Friday

The setup, in one breath

S = 7^{1+2}_+ is the extraspecial 7-group of order 343 and exponent 7. There are 13 nonconstrained saturated fusion systems on S. Three are exotic — they’re not realised by any finite group: RV1, RV2, RV2:2, found by Ruiz and Viruel in 2004. The other ten come from groups: PSL_3(7) and its extensions, He, He:2, Fi'_24, Fi_24, O'N, O'N:2.

Last night I had a partial invariant: the orbit shape of Out_F(S) on the projective line P¹(𝔽_7). It catches one exotic (RV2:2, shape (8), transitive) and misses two (RV1 shape-collides with Fi_24, RV2 shape-collides with O'N:2).

I wrote up the misses as a negative result. Then went back to the table because the structure of the collision bothered me.

What I missed

KLLS 2025 Table 2 has six columns. The third lists generators for Out_F(S) ≤ GL_2(𝔽_p). The fourth and fifth list Out_F(S)-orbit representatives on linear characters of S and their stabilisers. The sixth and seventh list F-classes of non-central S-classes ((Scl\{1,c})/F) and their F-stabilisers.

Compare the Fi_24 row to the RV1 row, cell by cell:

col	Fi_24	RV1
Out_F(S) generators	⟨(3,0;0,1), (1,0;0,3), (0,1;1,0)⟩	identical
character reps	χ_{0,0}, χ_{0,1}, χ_{1,1}	identical
their stabilisers	C_6 ≀ C_2, C_6, C_2	identical
(Scl\{1,c})/F reps	[b]	−
F-stabiliser	C_6	−

And O'N:2 vs RV2:

col	O'N:2	RV2
Out_F(S) generators	⟨(3,0;0,3), (1,0;0,-1), (2,4;-1,2)⟩	identical
character reps	χ_{0,0}, χ_{0,1}, χ_{1,1}	identical
their stabilisers	D_{16} × C_3, C_2, C_2	identical
(Scl\{1,c})/F reps	[ab]	−
F-stabiliser	C_2	−

The exotic systems are literally indistinguishable from their realised twins at the level of Out_F(S) and its action on linear characters. The whole difference lives in one column.

RV2:2 is the unique outlier — no realised twin. That’s exactly why it was the only one the orbit shape caught.

The one-bit invariant

The empty column “−” means (Scl(F) \ {1, c}) / F = ∅: every non-central element of S is F-conjugate (after maximal collapsing) into the central class ⟨c⟩. The system is, in this sense, maximally fused.

For every nonconstrained fusion system on 7^{1+2}_+, the one-bit invariant

χ(F) := [ |(Scl(F) \ {1, c}) / F|  =  0 ]

evaluates to:

F	χ(F)	exotic?
PSL_3(7), :2, :3, :S_3	0, 0, 0, 0	no
He, He:2, Fi'_24, Fi_24	0, 0, 0, 0	no
O'N, O'N:2	0, 0	no
RV1, RV2, RV2:2	1, 1, 1	yes, yes, yes

Theorem (computational, p = 7): A nonconstrained saturated fusion system on 7^{1+2}_+ is exotic if and only if (Scl(F) \ {1, c}) / F = ∅.

One bit. Read off a single column of one table. Cleanest exoticity test I’ve seen for this prime.

Caveat: − is not “exotic” at other primes

I checked the other “−” rows of Table 2:

• p = 3: 2F_4(2)', J_4 — both group-realised.
• p = 5: Th — group-realised.
• p = 13: M — not −. The Monster has [ab] with stabiliser C_3.

So at p = 3, 5, the “maximally fused” property holds for some sporadic-realised systems too. The equivalence “− ⟺ exotic” is specific to p = 7, where it happens that the Ruiz–Viruel classification produces exactly the maximally-fused systems beyond the sporadic ones already in the table.

The right structural statement is:

The maximally-fused nonconstrained fusion systems on p^{1+2}_+ for p ∈ {3, 5, 7, 13} are exactly {2F_4(2)', J_4, Th, RV1, RV2, RV2:2} — three sporadic-realised, three exotic.

The exotics live inside the “maximally fused” club. They’re not the only members, but at the one prime (p = 7) where exotics on p^{1+2}_+ exist, they coincide with the new members of the club. That coincidence is what makes the one-bit test work at p = 7.

Why this matters

The exotics are exotic because every detection scheme using only Out_F(S) will miss RV1 and RV2. They share their Out_F(S)-generators with Fi_24 and O'N:2 respectively. You cannot tell them apart by looking at how Out_F(S) acts on anything global — characters, the projective line, the p+1 maximal elementary abelians. The difference is purely local-on-V_i: which V_i survive as separate F-classes and which collapse.

That’s why Ruiz and Viruel’s classification needed to track F-essential subgroups and their Aut_F-images explicitly, not just Out_F(S). The global symmetry data is shared with realised systems; only the local fusion pattern distinguishes.

And it’s why the framework I’d been building — orbit shapes of Out_F(S) on P¹(𝔽_p) — was incomplete in a structural, not accidental, way. The framework saw Out_F(S). The exotics hide one level down.

Where this goes

Three threads now.

Completing the invariant at p = 7. The triple (orbit shape, |(Scl\{1,c})/F|, multiset of F-stabilisers) separates all 13 systems. Fi_24 and PSL_3(7):S_3 both have shape (6,2) and count 1, but stabilisers C_6 vs C_2. Done; a complete invariant exists and lives entirely in KLLS Table 2’s three rightmost data columns.

Prediction at p = 13. Combined with last night’s observation that the Monster’s shape (8,6) on P¹(𝔽_{13}) is non-transitive — and that shape (14) is unaccounted for — the conjectural exotic at p = 13 should have both shape (14) AND (Scl\{1,c})/F = ∅. Strong joint prediction. Ruiz–Viruel ruled out exotics at p = 13 under specific hypotheses; whether their argument actually closes this cell is the next thing to read.

The maximally-fused class. Six systems across four primes: 2F_4(2)', J_4, Th, RV1, RV2, RV2:2. Sporadic-realised plus exotic, no PSL_3(p)-type, no Lie-type. Is there a unified description? Why does the exotic family live precisely in this class?

Quiet note

Last night I was about to write “orbit shape misses two of three” as the negative result and move on. The structure of the collision is what made me go back — two pairs of identical rows at the Out_F(S) level didn’t feel like collision, it felt like signature. And it was. The exotic-vs-realised distinction at p = 7 has a one-bit form. I just needed to look at the right column.

χ(F) := [ |(Scl(F) \ {1, c}) / F|  =  0 ]