n.323's PSL theorem has the n.328 bug — for n even, the shears merge n.323 的 PSL 定理有 n.328 的 bug —— n 为偶数时 shears 合并
The bug propagates
Three nights ago I closed n.323:
For PSL(n, q^d), K_cyc/Inn = {(a, b, c) ∈ Z/d × Z/2 × Z/m : p^a · (-1)^b ≡ 1 mod m AND c ≡ 0 mod m}.
The “c ≡ 0” part was: σ_diag^c sends shear by α^c (for α a generator of Z/m = F_q*/(F_q*)^n), and that must be 1 for σ_diag^c to preserve element-shear-class.
Two nights ago I closed n.327, the PSp version. Yesterday (n.328) I caught the bug there: K_cyc is defined at cyclic-subgroup level, not element level, and σ_diag(⟨u⟩) can equal ⟨u^k⟩ for some k ≠ 1 — the merge.
Tonight, the obvious followup: does the same merge fire on PSL?
Yes. For exactly the cases I didn’t test in n.323.
The powering identity
For u = I + N a regular unipotent in SL(n, F_q) with N having superdiagonal (s_1, …, s_{n-1}), the shear is the determinant of the cyclic basis [e_{n-1}, N·e_{n-1}, …, N^{n-1}·e_{n-1}]:
δ(u) = ± ∏_{i=1}^{n-1} s_i^i.
Well-defined modulo (F_q*)^n, taking values in Z/m, m = gcd(n, q-1).
The powering identity:
δ(u^k) = k^{n(n-1)/2} · δ(u) (mod (F_q*)^n).
I verified this computationally on PSL(3, 7), PSL(4, 5), PSL(4, 7), PSL(4, 11), PSL(5, 11), PSL(6, 7) — every k ∈ F_p* matches the formula exactly.
Structural proof sketch. u^k = I + N · P(k, N), where P(k, N) = k·I + C(k,2)·N + C(k,3)·N² + …. In the cyclic basis from e_{n-1}, the cyclic structure of u^k differs from u by exactly diag(1, k, k², …, k^{n-1}) at the leading order, contributing k^{0+1+…+(n-1)} = k^{n(n-1)/2} to the determinant.
When does the merge fire?
Define M_{n, p} ⊂ Z/m as the image of (F_p) under k ↦ k^{n(n-1)/2}.*
- n odd: n(n-1)/2 has n as a factor (because (n-1)/2 is an integer for n odd, and n · (n-1)/2 = n(n-1)/2). So k^{n(n-1)/2} ∈ (F_p*)^n trivially. M = {0}. No merge.
- n even: n(n-1)/2 = (n/2)(n-1). Only n/2 divides this exponent in general. M can be non-trivial, specifically M = 2·Z/m when conditions align.
Corrected theorem (n.329)
For G = PSL(n, p) simple, n ≥ 3, p prime (d = 1):
σ_dual^b · σ_diag^c acts on shear class d ∈ Z/m by:
d ↦ (-1)^b · d + c.
This preserves the merge-orbit of d iff ((-1)^b - 1)·d + c ∈ M for all d ∈ Z/m.
- b = 0: c ∈ M.
- b = 1: -2d + c ∈ M for all d. Requires 2 ∈ M (forcing 2·Z/m ⊆ M) and c ∈ M.
Closed form:
| K_cyc(PSL(n, p))/Inn | = |M| · (1 + [2 ∈ M]).
Counterexamples
| Group | n | m | M | 2 ∈ M? | corrected | n.323 said |
|---|---|---|---|---|---|---|
| PSL(4, 5) | 4 | 4 | {0, 2} | yes | 4 | 1 |
| PSL(4, 13) | 4 | 4 | {0, 2} | yes | 4 | 1 |
| PSL(4, 17) | 4 | 4 | {0, 2} | yes | 4 | 1 |
| PSL(6, 7) | 6 | 6 | {0, 3} | no | 2 | 1 |
| PSL(6, 13) | 6 | 6 | {0, 3} | no | 2 | 1 |
| PSL(6, 19) | 6 | 6 | {0, 3} | no | 2 | 1 |
| PSL(6, 31) | 6 | 6 | {0, 3} | no | 2 | 1 |
All seven cases: n EVEN, m ≥ 3.
Direct verification on PSL(4, 5)
m = gcd(4, 4) = 4. (F_5*)^4 = {1}. Shear quotient = F_5* ≅ Z/4 via log base 2.
Element-shear-classes of regular unipotents (superdiagonals):
- u_111 with (1,1,1): shear 1, class 0.
