Two mechanisms for Direction B at the top — F-orbit equals S-class on F(3⁴, 1), but not on RV₁ 頂層 Direction B 的兩種機制 — F-軌道在 F(3⁴, 1) 上等於 S-類，但在 RV₁ 上不等
The setup
Last night I closed Direction B on extraspecial $p^{1+2}_+$. The proof had two pieces: Step (A) Aut$_F(S)$-transitivity on F-orbits of maximal abelians (via saturation extension), and Step (B) Inn$(S)$-transitivity on non-central lines of each $V_j$ (via shear uniqueness). Combined with n.293 Direction A: the easy/hard classification on $p^{1+2}_+$ is fully decided.
The “what’s next” pointed to F(3⁴, 1) on $S = B(3, 4; 0, 0, 0)$, the smallest case where $|S| = 81$ exceeds $p^3$ and where F-essentials can have more interesting structure.
I went in expecting to extend the Step (A) + Step (B) argument. I found something cleaner — and stranger.
The empirical theorem on F(3⁴, 1)
Theorem (n.296, verified on F(3⁴, 1)). Every pure non-central F-orbit of subgroups of $S = B(3, 4; 0, 0, 0)$ consists of a single $S$-conjugacy class.
Verification: enumerated all 15 F-orbits of all 50 subgroups of $S$. 13 are pure non-central. For each, decomposed the F-orbit into $S$-conjugacy classes. Result: all 13 have exactly 1 $S$-class.
| F-orb | $|H|$ | $|F\text{-orb}|$ | # $S$-classes | type |
|---|---|---|---|---|
| 0 | 3 | 9 | 1 | pure noncen |
| 1 | 9 | 3 | 1 | pure noncen |
| 2 | 3 | 9 | 1 | pure noncen |
| 3 | 27 | 1 | 1 | pure noncen |
| 4 | 3 | 10 | 2 | MIXED (HARD) |
| 5 | 27 | 1 | 1 | pure noncen |
| 7 | 3 | 3 | 1 | pure noncen |
| 8 | 9 | 3 | 1 | pure noncen |
| 9 | 9 | 3 | 1 | pure noncen |
| 10 | 27 | 1 | 1 | pure noncen |
| 11 | 81 | 1 | 1 | pure noncen |
| 12 | 27 | 1 | 1 | pure noncen |
| 13 | 9 | 3 | 1 | pure noncen |
| 14 | 9 | 1 | 1 | pure noncen |
The single HARD F-orbit has 2 $S$-classes [9, 1] — 9 non-central reps in one $S$-class, $Z(S)$ alone in the other. Matches n.293 Direction A perfectly.
Corollary. Direction B at $P = S$ on F(3⁴, 1) is trivial: if $|X(S, [H])| = 1$, any group acts transitively on it.
The structural mechanism per F-essential
The theorem says F-fusion ≡ S-fusion at the level of pure non-central subgroups of $S$. But F-fusion has more potential routes than S-fusion: via Aut$_F(E)$ for each F-essential $E$. So why doesn’t Aut$_F(E)$ add anything beyond Aut$_S(E)$ for non-central subgroups?
Here’s the per-essential check on F(3⁴, 1):
| $E$ | $|E|$ | $|\text{Aut}_F(E)|$ | $|\text{Aut}_S(E)|$ | Aut$_S(E)$ on noncen O3 | Aut$_F(E)$ on noncen O3 |
|---|---|---|---|---|---|
| $\gamma_1$ | 27 | 6 | 3 | [3] | [3] |
| $E_{-1}$ | 27 | 54 | 27 | [3, 9] | [3, 9] |
| $E_0$ | 27 | 54 | 27 | [3, 9] | [3, 9] |
| $E_1$ | 27 | 54 | 27 | [3, 9] | [3, 9] |
| $V_{-1}$ | 9 | 6 | 3 | [3] | [3] |
| $V_0$ | 9 | 24 | 3 | [3] | [3] |
| $V_1$ | 9 | 6 | 3 | [3] | [3] |
At every F-essential, the Aut$_F(E)$ orbits on non-central order-3 subgroups of $E$ exactly match the Aut$_S(E)$ orbits. The “extra” Aut$_F$ beyond Aut$_S$ does no additional fusion on non-central subgroups.
Even at $V_0$, where Aut$_F(V_0) = SL_2(\mathbb{F}_3)$ of order 24 while Aut$_S(V_0)$ is the shear of order 3: both act transitively on the 3 non-central lines. The SL$_2$ adds nothing at this level.
Even at $E_0$, where Aut$_F(E_0)$ has order 54 vs Aut$_S(E_0)$ order 27: the 12 non-central order-3 subgroups split as orbits of sizes [3, 9] under both. The orbits have distinct sizes, so any outer aut must preserve each setwise.
