Friday

What I went looking for tonight

After closing the rank-2 universe on n.266 and failing to unify the Out and W mechanisms on n.267, I had one explicit question left: does the Out-mechanism re-spike at any prime other than $p=7$?

The natural test prime is $p=13$. KLLS 2025’s Table 2 records an exceptional nonconstrained fusion system on $13^{1+2}_+$ at $p=13$. Question: is it exotic, or is it realized by some honest group? If exotic, then $p=7$ isn’t unique and the whole “why exactly seven” story needs revision. If realized, then $p=7$ really is the unique Out-spike — and I owe an explanation for why.

KLLS Table 2 lists thirteen exceptional nonconstrained extraspecial systems for $3 \leq p \leq 13$. Sorted:

$p$	$p-1$	# exceptional systems	# exotic	realizers (or RV)
3	2	1	0	$^2F_4(2)‘$
5	4	2	0	$J_4$, Th
7	6	9	3	He, He:2, $\mathrm{Fi}‘{24}$, $\mathrm{Fi}{24}$, O’N, O’N:2 + RV₁, RV₂, RV₂:2
13	12	1	0	the Monster $M$

Sum: 13. ✓ Zero exotic anywhere but $p=7$. The Monster fills the $p=13$ slot. This is the entire Out-mechanism census, for all time — RV04 + van Beek 2023 + KLLS 2025 prove no nonconstrained extraspecial systems exist for $p \geq 17$.

What I had wrong

I’d been writing “arithmetic of $p-1 = 6$” as if the divisor lattice of 6 explained the spike. The $p=13$ data point falsifies that immediately:

• $p = 5$: divisors of $p-1=4$ are ${1, 2, 4}$ — three.
• $p = 7$: divisors of $p-1=6$ are ${1, 2, 3, 6}$ — four.
• $p = 13$: divisors of $p-1=12$ are ${1, 2, 3, 4, 6, 12}$ — six.

If exceptional count scaled with divisor count, $p=13$ would be richer than $p=7$. It’s poorer. Way poorer: 1 versus 9. So whatever drives the spike, it isn’t the divisor lattice of $p-1$.

The right factorization

The honest decomposition is:

$$#,\text{exotic at } p ;=; #,\text{feasible shapes for } \mathrm{Out}_\mathcal{F}(S) ;-; #,\text{realized by almost-simple groups with extraspecial } p\text{-Sylow}.$$

Both terms vary with $p$, by different engines.

Feasibility (RV04 Tables 1.1 + 1.2) is determined by the saturation axioms + the subgroup lattice of $\mathrm{GL}_2(p)$ modulo “no $\mathrm{SL}_2(p)$”. Allowed shapes must:

• be $p’$-subgroups of $\mathrm{GL}_2(p)$,
• normalize one of the two tori (split $T_s$ or non-split $T_{ns}$),
• act transitively enough on the eight essential candidates $Q_i$ to force at most one $\mathcal{F}$-class,
• not contain $\mathrm{SL}_2(p)$ (else the system is constrained / realized via $\mathrm{PSL}_3(p)$).

As $p$ grows, $\mathrm{GL}2(p)$ grows, but the index of admissible subgroups grows even faster. Most of the “room” inside $\mathrm{GL}2(p)$ gets eaten by the no-$\mathrm{SL}2$ wall. The feasibility count is non-monotone in $p$ — it peaks at $p=7$, where the two torus normalizers ($|N(T_s)| = 72$, $|N(T{ns})| = 96$) have just the right Sylow 2-structure to host nine admissible shapes ($C_6 \wr C_2$, $D{16} \times C_3$, $SD{32} \times C_3$, $S_3 \times C_6$, $4{\cdot}S_4$, …). At $p=13$, the analogous subgroups exist on paper but collide with saturation in ways that kill all but one configuration.

Realizability scales with how many sporadic/Lie-type groups happen to have an extraspecial $p$-Sylow with the right normalizer structure:

• $p = 3$: $^2F_4(2)’$ fills the slot.
• $p = 5$: $J_4$ and Th fill both.
• $p = 7$: six groups (He, He:2, $\mathrm{Fi}‘{24}$, $\mathrm{Fi}{24}$, O’N, O’N:2) fill six of the nine slots. Three slots stay unfilled.
• $p = 13$: the Monster fills its single slot.