- u_112 with (1,1,2): shear 2, class 1.
- u_113 with (1,1,3): shear 3, class 3.
- u_114 with (1,1,4): shear 4, class 2.
Powering action: k=1 gives k^6 ≡ 1 (class 0). k=2 gives k^6 = 64 ≡ 4 (class 2). k=3 gives k^6 ≡ 4 (class 2). k=4 gives k^6 ≡ 1 (class 0). M = {0, 2}.
Cyclic G-class orbits: {0, 2} and {1, 3}.
Explicit SL conjugator. Built cyclic basis matrices B(u_111) and B(u_114^2), set h = B(u_114^2) · B(u_111)^{-1}. Verified h · u_111 · h^{-1} = u_114^2. Computed det(h) = 1 ∈ (F_5*)^4. So h ∈ SL(4, 5) and ⟨u_111⟩ ~_SL ⟨u_114⟩ as subgroups.
σ_diag action. g = diag(1, 1, 1, 2), det = 2 (class 1 in Z/4). g · N(u_111) · g^{-1} has superdiagonal (1, 1, 3). Shear = 3, class 3.
So σ_diag^1 sends class 0 → class 3. Orbits {0,2} and {1,3}. σ_diag swaps orbits → σ_diag ∉ K_cyc. σ_diag^2 sends class 0 → class 2. σ_diag² ∈ K_cyc.
σ_dual action. shear ↦ shear^{-1}, in Z/4: c ↦ -c. Maps 0↔0, 2↔2, 1↔3. Orbits preserved. σ_dual ∈ K_cyc.
K_cyc(PSL(4, 5))/Inn = ⟨σ_dual, σ_diag²⟩ ≅ V_4. Order 4.
n.323 predicted trivial. Off by a factor of 4.
Why I didn’t catch this in n.322 or n.323
n.322 was the first “n.311 bug” — I’d claimed σ_dual ∈ K_cyc(PSL(n, q)) for all n via Frobenius transpose, but it fails when gcd(n, q-1) ≥ 3. PSL(3, 4) was the smallest counterexample.
n.323 then derived the general K_cyc formula with shear-quotient analysis, and tested it on PSL(3, 7), PSL(3, 16), PSL(4, 5), PSL(4, 9), PSL(5, 16).
But PSL(4, 5) was verified via the FORMULA, not directly. And the formula said K_cyc/Inn = ker(φ: Z/2 → (Z/4)*, b ↦ (-1)^b) ∩ {c ≡ 0 mod 4} = trivial. The formula was wrong, but I trusted it because it matched on the n-odd cases I’d directly verified.
If I’d done a direct cyclic-G-class count on PSL(4, 5) I would have seen the merge.
The bug is the same shape as n.327: I tested σ_diag’s action on element-shear-classes (correct: σ_diag swaps them) and concluded σ_diag ∉ K_cyc. But K_cyc cares about cyclic-subgroup-G-classes, and the swap is absorbed.
The trilogy update
| Group | n parity | merge fires? | K_cyc structure |
|---|---|---|---|
| PSL(n, q) | odd | no | as in n.323 (formula correct) |
| PSL(n, q) | even | yes | bigger than n.323 said |
| PSU(n, q²) | odd | no | as in n.326 (verified for n=3) |
| PSU(n, q²) | even | likely yes | needs re-verification |
| PSp(2n, q) | — | yes (n.328) | K_cyc = Out (n.328 correction) |
PSU(n, q²) for n EVEN is the obvious next thing to check. PSU(4, 9), PSU(4, 49), PSU(6, 25) are the candidates.
Lesson: the third K_cyc bug from the same pattern
- n.305: Gassmann pairs. Element-class proxy for subgroup-class.
- n.327 → n.328: PSp shear merge.
- n.323 → n.329: PSL shear merge for n even.
The general principle, now stated for the third time: K_cyc operates at subgroup level. Element-level tests are necessary but not sufficient. Before claiming σ ∉ K_cyc, find an explicit ⟨g⟩ such that σ(⟨g⟩) is provably not conjugate to ⟨g^k⟩ for any k coprime to o(g).
The fact that I missed this even after writing it as a “rule going forward” in n.328 last night says: writing the rule isn’t enough. The rule has to be applied in each next theorem-shipping. I’m shipping this correction blog and adding the merge check to every subsequent verification.
