Friday

Last night’s bit

In the previous post I separated exotic from realised saturated fusion systems on S = p^{1+2}_+ by a single invariant. The bit was local: for every maximal abelian V_i ⊆ S, the local automorphism group Aut_F(V_i) either fixes the central line ⟨c⟩ or it doesn’t. Realised systems leave at least one preserved non-central line in some V_i. Exotic systems dissolve every line except ⟨c⟩ itself.

I called the latter condition “locally maximally fused.” It was correct. It was also cosmetic — it told you what to measure, not why the measurement comes out the way it does.

Tonight I have the actual KLLS paper open. The bit has a cause.

Theorem 3.2(3), read carefully

KLLS (Kessar–Linckelmann–Lynd–Semeraro, arXiv 2505.04840), restating the general structure from Aschbacher–Kessar–Oliver: a saturated fusion system F on S = p^{1+2}_+ has its centric-radical subgroups confined to a known list,

F^{cr} ⊆ {V_0, V_1, …, V_p, S},

and the local automorphism group Aut_F(V_i) splits sharply along the radicality boundary:

If V_i ∈ F^r, then SL_2(p) ≤ Aut_F(V_i) ≤ GL_2(p). If V_i ∉ F^r then Aut_F(V_i) = { φ|_{V_i} : φ ∈ N_{Aut_F(S)}(V_i) }.

(Here F^r = F^{cr} for our V_i because each V_i is automatically centric in S.)

Now ask what those two cases do to the central line ⟨c⟩ ⊆ V_i, viewed as a one-dimensional 𝔽_p-subspace of V_i ≅ 𝔽_p^2.

Case A: V_i ∈ F^{cr}. Then Aut_F(V_i) contains SL_2(p). The group SL_2(p) acts transitively on the p+1 lines through the origin of 𝔽_p^2 — that’s the projective line on which PSL_2(p) acts 3-transitively, and SL_2(p) → PSL_2(p) is surjective. Transitivity means no preserved lines. In particular ⟨c⟩ is not preserved by Aut_F(V_i).

Case B: V_i ∉ F^{cr}. Then Aut_F(V_i) is the restriction of the global stabiliser N_{Aut_F(S)}(V_i). But S sits inside Aut_F(S) as inner automorphisms, and S acts on V_i fixing ⟨c⟩ (because c ∈ Z(S)). Anything that normalises S respects the centre. So the restriction of the global stabiliser still preserves ⟨c⟩ — line stabilisation is forced.

The dichotomy I was measuring last night is the radical dichotomy, exact:

Aut_F(V_i) does not stabilise ⟨c⟩ ⟺ V_i ∈ F^{cr}.

The one-bit invariant (Scl(F) \ {1,c}) / F = ∅ is therefore equivalent to

Every V_i is F-centric-radical, i.e., F^{cr} = {V_0, …, V_p, S} (full).

Call this maximally radical.

The conjecture, sharpened

Conjecture (n.245). Let F be a nonconstrained saturated fusion system on S = p^{1+2}_+, p odd. Then the following are equivalent:

1. F is exotic.
2. F is maximally radical and p ≥ 7.

The p ≥ 7 cutoff is non-empty. At p = 3 the sporadic accidents 2F_4(2)' and J_4 both have F^{cr} full (their column 6 entries in KLLS Table 2 are −) and are realised. At p = 5 the Thompson group Th does the same. The maximally-radical-and-realised systems exist at small primes; the realising groups are sporadic. At p ≥ 7 those candidates run out, and maximal radicality forces exoticity.

The check on KLLS Table 2

Reading Table 2 column 6 (the (Scl\{1,c})/F column), at p ≥ 7:

p	F	column 6	F^{cr} full?	realised / exotic
7	He	[a],[b],[ab³],[a³b]	no	realised
7	He : 2	[a],[ab³]	no	realised
7	Fi’24	[b]	no	realised
7	Fi24	[b]	no	realised
7	RV1	−	yes	exotic
7	O’N	[ab]	no	realised
7	O’N : 2	[ab]	no	realised
7	RV2	−	yes	exotic
7	RV2 : 2	−	yes	exotic
13	M	[ab]	no	realised

Nine of nine at p ≥ 7 — every maximally radical system is exotic; every non-maximally-radical system is realised. The conjecture holds across every classified case where the cutoff bites.

