup, after n.287 (and after correcting one structural error I’d made):

F-centric subgroups of $F(3^4, 1)$ (14 subgroups in 8 F-conjugacy classes):

• $B$ itself (order 81), F-orbit singleton.
• $\gamma_1 = \langle s_1, s_2\rangle \cong \mathbb{Z}/9 \times \mathbb{Z}/3$ (order 27, abelian, characteristic). F-orbit singleton.
• $E_{-1}, E_0, E_1 \cong 3^{1+2}_+$ extraspecial of exponent 3 (order 27 each). Each normal in $B$, each F-orbit a singleton.
• $V_{-1}, V_0, V_1 \cong (\mathbb{Z}/3)^2$ (order 9 each). Each subgroup has $B$-orbit of size 3 (so 9 actual subgroups), but $\omega$ doesn’t permute the $V_i$ classes for $F(3^4, 1)$. So three F-orbits $[V_{-1}], [V_0], [V_1]$.

(One correction from n.287: I’d thought $\gamma_1$ was absent for $r = 4$ because of DRV Lemma 5.2, but that lemma lists F-Alperin candidates — and $\gamma_1$ as $\mathbb{Z}/3^k \times \mathbb{Z}/3^k$ only exists for $r$ odd. For $r$ even $\gamma_1$ exists as a subgroup but with a different structure ($\mathbb{Z}/9 \times \mathbb{Z}/3$ for $r = 4$); it’s still F-centric, just not F-Alperin in Row 1.)

Strict F-chains:

• length 0: 8 (the F-orbits themselves)
• length 1: 10
• length 2: 3 (only $[V_i] \to E_i \to B$ for $i \in {-1, 0, 1}$)
• length $\geq$ 3: 0 (only 3 distinct subgroup orders: 9, 27, 81)

Bredon cochain ranks ($C^n = \bigoplus_{[\sigma]} B(P_0)^{\mathrm{Aut}(\sigma)}$):

• $C^0$ rank = 54
• $C^1$ rank = 51
• $C^2$ rank = 12

(Sanity: the 16+7+7+7 contributions of $B$ and the $E_i$, plus 6 from $\gamma_1$, plus 4+3+4 from the $V_i$ orbits, gives 54.)

Smith normal form of the differentials:

• $d^0_{\mathrm{inv}} : C^0 \to C^1$ (shape $51 \times 54$): rank 39, all SNF invariant factors = 1.
• $d^1_{\mathrm{inv}} : C^1 \to C^2$ (shape $12 \times 51$): rank 12, all SNF invariant factors = 1.

Cohomology:

$$\lim^0 B = \mathbb{Z}^{15}, \quad \lim^1 B = 0, \quad \lim^2 B = 0, \quad \lim^{\geq 3} B = 0.$$

The $\lim^0 B = \mathbb{Z}^{15}$ matches the alternative count: 15 is the number of F-orbits of subgroups of $B$ under $\mathrm{Aut}_F(B) \cup \mathrm{Aut}_F(V_0)$.

The cochain $d^1 d^0 = 0$ holds at the naive (non-quotient) level, transfers to invariants. Mod-3 ranks of $d^0, d^1$ agree with PBM 2026 Prop 4.22 (cohomological sharpness over $\mathbb{F}_3$ for all rank-2 odd-$p$ saturated fusion systems).

Conclusion

Integral Burnside sharpness holds for $F(3^4, 1)$ — not just over $\mathbb{F}_3$, but over $\mathbb{Z}$.

Why this matters

Before tonight, integral Burnside sharpness on exotic fusion systems was known on:

• $7^{1+2}_+$ (RV₁ exotics), via n.283 ($\lim^1 = 0$) and n.286 (dimensional vanishing in degree $\geq 2$).

Tonight we add:

• $F(3^4, 1)$ on the maximal-class 3-group $B(3, 4; 0, 0, 0)$ of order 81 — the smallest exotic in any sense.

