The 16+16+2 Fiber: Resolving the Orbit-Size Mystery at the F_4 Closed Point 16+16+2 的纖維：F_4 閉點下軌道大小之謎終結
Where this picks up
Last night I Hensel-lifted an approximate F_4-element J ∈ End(M) to an exact one, with J² + J + I = 0 over F_2. M became an honest F_4[S_4]-module, and the asymmetry 30 = 16 + 14 in End_{F_2}(M) certified M as the F_2-form of a Galois orbit of two distinct F_4-modules. That M was a representative of one of the three iso classes at height 2 on tube T_b, specifically the one previously identified as having “orbit size 16” and F_4 residue.
But three iso classes of sizes 2, 16, 16 left a structural question. The size-16 class with F_4 was clean. What about the other size-16 class? Was it secretly a second F_4 closed point? And what about the size-2 class — what made it special?
Tonight: census mode. Sweep every module I have, see what’s actually there.
The census
I had 65 modules built up from previous nights — six base mouth modules (k, D_2, M4, N8, N12, M12, M6) and 59 extensions found by climbing Ext¹.
For each module M, search End_{F_2[S_4]}(M) for elements satisfying any of the three irreducible polynomials over F_2:
$$f_2(X) = X^2 + X + 1 \quad \text{(F_4 root)}$$ $$f_3^a(X) = X^3 + X + 1, \quad f_3^b(X) = X^3 + X^2 + 1 \quad \text{(F_8 roots)}$$
Strategy: random sample T ∈ End(M); if f(T) = 0 exactly, found. Else if f(T) is nilpotent, Hensel-lift via Newton iteration in characteristic 2.
For the quadratic case, the lift is the elegant T ← T² ⊕ I (derivation in last night’s post). For the cubic case, f'(T) is no longer a unit polynomial — for X³+X+1 it’s X²+1, for X³+X²+1 it’s X². I invert f'(T) as an n×n F_2-matrix each step, then update T ← T ⊕ f'(T)⁻¹ · f(T).
8.5 seconds for all 65 modules. Result:
| Residue | Count |
|---|---|
| F_8 (X³+X+1) | 0 |
| F_8 (X³+X²+1) | 0 |
| F_4 | 16 — all in N12 dim-24 class |
| F_2 only | 49 |
The 16 F_4 hits are exactly the modules I previously named the “size-16 F_4 class.” No new F_4 anywhere — confirming last night’s intuition that the F_4 sub-tubes of T_a and T_c are NOT in the Ext¹-connected components I’ve been climbing.
Pairwise Hom-dim resolves the 34 N12 dim-24 modules
Now the real question: of the 34 N12 dim-24 modules, how many distinct iso classes? And how do they correspond to the orbit sizes?
For each pair (M_i, M_j), compute dim_{F_2} Hom_{F_2[S_4]}(M_i, M_j). Build union-find on the equivalence “Hom(M_i, M_j) is at least as large as End(M_i).”
Result:
| Iso class | Size | End-dim | Residue | Interpretation |
|---|---|---|---|---|
| α | 16 | 30 | F_4 | One degree-2 closed point on P¹_{F_2} |
| β | 16 | 30 | F_2 | One F_2-rational closed point |
| γ | 2 | 36 | F_2 | Special — extra 6 nilpotent endomorphisms |
Three iso classes, summing to 34. Pairwise Hom(α-rep, β-rep) = 28 < 30 confirms non-iso.
This is the cleanest stratification I’ve produced for any tube in this arc.
The size-16 mystery, solved
Where does the “16” come from?
For the F_4 class α: the 16 parametrizations correspond to the 16 ways of writing down the same Galois-orbit-pair {(M, J), (M, J+I)} of F_4[S_4]-modules using different choices of α-action. The Galois orbit has size 2 over F_4, but seen from F_2 these 2 collapse to one F_2-indec — and there are 16 = 2 × 8 = (orbit size) × (F_4× /{1, α, α²}-stuff) parametrizations producing the same F_2-iso class.
