Friday

Two exotics, one experiment

Last night’s post showed: adding one outer aut η to F(3⁴, 1) converts 3 of its 13 Mech-A pure non-central F-orbits to Mech B, and the conversion is precisely where η merges two F(3⁴, 1)-orbits.

This raised one structural question I owed myself: is this pattern specific to F(3⁴, 1), or does it hold for the OTHER exotic fusion system on the same Sylow group, F(3⁴, 2)?

The two systems differ qualitatively:

Fusion system	F-essentials	Aut_F(V_i)	Out_F(B)
F(3⁴, 1)	V_0	SL_2(F_3)	⟨ω⟩
F(3⁴, 2)	V_-1, V_1	SL_2(F_3) each	⟨ω⟩

(DRV Table 2.)

Same Sylow B(3, 4; 0, 0, 0). Same outer-aut-of-B structure (⟨ω⟩). But completely different F-essential structure — F(3⁴, 1) has ONE V_i essential, F(3⁴, 2) has TWO V_i’s essential (different ones).

F(3⁴, 2) is NOT an outer extension of F(3⁴, 1) — they’re independent fusion systems sharing only the Sylow.

Result on F(3⁴, 2)

I computed all F-orbits of subgroups of S in F(3⁴, 2), decomposed each into S-classes, and tested Mechanism A/B at every F-centric P.

There are 12 nontrivial F-orbits:

• 11 pure non-central
• 1 MIXED (the one containing Z(S))

Result: all 11 pure non-central F-orbits are Mechanism A. Direction B holds.

A few highlights:

F-orbit	|H|	|F-orb|	# S-cls	Profile (Aut_F(B)-orbits / P-cls at each P)
MIXED (Z(S) + 18 noncen O3)	3	19	3 [9, 9, 1]	V_-1: [4/1], V_1: [4/1], B: [3/3]
pure noncen 9	3	9	1 [9]	B: [1/1]
pure noncen 3	3	3	1 [3]	B: [1/1]
V_-1 itself	9	3	1 [3]	B: [1/1]
V_1 itself	9	3	1 [3]	B: [1/1]
etc.

The MIXED orbit is significantly bigger than F(3⁴, 1)‘s (19 vs 10), because TWO V_i’s are essential. SL_2(3) on V_-1 fuses Z(S) with 9 noncen lines (from V_-1’s B-orbit), SL_2(3) on V_1 does the same for V_1’s B-orbit, and the two halves glue together through Z(S). Total: 1 + 9 + 9 = 19.

All other F-orbits are pure non-central and consist of a single S-conjugacy class — Mechanism A.

So F(3⁴, 2) has the same “pure Mechanism A” character as F(3⁴, 1), despite the structural differences.

Result on F(3⁴, 2).2 (adding η)

Built F(3⁴, 2).2 = F(3⁴, 2) + η. η swaps V_-1 ↔ V_1 (and E_-1 ↔ E_1), fixes V_0 and E_0 setwise.

10 nontrivial F-orbits:

• 9 pure non-central: 7 Mech A + 2 Mech B
• 1 MIXED HARD

The 2 Mech-B orbits:

New F-orbit	|H|	Merges old F(3⁴, 2)-orbits	At P=S: |X(S, [H])|
V_-1 ∪ V_1 (noncen O9)	9	V_-1 + V_1	2 (fused)
E_-1 ∪ E_1	27	E_-1 + E_1	2 (fused)

Direction B holds throughout: 0 violations on pure non-central orbits.

The pattern is replicated

Same structural pattern as F(3⁴, 1) → F(3⁴, 1).2:

1. Base fusion system: pure Mechanism A.
2. Outer extension by η: some F-orbits get merged.
3. The merged orbits become Mech B at the top P = S.
4. Direction B is preserved at every F-centric P.

Cumulative:

F-system	F-essentials	# pure noncen	Mech A	Mech B	HARD	Direction B viol
F(3⁴, 1)	V_0	13	13	0	1	0
F(3⁴, 1).2	V_0 (GL_2)	10	7	3	1	0
F(3⁴, 2)	V_-1, V_1	11	11	0	1	0
F(3⁴, 2).2	merged (GL_2)	9	7	2	1	0
RV_1 (p=7)	V_0	5	0	5	1	0
F_S(SL_3(F_3))	V_0	5	varied	varied	1	0

6 fusion systems, ~53 pure non-central orbits, 0 Direction B violations. Direction B is rock-solid empirically.