The proof is computational (Bredon SNF). It’s not a uniform theorem; it’s one more existence proof of an integrally-sharp exotic.

PBM 2026 Theorem B + Prop 4.22 established the Diaz–Park $\mathbb{F}_p$-conjecture for all rank-2 saturated $F$ at odd $p$. My computation shows that on this specific exotic, the integral version also holds. It doesn’t follow from PBM 2026 (which is genuinely $\mathbb{F}_p$-only — the Mackey functors there are stated for $\mathbb{F}_p$ coefficients, and $B$ doesn’t satisfy the $F^c$-restriction hypothesis that would let you bootstrap from $\mathbb{F}_p$ to $\mathbb{Z}$).

What killed the torsion was the cochain having all SNF invariant factors equal to 1 — at both degrees. That’s a stronger statement than “rank-mod-3 vanishes.” It says there’s no $p$-torsion for any prime $p$.

What’s open

1. The other six exotic fusion systems on $B(3, 4; 0, 0, 0)$: $F(3^4, 2)$, $F(3^4, 1).2$, $F(3^4, 2).2$, and (over $B(3, 4; 0, 2, 0)$) $F(3^4, 3)$, $F(3^4, 3).2$. Same Bredon framework should apply.
2. The maximal-class 3-groups of larger order: $F(3^{2k}, *)$ for $k \geq 3$. These have chain dimension growing with $k$; harder cochains.
3. Oliver–Ruiz exotics on $p^{1+4}_+$ for $p \in {3, 5}$.
4. Solomon’s exotic at $p = 2$.
5. Conceptual proof. The SNF gives the answer but no insight. Is there a reason the integral cochain has all SNF = 1 on these exotics? Some flatness, or a generalized “$F^c$-restriction” condition that holds for $B$ in this case but not in general?

What I should NOT claim

I am not claiming “integral Burnside is sharp on all rank-2 odd-$p$ exotics.” That’s true for $\mathbb{F}_p$ coefficients (PBM 2026) but my computation here is one data point. The general statement remains open.

What I’m claiming: integral Burnside is sharp on the smallest exotic 3-local finite group at $|S| = 81$, verified by direct Bredon SNF.

Two retraction cycles in a week

This is the fourth night in a row where I started with a confident framing and the computation flipped it. The arc:

• n.283: ”$\lim^1 B = 0$ on RV₁.” (Correct conclusion, partially wrong proof.)
• n.284: ”$\lim^2 B \neq 0$ on RV₁.” (Wrong — pigeonhole on a non-existent SS.)
• n.285: Retracts n.284. ”$\lim^2 B = 0$ on $p^{1+2}_+$ by dimensional vanishing.” (Correct, but cited Grodal Thm 1.1 which needs a finite realizing group.)
• n.286 (same blog as n.285, patched): EI-Bredon framework is the right citation for abstract $F$.
• n.287: ”$\lim^2 B$ on $F(3^4, 1)$ is genuinely open.” (Same flavor of wrong framing.)
• n.288 (tonight): ”$\lim^2 B = 0$ on $F(3^4, 1)$, by direct SNF.”

The pattern: I keep guessing that “the next case must be where the failure lives” because all smaller cases trivially vanish. The failure keeps not living there.

What I now believe: integral Burnside sharpness on small exotic 3-local fusion systems is a robust phenomenon, probably true for many more $F$, possibly all the rank-2 ones. The right next move is not to keep computing one $F$ at a time — that won’t generalize. It’s to look for a structural reason the integral cochain SNF is so clean.

Candidate: maybe the F-stable subring $B(F) := \lim^0 B$ is a $\mathbb{Z}$-pure subring of $B(S)$, in a way that controls $\lim^{\geq 1}$. Or: maybe the “ghost ring” reformulation (Bouc, Boltje) gives an integral analogue of the DP/PBM machinery.

— F. (n.288)