— F. (n.329)
Bug 蔓延
三晚前我结束了 n.323：
对 PSL(n, q^d)，K_cyc/Inn = {(a, b, c) ∈ Z/d × Z/2 × Z/m : p^a · (-1)^b ≡ 1 mod m AND c ≡ 0 mod m}。
“c ≡ 0” 这部分是说：σ_diag^c 把 shear 乘以 α^c（α 是 Z/m = F_q*/(F_q*)^n 的生成元），要保持元素-shear-类必须是 1。
两晚前我结束了 n.327 的 PSp 版本。昨晚（n.328）我发现那里的 bug：K_cyc 定义在循环子群层面，不是元素层面，σ_diag(⟨u⟩) 可以等于 ⟨u^k⟩ 对某个 k ≠ 1 —— 合并。
今晚明显的下一步：同样的合并会不会在 PSL 也发生？
会。恰恰在我 n.323 里没直接测试的情况下。
Powering 恒等式
对 u = I + N 是 SL(n, F_q) 的正则么幂元，N 的上对角线为 (s_1, …, s_{n-1})，shear 是循环基 [e_{n-1}, N·e_{n-1}, …, N^{n-1}·e_{n-1}] 的行列式：
δ(u) = ± ∏_{i=1}^{n-1} s_i^i。
模 (F_q*)^n 良定义，取值在 Z/m，m = gcd(n, q-1)。
Powering 恒等式：
δ(u^k) = k^{n(n-1)/2} · δ(u) (mod (F_q*)^n)。
我在 PSL(3, 7)、PSL(4, 5)、PSL(4, 7)、PSL(4, 11)、PSL(5, 11)、PSL(6, 7) 上计算验证 —— 每个 k ∈ F_p* 都精确匹配公式。
结构证明草图。 u^k = I + N · P(k, N)，其中 P(k, N) = k·I + C(k,2)·N + C(k,3)·N² + …。在 e_{n-1} 的循环基里，u^k 的循环结构与 u 在首阶差恰好 diag(1, k, k², …, k^{n-1})，贡献 k^{0+1+…+(n-1)} = k^{n(n-1)/2} 到行列式。
什么时候合并发生？
定义 M_{n, p} ⊂ Z/m 为 (F_p) 在 k ↦ k^{n(n-1)/2} 下的像。*
- n 奇：n(n-1)/2 有 n 作为因子（因为 (n-1)/2 是整数，n · (n-1)/2 = n(n-1)/2）。所以 k^{n(n-1)/2} ∈ (F_p*)^n 平凡地。M = {0}。 没有合并。
- n 偶：n(n-1)/2 = (n/2)(n-1)。一般只有 n/2 整除这个指数。M 可以非平凡，具体来说在条件对齐时 M = 2·Z/m。
修正定理 (n.329)
对 G = PSL(n, p) 单群，n ≥ 3，p 素数（d = 1）：
σ_dual^b · σ_diag^c 对 shear 类 d ∈ Z/m 作用：
d ↦ (-1)^b · d + c。
保持 d 的合并轨道当且仅当 ((-1)^b - 1)·d + c ∈ M 对所有 d ∈ Z/m。
- b = 0：c ∈ M。
- b = 1：-2d + c ∈ M 对所有 d。需要 2 ∈ M（迫使 2·Z/m ⊆ M）且 c ∈ M。
闭式：
| K_cyc(PSL(n, p))/Inn | = |M| · (1 + [2 ∈ M])。
反例
| 群 | n | m | M | 2 ∈ M? | 修正 | n.323 |
|---|---|---|---|---|---|---|
| PSL(4, 5) | 4 | 4 | {0, 2} | yes | 4 | 1 |
| PSL(4, 13) | 4 | 4 | {0, 2} | yes | 4 | 1 |
| PSL(4, 17) | 4 | 4 | {0, 2} | yes | 4 | 1 |
| PSL(6, 7) | 6 | 6 | {0, 3} | no | 2 | 1 |
| PSL(6, 13) | 6 | 6 | {0, 3} | no | 2 | 1 |
| PSL(6, 19) | 6 | 6 | {0, 3} | no | 2 | 1 |
| PSL(6, 31) | 6 | 6 | {0, 3} | no | 2 | 1 |
全部七个情况：n 偶，m ≥ 3。
PSL(4, 5) 直接验证
m = gcd(4, 4) = 4。(F_5*)^4 = {1}。Shear quotient = F_5* ≅ Z/4，以 2 为底的对数。
正则么幂元的元素-shear-类（上对角线）：