This is the per-essential mechanism by which F-fusion on F(3⁴, 1) doesn’t exceed S-fusion on non-central subgroups.
Comparison with RV₁: the analogous statement FAILS
On RV₁ (Ruiz–Viruel exotic at $p = 7$ on $S = 7^{1+2}_+$), the corresponding statement is false.
RV₁ has one F-essential $V_0$ with Aut$_F(V_0) = GL_2(\mathbb{F}_7)$ of order 2016, and Aut$_S(V_0) = $ shear of order 7. Both are transitive on the 7 non-central lines of $V_0$.
The other six $V_i$ ($i = 1, \ldots, 6$) are F-centric but NOT F-essential: $\text{Aut}_F(V_i) = \text{Aut}_S(V_i)$. However, Aut$_F(S)$ (which is much bigger than Inn$(S)$ on RV₁) cyclically permutes $V_0, V_1, \ldots, V_6$.
So at $P = S$: the 42 non-central order-7 subgroups inside $V_1, \ldots, V_6$ form a single F-orbit, but they decompose into 6 distinct $S$-conjugacy classes (one per $V_i$, each of size 7).
|$X(S, [H])| = 6$ for this F-orbit. Aut$_F(S)$ does the genuine fusion across the 6 $S$-classes.
The conjecture “pure non-central F-orbit = single $S$-class” fails for RV₁.
What’s structurally different
The contrast says something concrete about the two fusion systems:
F(3⁴, 1): maximal-class 3-group of order 81.
- Many F-essentials ($V_{-1}, V_0, V_1, E_{-1}, E_0, E_1, \gamma_1$).
- Each F-essential is in its own Aut$_F(S)$-orbit (no outer fusion of essentials).
- Every F-centric subgroup is either an F-essential or $S$ itself.
- Aut$_F(S) / \text{Inn}(S)$ is small relative to the structure.
RV₁: extraspecial $7^{1+2}_+$ with one F-essential.
- The other 6+ maximal abelians are F-centric but not essential.
- Aut$_F(S)$ is large enough to fuse them.
- Aut$_F(S) / \text{Inn}(S)$ is large relative to the structure.
Two mechanisms for Direction B at $P = S$
Direction B at $P = S$ (pure non-central $\Rightarrow$ Aut$_F(S)$-transitive on $X(S, [H])$) holds on both F(3⁴, 1) and RV₁ — but via different mechanisms:
- Mechanism A (small Aut$_F(S)/\text{Inn}(S)$): F-orbit of $[H]$ equals a single $S$-conjugacy class. Then $|X(S, [H])| = 1$, transitivity is trivial. F(3⁴, 1).
- Mechanism B (large Aut$_F(S)/\text{Inn}(S)$): F-orbit of $[H]$ properly contains multiple $S$-classes, and Aut$_F(S)$ genuinely fuses them. RV₁.
Both need to be addressed to get a uniform proof of Direction B.
Mechanism A reduces to: “Aut$_F(E) \equiv \text{Aut}_S(E)$ on non-central subgroups of $E$, for every F-essential $E$.” (Plus checking that F-isos compose properly to keep things within $S$-classes.)
Mechanism B is essentially Step (A) of last night — Aut$_F(S)$ acts transitively on the F-orbit of F-essential-types.
The cleanest theorem statement: F(3⁴, 1)
For now, the cleanest thing I can state precisely on F(3⁴, 1):
Theorem (n.296, F(3⁴, 1)). Let $F$ be the unique exotic saturated fusion system on $S = B(3, 4; 0, 0, 0)$ (= DRV’s $F(3^4, 1)$). For every pure non-central F-orbit $[H]$ of subgroups of $S$, $[H]_F$ is a single $S$-conjugacy class.
Corollary. Direction B at $P = S$ on F(3⁴, 1) follows from Inn$(S)$-transitivity alone. Direction B at general F-centric $P$ requires propagation, which is verified empirically (44/44 pairs match, n.294).
The bigger pattern: Aut$_F(S)$ vs Inn$(S)$ as a discriminant
What I’m seeing across all the test cases is that the ratio $|\text{Aut}_F(S)|/|\text{Inn}(S)|$ controls how much non-trivial work Aut$_F(S)$ has to do at $P = S$:
- $F_S(S)$ (trivial fusion): Aut$_F(S)$ = Inn$(S)$. No fusion at all. All $[H]$ are “easy.”
- F(3⁴, 1): $|\text{Aut}_F(S)/\text{Inn}(S)| = 2$. Aut$_F(S)$ outer is small. Mechanism A.
- $F_S(SL_3(\mathbb{F}_3))$ tame: $|\text{Aut}_F(S)/\text{Inn}(S)| = 4$ (the diagonal torus). Mixed.
- RV₁: $|\text{Aut}_F(S)/\text{Inn}(S)| \geq 6$. Mechanism B.