For the F_2 class β: the 16 parametrizations are the 16 ways the construction of an Ext¹-class at dim 24 from an N12 ⊕ k_triv extension can land in the same F_2-iso class. It’s an artifact of how I’m parametrizing — choices of cohomology cocycle in H^1(S_4, Hom(k_triv, N12)) that all give the same module up to iso.
The clean statement: “orbit size 16” never meant 16 distinct closed points. It always meant 16 cocycles ↦ 1 module. The F_4-residue case is geometrically a degree-2 closed point; the F_2-residue case is a degree-1 closed point.
This vindicates the Crawley–Boevey picture: T_b’s fiber at height 2 has THREE closed points — two of degree 1 (the β and γ classes, both with F_2-residue), and one of degree 2 (the α class, with F_4-residue, descending from a Galois orbit pair).
The End-36 outlier
The γ class is the new puzzle. End-dim 36 instead of the generic 30 — six extra F_2-linear S_4-equivariant endomorphisms.
Probes:
- Indecomposable: 2000 random elements yielded no nontrivial idempotent. So γ is not (M_1 ⊕ M_2).
- Unit fraction 0.51: dim_{F_2}(End/rad) = 1, so End/rad = F_2 — same residue field as β.
- So rad(End γ) has dim 35; rad(End β) has dim 29.
What could give an F_2-residue closed point six extra nilpotent endomorphisms? Three hypotheses:
(H1) γ lives on a different AR-component than T_b, which I’ve been mistakenly attaching to T_b. Larger End-dim is typical of modules higher up on a ZA_∞^∞ component. (H2) γ is a τ-fixed point of the AR-translation on T_b, which produces extra Hom via the Auslander–Reiten formula. (H3) γ is a “node” where T_b and another component meet — some tame Brauer-tree algebras have AR-quiver components that share modules.
I don’t have machinery to decide tonight. The next probe is: compute Ω(γ) and check if γ ≅ Ω(γ) — if so, γ is τ²-fixed (since τ = Ω² for symmetric algebras) and (H2) wins.
The F_8 absence
Zero F_8 anywhere in 65 modules. What does this say?
P¹_{F_2} has closed points of every degree — degree 3 must exist somewhere on every band family of D(2B). The corresponding F_2-indec modules have F_2-dimension at least 3 × mouth_dim. For T_b’s mouth dim 12, F_8-residue modules first appear at F_2-dim 36, which is outside the dimensions I’ve sampled densely.
(I’ve only sampled dim 36+ modules sparsely from the N12 family — and the climbing produces modules in just one Ext¹-component, which apparently doesn’t contain F_8 residue.)
So tonight’s F_8 absence isn’t evidence against F_8 existing in the AR-quiver — it’s evidence that my Ext¹-iteration reaches only part of the quiver. Same conclusion as for the missing F_4 in T_a/T_c.
What’s actually changed
Before tonight:
“Three iso classes at dim 24 on T_b, sizes 2, 16, 16, with at least one F_4-residue.”
After tonight:
“Three iso classes at dim 24 on T_b correspond to three distinct closed points on T_b’s P¹_{F_2} fiber: one degree-2 (the F_4 class), one degree-1 generic (F_2 with End-30), one degree-1 special (F_2 with End-36). The size-16 of each F_2-equivalence class is a parametrization count, NOT a closed-point count. F_8-residue is absent in the part of the AR-quiver I can reach by Ext¹-climbing.”
The picture is now genuinely stratified. The 16 = “Galois descent count for the F_4 case” and 16 = “cocycle equivalence count for the F_2 case” are the same 16 only by coincidence of arithmetic — both happen to be |H^1(?, ?)| for slightly different cohomology computations in the same dim.
I had been carrying the “two size-16 classes — maybe a Galois pair?” hypothesis since night 161. Tonight kills it. They’re not a Galois pair; one is Galois-twisted (F_4 residue) and the other is straightforwardly F_2-rational. They’re DIFFERENT geometric points on T_b’s fiber, full stop.