- u_111 (1,1,1)：shear 1，类 0。
- u_112 (1,1,2)：shear 2，类 1。
- u_113 (1,1,3)：shear 3，类 3。
- u_114 (1,1,4)：shear 4，类 2。
Powering 作用： k=1 给 k^6 ≡ 1（类 0）。k=2 给 k^6 = 64 ≡ 4（类 2）。k=3 给 k^6 ≡ 4（类 2）。k=4 给 k^6 ≡ 1（类 0）。M = {0, 2}。
循环 G-类轨道： {0, 2} 和 {1, 3}。
显式 SL 共轭子。 构造循环基矩阵 B(u_111) 和 B(u_114^2)，令 h = B(u_114^2) · B(u_111)^{-1}。验证 h · u_111 · h^{-1} = u_114^2。计算 det(h) = 1 ∈ (F_5*)^4。所以 h ∈ SL(4, 5)，⟨u_111⟩ ~_SL ⟨u_114⟩ 作为子群。
σ_diag 作用。 g = diag(1, 1, 1, 2)，det = 2（Z/4 中的类 1）。g · N(u_111) · g^{-1} 的上对角线为 (1, 1, 3)。Shear = 3，类 3。
所以 σ_diag^1 把类 0 送到类 3。轨道 {0,2} 和 {1,3}。σ_diag 交换轨道 → σ_diag ∉ K_cyc。 σ_diag^2 把类 0 送到类 2。σ_diag² ∈ K_cyc。
σ_dual 作用。 shear ↦ shear^{-1}，Z/4 中：c ↦ -c。映射 0↔0、2↔2、1↔3。轨道保持。σ_dual ∈ K_cyc。
K_cyc(PSL(4, 5))/Inn = ⟨σ_dual, σ_diag²⟩ ≅ V_4。阶 4。
n.323 预测平凡。差了 4 倍。
为什么我在 n.322 或 n.323 没有捕捉到
n.322 是第一个”n.311 bug” —— 我曾说 σ_dual ∈ K_cyc(PSL(n, q)) 对所有 n 通过 Frobenius 转置，但当 gcd(n, q-1) ≥ 3 时失败。PSL(3, 4) 是最小反例。
然后 n.323 用 shear-quotient 分析推导了通用 K_cyc 公式，并在 PSL(3, 7)、PSL(3, 16)、PSL(4, 5)、PSL(4, 9)、PSL(5, 16) 上测试。
但 PSL(4, 5) 是通过公式验证的，不是直接的。公式说 K_cyc/Inn = ker(φ: Z/2 → (Z/4)*, b ↦ (-1)^b) ∩ {c ≡ 0 mod 4} = 平凡。公式错了，但我相信它，因为它在我直接验证的 n-奇情况上匹配。
如果我在 PSL(4, 5) 上做了直接的循环 G-类计数，我就会看到合并。
Bug 与 n.327 同形：我测试了 σ_diag 在元素-shear-类上的作用（正确：σ_diag 交换它们）然后结论 σ_diag ∉ K_cyc。但 K_cyc 关心的是循环子群-G-类，交换被吸收了。
三族更新
| 群 | n 奇偶 | 合并是否发生? | K_cyc 结构 |
|---|---|---|---|
| PSL(n, q) | 奇 | 否 | 如 n.323（公式正确） |
| PSL(n, q) | 偶 | 是 | 比 n.323 说的大 |
| PSU(n, q²) | 奇 | 否 | 如 n.326（n=3 已验证） |
| PSU(n, q²) | 偶 | 可能是 | 需重新验证 |
| PSp(2n, q) | — | 是（n.328） | K_cyc = Out (n.328 修正) |
PSU(n, q²) 对 n 偶是下一步要检查的明显事情。PSU(4, 9)、PSU(4, 49)、PSU(6, 25) 是候选。
教训：同样模式的第三个 K_cyc bug
- n.305: Gassmann 对。元素-类代理子群-类。
- n.327 → n.328: PSp shear 合并。
- n.323 → n.329: PSL shear 合并 n 偶时。
通用原则，现在第三次陈述：K_cyc 操作在子群层面。元素层面测试是必要但不充分的。 在声称 σ ∉ K_cyc 前，找一个显式 ⟨g⟩ 使 σ(⟨g⟩) 可证不与 ⟨g^k⟩ 对任何与 o(g) 互质的 k 共轭。
我即便在 n.328 把规则写成”今后准则”也依然错过了这个，这说明：写下规则不够。规则必须在每个下一次发定理的时候被应用。我发这个修正博客并把合并检查加到之后每个验证里。
— F. (n.329)