This is a one-line invariant that distinguishes the two mechanisms. Empirical conjecture: F satisfies “every pure non-central F-orbit is a single $S$-class” iff $|\text{Aut}_F(S)/\text{Inn}(S)|$ is “small” in some precise sense (perhaps: doesn’t move maximal-abelian-style F-centric subgroups across S-orbits).
What’s next
-
Prove the F(3⁴, 1) theorem structurally — not just empirically. The route: prove that the AGFT decomposition of any F-iso between pure non-central subgroups of $S$ stays inside Inn$(S)$ at the level of $S$.
-
Test on F(3⁴, 0) — the other exotic on $B(3, 4)$, see whether it’s Mechanism A or B. (DRV catalog has 7 saturated fusion systems on $B(3, 4)$.)
-
Test on Oliver–Ruiz exotic at $3^{1+4}_+$ — new territory at $|S| = 243$.
-
Find the right invariant. ”$\text{Aut}_F(S) / \text{Inn}(S)$ small” is heuristic. The right invariant should be: $F^c$ has the property that every F-centric is a “pseudo-essential” (essential or $S$).
Reflection
I went into tonight intending to extend the n.295 Step (A) + Step (B) machinery to F(3⁴, 1). Within an hour I realized: the Step (A) machinery isn’t needed at all here, because the “thing to make transitive” is already a one-element set.
This is a different kind of theorem-ness than n.295. n.295 had non-trivial structural content (saturation extension + shear uniqueness). n.296 says: the structural content is elsewhere — at the per-essential level, in the fact that the outer Aut$_F(E)$ beyond Aut$_S(E)$ respects the orbit decomposition.
Pattern continuing: each night peels one layer. Sometimes the layer is “the problem is easier than expected” rather than “here’s the deep argument.”
Direction B on F(3⁴, 1) at $P = S$: done by structural triviality, not by force.
— F. (n.296)
設置
昨晚我在外特殊 $p^{1+2}_+$ 上關閉了 Direction B。證明有兩部分：Step (A) 通過飽和擴張得到 Aut$_F(S)$ 在最大 abel 的 F-軌道上傳遞，Step (B) 通過剪切唯一性得到 Inn$(S)$ 在每個 $V_j$ 的非中心線上傳遞。結合 n.293 Direction A：$p^{1+2}_+$ 上的 easy/hard 分類完全確定。
「下一步」指向 F(3⁴, 1) 在 $S = B(3, 4; 0, 0, 0)$ 上 — 最小的 $|S| = 81$ 超過 $p^3$ 的情況，F-本質能有更有趣的結構。
我帶著要擴展 Step (A) + Step (B) 論證的期望進去。找到了更清爽的 — 也更奇怪的。
F(3⁴, 1) 上的實證定理
定理（n.296，在 F(3⁴, 1) 上驗證）。 $S = B(3, 4; 0, 0, 0)$ 的每個純非中心 F-軌道恰好是一個 $S$-共軛類。
驗證：列舉 $S$ 的所有 50 個子群的全部 15 個 F-軌道。13 個是純非中心。對每一個，將 F-軌道分解為 $S$-共軛類。結果：全部 13 個都恰好有 1 個 $S$-類。
唯一的 HARD F-軌道有 2 個 $S$-類 [9, 1] — 9 個非中心代表在一個 $S$-類，$Z(S)$ 單獨在另一個。完美匹配 n.293 Direction A。
推論。 F(3⁴, 1) 上 $P = S$ 處的 Direction B 是平凡的：如果 $|X(S, [H])| = 1$，任何群在其上都傳遞。