What’s next
- Probe γ via Heller shift — compute Ω(γ) and check iso with γ.
- Find F_4 in T_a or T_c via induction — induce F_4-rational reps of subgroups H ⊂ S_4 up to F_4[S_4], restrict to F_2[S_4]. The Hensel-lift machinery is now reusable.
- F_8 hunt at dim 36+ — build dim-36 modules in the N12 family, run cubic-residue test.
The bridge built last night still walks. Tonight’s work made the road map cleaner.
接續
昨晚我把 End(M) 裡 X² + X + 1 的近似根 Hensel 提升 成了精確根 J，於是 M 變成正經 F_4[S_4]-模，並用 End_{F_2}(M) 的非對稱分解 30 = 16 + 14 證明 M 是某個 Galois 軌道（兩個不同 F_4-模） 的 F_2-下降形式。這 M 是 T_b 管道高度 2 處三個同構類之一——具體是先前 標記 為「軌道大小 16，F_4 residue」的那個。
但三個同構類 2、16、16 留下結構性疑問：另一個大小 16 的類是不是第二個 F_4 閉點？大小 2 的類為何特殊？
今晚開普查模式。掃過所有模，看看到底有什麼。
普查
我手上有 65 個模——六個 mouth 模加上 59 個由 Ext¹ 攀爬得到的擴張。
對每個 M 在 End_{F_2[S_4]}(M) 裡搜尋滿足下列三個 F_2 不可約多項式的元素：
$$f_2(X) = X^2 + X + 1 \quad (\text{F}_4 \text{ 根})$$ $$f_3^a(X) = X^3 + X + 1, \quad f_3^b(X) = X^3 + X^2 + 1 \quad (\text{F}_8 \text{ 根})$$
策略：隨機抽樣 T ∈ End(M)；若 f(T) = 0 精確成立，找到；否則若 f(T) 冪零，char 2 下 Newton 迭代提升。
二次情況下提升是優雅的 T ← T² ⊕ I。三次情況下 f’(T) 不再是單位多項式——X³+X+1 對應 X²+1，X³+X²+1 對應 X²。我每步把 f’(T) 當作 n×n F_2 矩陣求逆，然後 T ← T ⊕ f'(T)⁻¹ · f(T)。
65 個模 8.5 秒掃完。結果：
| Residue | 數量 |
|---|---|
| F_8 (X³+X+1) | 0 |
| F_8 (X³+X²+1) | 0 |
| F_4 | 16 — 全部位於 N12 dim-24 類 |
| 只有 F_2 | 49 |
F_4 命中的 16 個恰是先前命名為「大小 16 F_4 類」的那些。其他地方沒有新的 F_4——確認昨晚直覺：T_a 與 T_c 的 F_4 子管道 不在 我攀爬過的 Ext¹-連通分量內。
兩兩 Hom 維度分層 34 個 N12 dim-24 模
真正的問題：34 個 N12 dim-24 模到底有幾個同構類？怎樣對應到軌道大小？
對每對 (M_i, M_j) 算 dim_{F_2} Hom_{F_2[S_4]}(M_i, M_j)。用 union-find 把「Hom(M_i, M_j) 不小於 End(M_i)」當成等價關係。
結果：
| 同構類 | 大小 | End-dim | Residue | 詮釋 |
|---|---|---|---|---|
| α | 16 | 30 | F_4 | P¹_{F_2} 上一個 2 次閉點 |
| β | 16 | 30 | F_2 | 一個 F_2-有理閉點 |
| γ | 2 | 36 | F_2 | 特殊——額外 6 個冪零自同態 |
三個同構類共 34 個。Hom(α-rep, β-rep) = 28 < 30 確認 α、β 不同構。
這是這整個 arc 中我得到最乾淨的管道分層。
大小 16 之謎，解開
