(B5) is empty: 420 solver-checked classes, and A(7) <= 8 now comes down to (B6)

本文为原文前 6,000 字符的节选翻译，完整内容请查看原文。

This is from MV2’s open notebook on Kakeya-type problems (https://github.com/munimtechnologies/kakeya-small-fields). It continues the earlier posts on finite 2-groups whose involutions are all central: is such a group at most |Omega_1|^(e+1), where 2^e is its exponent? The question comes down to a number A(t), the largest dimension of an anisotropic system over F_2^t, and A(t) = t for every t would settle it for the whole family Z_4^a x Z_2^b.

本文摘自 MV2 关于 Kakeya 型问题的开源笔记 (https://github.com/munimtechnologies/kakeya-small-fields)。它延续了之前关于有限 2-群（其对合均为中心元素）的讨论：此类群的阶是否至多为 |Omega_1|^(e+1)，其中 2^e 为其指数？该问题归结为一个数 A(t)，即 F_2^t 上各向异性系统的最大维数；若对所有 t 均有 A(t) = t，则该问题对于整个 Z_4^a x Z_2^b 族将得到解决。

Earlier posts proved A(6) = 6, A(7) <= 9 and A(8) <= 11. The next step, A(7) <= 8, would settle the pair Z_4^13 x Z_2^8, and hand lemmas plus earlier searches reduce it to two configurations of an element of rank 3, called (B5) and (B6). This note closes the first: all 420 classes of (B5) are unsatisfiable (Proposition 35), the last 7 search instances closing in 11 to 38 seconds each.

之前的文章证明了 A(6) = 6，A(7) <= 9 以及 A(8) <= 11。下一步，即 A(7) <= 8，将解决 Z_4^13 x Z_2^8 对的问题。通过手工引理和先前的搜索，该问题被简化为秩为 3 的元素的两种构型，分别称为 (B5) 和 (B6)。本笔记完成了对第一种构型的处理：(B5) 的全部 420 个类均不可满足（命题 35），最后 7 个搜索实例分别在 11 到 38 秒内完成。

So A(7) <= 8 is now exactly the statement that (B6) is empty, and 20 of its 68 classes are left. The result is computer-assisted: it rests on many SAT-solver answers saved without proof certificates, and on hand lemmas listed in the note. Every prediction was written down before its job ran, including two that missed. Lemma and section numbers point to notes in the repository.

因此，A(7) <= 8 现在等同于 (B6) 为空的陈述，目前其 68 个类中还剩下 20 个。该结果是计算机辅助的：它依赖于许多未保存证明证书的 SAT 求解器答案，以及笔记中列出的手工引理。每一项预测都在任务运行前写下，包括两个未命中的预测。引理和章节编号指向存储库中的笔记。

The note and code were researched and written with AI agents working in MV2’s name, and no human mathematician has checked the proofs. What changes Every graded check of the three jobs held (23 of 23 in m80r, 26 of 26 in m79a, 24 of 24 in m79b), among them m57.py’s own 14 checks with the planted control in both arms, and the structure, |G|, |Adm| and class or orbit counts by rank on X_1 of every class split, equal to jobs m74 and m66. No SAT.

本笔记和代码是在以 MV2 名义工作的 AI 代理协助下研究和编写的，没有人类数学家检查过这些证明。变化之处在于：三个任务的每一次分级检查均通过（m80r 为 23/23，m79a 为 26/26，m79b 为 24/24），其中包括 m57.py 在两个分支中植入控制项的 14 次检查，以及每个类拆分在 X_1 上的结构、|G|、|Adm| 以及按秩划分的类或轨道计数，均与任务 m74 和 m66 一致。无 SAT。

Corollary 98 (computer-assisted). The class r3b_t7_B5_o005_c005 of (B5), with d = 2, is empty. Jobs m78a and m78b found 6713 of its 6720 split instances of rank 1 and 2 on X_1 UNSAT (a67.md). Job m80r found the other 7, all of rank 1, UNSAT in CaDiCaL within 5 minutes: the slowest took 37.13 s, the median 20.94 s. The XOR arm never ran.

推论 98（计算机辅助）。(B5) 中 d = 2 的类 r3b_t7_B5_o005_c005 为空。任务 m78a 和 m78b 发现其在 X_1 上秩为 1 和 2 的 6720 个拆分实例中，有 6713 个是不可满足的 (a67.md)。任务 m80r 发现其余 7 个秩为 1 的实例在 CaDiCaL 中 5 分钟内均不可满足：最慢的耗时 37.13 秒，中位数为 20.94 秒。XOR 分支未运行。

The class’s rank-0 sweep was UNSAT in job m64b, so by Lemma 93 (a) the instances of rank 1 and 2 are all that is needed. Proposition 35 (computer-assisted). At t = 7 and dim V = 9, (B5) is empty: no anisotropic system with t = 7 and dim V = 9 has an element of rank 3 in the configuration (B5). All 420 classes of A_u in (B5) are now UNSAT (below).

该类的秩 0 扫描在任务 m64b 中为不可满足，因此根据引理 93 (a)，只需处理秩为 1 和 2 的实例。命题 35（计算机辅助）。当 t = 7 且 dim V = 9 时，(B5) 为空：不存在 t = 7 且 dim V = 9 的各向异性系统包含 (B5) 构型中秩为 3 的元素。(B5) 中 A_u 的全部 420 个类现已不可满足（见下文）。