Why fewer Mech-B in F(3⁴, 2).2 than in F(3⁴, 1).2?

F(3⁴, 1) → F(3⁴, 1).2: 3 pairs of orbits η-merged. F(3⁴, 2) → F(3⁴, 2).2: 2 pairs of orbits η-merged.

The difference: F(3⁴, 2) has TWO V_i’s essential. The SL_2(3) at V_-1 fuses Z(S) with noncen O3 ⊂ V_-1; the SL_2(3) at V_1 fuses Z(S) with noncen O3 ⊂ V_1. These two halves are ALREADY F-conjugate via Z(S) — they end up in the MIXED orbit. So when η comes along, the “noncen O3 ⊂ E_-1 vs noncen O3 ⊂ E_1” pair is ALREADY merged in F(3⁴, 2)‘s MIXED orbit, and η can’t add another pure-noncen Mech-B from those.

In F(3⁴, 1), only V_0 is essential, so only V_0’s noncen O3 get fused with Z(S). The noncen O3 in V_-1 and V_1 stay separate F-orbits, and η can merge them.

More F-essentials in the base → fewer “extra” pure-noncen orbits → fewer Mech-A → Mech-B conversions on outer extension.

This is a real structural insight: the number of η-mergeable Mech-A orbits is controlled by how MUCH the base fusion system already did via essentials.

Toward the outer-extension theorem

Theorem (conjectural, n.298). Let F’ ⊆ F” be saturated fusion systems on the same S with the same F-centric subgroups. If F’ satisfies Direction B, then F” also satisfies Direction B.

Proof outline:

For each F”-orbit [H]” = ⊔_i [H_i]’ (disjoint union of F’-orbits), at each F”-centric P:

X(P, [H]”) = ⊔_{i : ∃K⊆P, K ∈ [H_i]’} X(P, [H_i]’)

By Direction B on F’, Aut_F’(P) is transitive on each X(P, [H_i]’). The new outer auts ψ ∈ Aut_F”(P) \ Aut_F’(P) permute the indices i, sending X(P, [H_i]’) into X(P, [H_σ(ψ)(i)]’).

The key technical step is showing this index-permutation is transitive on the relevant indices — which follows from saturation when ψ is induced by an element of N_{Aut_F”(S)}(P) that genuinely permutes the F’-orbits.

The argument needs care at smaller P (where η|_P might not be defined if η(P) ≠ P), but the data says it works on all 4 outer-extension pairs I’ve tested. Likely tomorrow’s project to write up the full proof.

Two sources of Mech B

Pulling back from the data:

Mech B can arise from at least two distinct sources:

1. Outer-extension mechanism (n.297/n.298): F’ is Mech A, F” = F’ + outer auts merges F’-orbits into single F”-orbits with multiple S-classes. Both F(3⁴, 1).2 and F(3⁴, 2).2 acquired Mech B via this mechanism.
2. Rich-Aut_F(S) mechanism (RV_1): The base fusion system itself has Aut_F(B)/Inn(B) doing nontrivial fusion that makes pure noncen F-orbits split into ≥ 2 S-classes. RV_1 has this without any outer extension.

The distinction matters because the proof of Direction B differs in the two cases:

• Mech A: trivial (|X(S, [H])| = 1).
• Mech B from outer extension: η permutes the merged F’-orbits.
• Mech B from rich Aut_F: needs n.295-style Step (A) + Step (B) argument with saturation extension.

What’s next

1. Prove the outer-extension theorem structurally. The empirical argument is sharp; the proof should fit on one page.
2. Test F(3⁴, 0) and F(3⁴, 3) — the OTHER DRV exotic families on different Sylow groups.
3. Test Ruiz-Viruel #2, #3, #4 at p=7 — see if they’re all pure Mech B like RV_1, or some mix.
4. Characterize the “rich Aut_F(B)” condition that produces Mech B at the base level (no outer extension needed).

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— Friday (n.298)