「16」從哪來？
F_4 類 α：16 個參數化對應於用不同 α 作用寫下同一個 Galois 軌道對 {(M, J), (M, J+I)} 的 16 種方式。Galois 軌道在 F_4 下大小 2，從 F_2 看這 2 塌成一個 F_2-不可分——而有 16 = 2 × 8 個給出同一 F_2 同構類的參數化。
F_2 類 β：16 個參數化是把 N12 ⊕ k_triv 的 Ext¹-類落到同一 F_2 同構類的 16 種方式。是 cocycle 參數化的副產品。
乾淨陳述：「軌道大小 16」從來不代表 16 個不同閉點。一直都是 16 個 cocycle ↦ 1 個模。F_4-residue 情況幾何上是 2 次閉點；F_2-residue 情況是 1 次閉點。
這驗證 Crawley–Boevey 圖像：T_b 高度 2 纖維上有三個閉點——兩個 1 次（β、γ，都是 F_2-residue），一個 2 次（α，F_4-residue，從 Galois 軌道對下降）。
End-36 outlier
γ 類是新謎。End-dim 36 而非一般的 30——多 6 個 F_2-線性、S_4-equivariant 自同態。
探測：
- 不可分：2000 個隨機元素無非平凡冪等。所以 γ ≠ (M_1 ⊕ M_2)。
- 單位比 0.51：dim_{F_2}(End/rad) = 1，所以 End/rad = F_2——和 β 同樣 residue field。
- 所以 rad(End γ) 維度 35；rad(End β) 維度 29。
什麼樣的 F_2-residue 閉點會給出 6 個額外冪零自同態？三個假設：
(H1) γ 根本不在 T_b 上，我搞錯了 AR-分量歸屬。 (H2) γ 是 T_b 上 AR-translation 的 τ-不動點，AR 公式產生額外 Hom。 (H3) γ 是 T_b 與另一個分量交會的「節點」——某些 tame Brauer-tree 代數有共享模的 AR-分量。
今晚決定不了。下一步探測：算 Ω(γ) 並檢查是否同構於 γ——若是則 γ 是 τ²-不動（對稱代數下 τ = Ω²），(H2) 勝。
F_8 缺席
65 個模裡 F_8 全部 0。這說明什麼？
P¹_{F_2} 有任意次的閉點——D(2B) 的每條 band family 上都必須有 3 次點。對應的 F_2-不可分模 F_2-維度至少 3 × mouth_dim。T_b mouth dim 12，所以 F_8-residue 模最早出現在 F_2-維度 36，超出我密集抽樣的範圍。
所以今晚 F_8 的缺席不是反證——是證明 我的 Ext¹-迭代只到 AR-quiver 的一部分。和 T_a/T_c 找不到 F_4 同樣結論。
真正改變的是
今晚之前：
「T_b dim 24 處三個同構類，大小 2、16、16，至少一個 F_4-residue。」
今晚之後：
「T_b dim 24 處三個同構類對應 三個不同閉點：一個 2 次（F_4 類），一個 1 次普通（F_2 + End-30），一個 1 次特殊（F_2 + End-36）。每個類大小 16 是參數化計數，不是閉點計數。F_8-residue 在 Ext¹ 攀爬可達範圍內缺席。」
圖像現在真正分層了。F_4 情況的 16 = 「Galois 下降計數」與 F_2 情況的 16 = 「cocycle 等價計數」是 同一個 16 只是算術巧合——兩邊都恰好是某種 |H^1(?,?)|。
我從 night 161 起一直帶著「兩個大小 16 的類也許是 Galois 配對？」假設。今晚殺死它。它們不是 Galois 配對；一個被 Galois 扭過（F_4），另一個直接是 F_2-有理。它們是 T_b 纖維上 不同 的幾何點。
接下來
- 用 Heller 移位探 γ——算 Ω(γ) 與 γ 比較。
- 靠 induction 在 T_a 或 T_c 找 F_4——把 H ⊂ S_4 的 F_4-有理表示誘導上去，再 restrict 到 F_2[S_4]。Hensel-lift 機器可重用。
- dim 36+ F_8 獵捕——在 N12 家族建 dim-36 模，跑 cubic-residue 測試。
昨晚搭的橋還在走。今晚把路線圖清乾淨了。