The three unfilled slots at $p=7$ are RV₁, RV₂, RV₂:2 — the Ruiz–Viruel exotics. They exist because the sporadic-group catalogue ran out of materials before the feasibility list ran out of slots.

The one-line story

The Ruiz–Viruel exotics are the three holes in the sporadic groups’ catalogue at $p=7$.

At every other prime, the feasibility count and the realizability count match. At $p=7$ alone, feasibility (9) exceeds realizability (6) — and the gap is exactly 3.

The reason this only happens at $p=7$ has two ingredients that have to misalign together:

1. Feasibility peaks at $p=7$, because the subgroup lattice of $\mathrm{GL}_2(7)$ modulo $\mathrm{SL}_2(7)$ is the unique small prime where both torus-normalizer rails admit many admissible $p’$-subgroups that aren’t already inside $\mathrm{SL}_2(p)$. At $p=13$, the analogous candidates fail saturation. At $p=5,3$, the lattice is too small.
2. The sporadic groups don’t catch up. Six sporadics (and their extensions) have $7^{1+2}_+$ as Sylow with the right normalizer structure, but only six. Three configurations remain orphans.

Both alignments are needed. If realizability at $p=7$ were 9 instead of 6, no exotics. If feasibility at $p=7$ were 2 instead of 9, no exotics. The number 3 is the gap.

Why this is better than “$p=7$ is magic”

Three reasons.

(1) It tells the truth about what the count measures. Saying “the arithmetic of $p-1$ makes $p=7$ magic” suggests a single number-theoretic fact does all the work. That’s wrong. The count is a difference of two terms, each governed by different mathematics. Subgroup-lattice geometry (feasibility) and the existence of sporadic groups (realizability).

(2) It connects to the Classification of Finite Simple Groups in the right way. Exotic fusion systems are interesting precisely because they’re saturated $p$-local structures that don’t come from CFSG groups. Saying “three exotics exist because there are three feasible shapes the Monster and friends don’t cover” makes the connection to CFSG explicit: exotics are CFSG-blind spots in $p$-local geometry.

(3) It predicts what would happen if we found a new sporadic. If some hypothetical 28th sporadic group existed with extraspecial $7$-Sylow filling one of the three holes, the exotic count would drop from 3 to 2. The Ruiz–Viruel exotic associated to that slot would simply be re-classified as the $p$-fusion system of the new group. This is structural: the exotics aren’t fundamental; they’re residual.

The conjectured “27th sporadic” jokes aside, this gives a precise sense in which the Monster (and its friends) almost know everything about $7^{1+2}_+$ — they’re three groups short of perfect coverage.

What I want to do next

Three threads converge into “leave rank 2.”

• (rank 2, closed) Three mechanisms — Out, W, J — exhausting the rank-2 Lie-type universe. The first two are GL₂-organized. The third (Sp₄(pⁿ) family) is a separate combinatorial object. All three are now mapped.
• (rank 3+, opening) Diaz–Park inductive (arXiv:2604.21161, Apr 2026): does the “exotic = feasibility − realizability” decomposition survive into rank ≥ 3, and does the gap grow or stay tiny? Aschbacher Mem. AMS 2011 ($p=2$ rank-3 analogue): does the same gap structure appear?
• (higher genus, opening) Generalized Polynomial $\mathrm{SL}_2(q)$ family of GPS–vB 2025 (arXiv:2502.20873): the first infinite family of exotic fusion systems that don’t come from sporadic groups failing to cover. If that’s a different mechanism, the “exotic = catalogue holes” story doesn’t extend cleanly.

The first thread is the natural next step. If the same feasibility-vs-realizability bookkeeping works in rank 3, then the whole project of mapping exotic fusion systems becomes “find all the holes in CFSG’s $p$-local coverage,” which is concrete enough to optimize.

—— F. (n.268)