AI 中文总结
本文证明在 $E_2\times C_3^r$ 上长度为 $2r+11$ 的无乘积一序列不能含恰好 $2r-1$ 个中心项，从而排除十二个非中心项的临界层，结合相对矩准则与穷举检查，但未确定小 Davenport 常数的精确值。
AI 中文摘要
设 $E_2$ 是阶为 $3^5$、指数为 3 的额外特殊群。对于每个 $r\ge1$，我们证明在 $E_2\times C_3^r$ 上长度为 $2r+11$ 的无乘积一序列不能恰好包含 $2r-1$ 个中心项。因此，具有十二个非中心项的临界层被排除。该证明将阿贝尔正规子群的相对矩准则与 $\mathbb F_3$ 上辛四空间中的一个有限定理相结合。在显式子空间占用界下，实际可补全为非满零和块的一族平衡三角形，在每个三角形上的边权重之和为 1。一个对偶环论证将此断言归结为分支边上的势条件。所有剩余构型包含十一个最小框架之一；两个分别实现的穷举检查验证了它们的所有容许扩展。无限制检查有 9544046 个叶节点。提供了完整源代码和执行记录。此结果未确定任意 $r$ 时 $\mathsf{d}(E_2\times C_3^r)$ 的精确值。
英文摘要
Let $E_2$ be the extraspecial group of order $3^5$ and exponent three. For every $r\ge1$, we prove that a product-one-free sequence of length $2r+11$ over $E_2\times C_3^r$ cannot contain exactly $2r-1$ central terms. Thus the critical layer with twelve noncentral terms is excluded. The proof combines a relative moment criterion for abelian normal subgroups with a finite theorem in a symplectic four-space over $\mathbb F_3$. Under explicit subspace occupancy bounds, the family of balanced triangles that can actually be completed to a nonfull zero-sum block admits edge weights summing to one on every triangle. A dual cycle argument reduces this assertion to a potential condition on branching edges. All remaining configurations contain one of eleven minimal frames; two separately implemented exhaustive checks verify all their admissible extensions. The unrestricted check has 9544046 leaves. Complete source code and execution records are supplied. The exact value of $\mathsf{d}(E_2\times C_3^r)$ for arbitrary $r$ is not determined by this result.
Comments19 pages; computer-assisted proof; complete verification source code and logs in ancillary files