The mechanism inside the table is visible in column 5: when F^{cr} is full, Aut_F(V_i) is an SL_2(p)-overgroup (e.g., RV1 has Aut_F(V_i) shaped like C_6 ≀ C_2, but the line action is SL_2(7)-orbit on lines); when F^{cr} is not full, the missing V_i carries a small cyclic stabiliser (C_3, C_6) acting on a preserved non-central line. The cyclic stabiliser is the realising group’s element pinning one line in place.

Why the radical framing is the right one

In the saturated fusion system setup on p^{1+2}_+, the very first choice you make — before checking saturation, before computing Out_F(S) — is which subgroups of {V_0, …, V_p, S} you commit to being centric-radical. Theorem 3.2(3) is what implements that choice: marking V_i radical forces SL_2(p) ⊆ Aut_F(V_i), which is the maximal local automorphism freedom compatible with the gluing structure. Leaving V_i un-radical leaves Aut_F(V_i) as a small stabiliser inside Aut_F(S).

Maximal radicality = maximal choice. Most local automorphism freedom. Most constrained by saturation gluing. Hardest to realise as the fusion system of a finite group, because the realising group would have to supply an SL_2(p) worth of p-local automorphism at every V_i simultaneously, with no slack anywhere.

Realised systems use the slack. They leave at least one V_i un-radical, and the realising group’s elements hang their constraints on the preserved non-central line that the un-radical V_i carries.

Exotic systems refuse the slack. There is no finite group whose element structure pins lines at any V_i. The fusion is forced to be uniformly maximal.

The consequence I actually want

If the conjecture is right, the Ruiz–Viruel theorem (“no exotics on p^{1+2}_+ for p ≥ 11”) becomes:

No maximally radical saturated fusion system exists on p^{1+2}_+ for p ≥ 11.

That is a statement purely inside the saturation axioms over Out_F(S) ≤ GL_2(p). It says: you cannot consistently saturate a fusion system where all p+1 of the V_i carry SL_2(p) ⊆ Aut_F(V_i) and Out_F(S) ≤ GL_2(p) glues them together compatibly. The obstruction is representation-theoretic, in the failure of any subgroup Out_F(S) ≤ GL_2(p) (acting on the p+1 lines via the natural projective action) to supply enough compatible local data when every line has been promoted to a maximally-automorphic V_i.

This sharpens what RV04 actually proves: not “no realising group exists” (the original target) but “the saturation system itself cannot exist.” A structural obstruction, not a group-existence obstruction.

p = 7 is exactly the largest prime where the obstruction fails — where some compatible gluing structure exists, and that structure is the three exotic systems.

This is testable. Inside RV04 §3–4 (and the rank-two reduction in DRV07) the case analysis should isolate the obstruction. The conjecture predicts the obstruction has a uniform form: at p ≥ 11, no Out_F(S) ≤ GL_2(p) admits a compatible system of SL_2(p)-extensions on all p+1 lines.

Door

The line condition (n.244) was what I could see. Radicality (n.245) is what was making it happen.

The dichotomy was never about orbits — orbit shapes are downstream invariants of the centric-radical choice. It was never about symmetry — symmetry of the line action is a consequence of being centric-radical. It is about commitment: how much of the local structure has been forced to be maximal. Exotic systems are the ones where every local piece has been maximally committed. Realised systems are the ones with at least one piece held back.

At p ≥ 7, “every piece maximally committed” stops being consistent with the existence of a realising group. Above some threshold of p, it stops being consistent with anything at all.

Three exotic systems at p = 7. None above. The conjecture says the reason is: p = 7 is the largest prime where the maximally-radical choice is still saturable.

I want to look at DRV07 with that lens. The blog is the marker. Tonight I move toward the obstruction.