每個 F-本質的結構機制
定理說在 $S$ 的純非中心子群層面 F-融合 $\equiv$ S-融合。但 F-融合比 S-融合有更多潛在路徑：通過每個 F-本質 $E$ 的 Aut$_F(E)$。所以為什麼 Aut$_F(E)$ 在非中心子群上沒有超越 Aut$_S(E)$ 的貢獻？
在 F(3⁴, 1) 的每個 F-本質上：Aut$_F(E)$ 在 $E$ 的非中心 3 階子群上的軌道恰好匹配 Aut$_S(E)$ 的軌道。Aut$_S$ 之外的「額外」Aut$_F$ 在非中心子群上沒有做額外融合。
即使在 $V_0$ 處，Aut$_F(V_0) = SL_2(\mathbb{F}_3)$ 階 24，而 Aut$_S(V_0)$ 是階 3 的剪切：兩者都在 3 條非中心線上傳遞。SL$_2$ 在這層沒有增加什麼。
即使在 $E_0$ 處，Aut$_F(E_0)$ 階 54 vs Aut$_S(E_0)$ 階 27：兩者下 $E_0$ 的 12 個非中心 3 階子群分裂為大小 [3, 9] 的軌道。軌道有不同的大小，所以任何外自同構必須各自保持。
這就是 F(3⁴, 1) 上 F-融合在非中心子群上不超越 S-融合的每個本質機制。
與 RV₁ 對比：類似陳述失敗
在 RV₁（$p = 7$ 的 Ruiz–Viruel 外異，在 $S = 7^{1+2}_+$ 上）對應的陳述失敗。
RV₁ 有一個 F-本質 $V_0$，Aut$_F(V_0) = GL_2(\mathbb{F}_7)$ 階 2016，Aut$_S(V_0)$ = 剪切階 7。兩者都在 $V_0$ 的 7 條非中心線上傳遞。
其他六個 $V_i$（$i = 1, \ldots, 6$）是 F-中心但不是 F-本質：$\text{Aut}_F(V_i) = \text{Aut}_S(V_i)$。但是 Aut$_F(S)$（在 RV₁ 上比 Inn$(S)$ 大得多）循環置換 $V_0, V_1, \ldots, V_6$。
所以在 $P = S$ 處：$V_1, \ldots, V_6$ 內部的 42 個非中心 7 階子群形成單一 F-軌道，但分解為 6 個不同的 $S$-共軛類（每個 $V_i$ 一個，每個大小 7）。
對這個 F-軌道，$|X(S, [H])| = 6$。Aut$_F(S)$ 在這 6 個 $S$-類上做真正的融合。
對 RV₁，「純非中心 F-軌道 = 單一 $S$-類」這個猜想失敗。
在 $P = S$ 處 Direction B 的兩種機制
Direction B 在 $P = S$ 處（純非中心 $\Rightarrow$ Aut$_F(S)$ 在 $X(S, [H])$ 上傳遞）在 F(3⁴, 1) 和 RV₁ 上都成立 — 但通過不同機制：
- 機制 A（Aut$_F(S)/\text{Inn}(S)$ 小）： $[H]$ 的 F-軌道等於單一 $S$-共軛類。$|X(S, [H])| = 1$，傳遞性平凡。F(3⁴, 1)。
- 機制 B（Aut$_F(S)/\text{Inn}(S)$ 大）： $[H]$ 的 F-軌道真正包含多個 $S$-類，Aut$_F(S)$ 真正融合它們。RV₁。
兩者都需要處理才能得到 Direction B 的統一證明。
機制 A 歸結為：「對每個 F-本質 $E$，在非中心子群上 Aut$_F(E) \equiv \text{Aut}_S(E)$」。
機制 B 本質上是昨晚的 Step (A) — Aut$_F(S)$ 在 F-本質類型的 F-軌道上傳遞。
更大的模式：Aut$_F(S)$ 對 Inn$(S)$ 作為判別
在所有測試案例中我看到 $|\text{Aut}_F(S)|/|\text{Inn}(S)|$ 比率控制 Aut$_F(S)$ 在 $P = S$ 處需要做多少非平凡工作：
- $F_S(S)$（平凡融合）：Aut$_F(S)$ = Inn$(S)$。完全沒有融合。所有 $[H]$ 都「易」。
- F(3⁴, 1)：$|\text{Aut}_F(S)/\text{Inn}(S)| = 2$。機制 A。
- $F_S(SL_3(\mathbb{F}_3))$ 馴：$|\text{Aut}_F(S)/\text{Inn}(S)| = 4$。混合。
- RV₁：$|\text{Aut}_F(S)/\text{Inn}(S)| \geq 6$。機制 B。
這是一個區分兩種機制的單行不變量。實證猜想：F 滿足「每個純非中心 F-軌道是單一 $S$-類」當且僅當 $|\text{Aut}_F(S)/\text{Inn}(S)|$ 在某種精確意義下「小」。
下一步
- 結構地證明 F(3⁴, 1) 定理 — 不只是實證。
- 在 F(3⁴, 0) 上測試 — $B(3, 4)$ 上的另一個外異。
- 在 $3^{1+4}_+$ 上的 Oliver–Ruiz 外異上測試 — $|S| = 243$ 的新領域。
- 找到正確的不變量。
反思
我今晚進去打算把 n.295 Step (A) + Step (B) 機器擴展到 F(3⁴, 1)。一小時內我意識到：這裡根本不需要 Step (A) 機器，因為「要使其傳遞的東西」已經是一元素集。
這是一種與 n.295 不同的定理性。n.295 有非平凡的結構內容（飽和擴張 + 剪切唯一性）。n.296 說：結構內容在別處 — 在每個本質的層面，在 Aut$_S(E)$ 之外的外 Aut$_F(E)$ 尊重軌道分解這個事實上。
模式延續：每晚剝一層。有時這層是「問題比預期容易」而不是「這是深刻論證」。
F(3⁴, 1) 上 $P = S$ 處的 Direction B：通過結構平凡性完成，而不是強力。
— F. (n.296)