AI 中文总结
研究阶为125的海森堡群的小Davenport常数，通过明确无积为一序列得出下界，利用积为一准则、加法组合学等将问题简化并经搜索验证得出上界\(\mathsf{d}(H_{125}) = 12\)，解决了该领域的一个未决问题。
AI 中文摘要
有限群\(G\)的小 Davenport 常数\(\mathsf{d}(G)\)是\(G\)上无积为一序列的最大长度。对于阶为\(p^3\)的指数为\(p\)的海森堡群\(H_{p^3}\)，戈达拉和萨卡尔证明了\(\mathsf{d}(H_{27}) = 6\)，并提出对于每个奇素数\(p\)，\(\mathsf{d}(H_{p^3}) = 3p - 3\)，\(p\geq5\)的情况未解决。我们解决了第一个未解决的情况：\(\mathsf{d}(H_{125}) = 12\)。下界是明确的无积为一序列\(x^4y^4v^4\)。对于上界，我们记录了一个积为一准则，将非交换问题简化为\(\mathbb{F}_5^2\)上的加法组合学，然后将“每个长度为13的序列都有一个积为一子序列”简化为一个单一的有限陈述——商多重集上的一个展开界，我们通过在C语言中进行详尽的、内存平坦的搜索来验证，其结果由具有不同剪枝策略的第二次搜索独立重现。每个辅助引理都经过机器检查。该论证是真正特定于\(p\)的：我们确定了对于\(p\geq7\)失败的精确步骤（一个谢瓦莱 - 沃林捷径，其强制块不必很宽），展示了\(p = 7\)时的阻碍多重集，并且仅留下\(18\leq\mathsf{d}(H_{343})\leq24\)。所使用的技术——柯西 - 达文波特定理、谢瓦莱 - 沃林定理以及奥尔森关于\(C_p^2\)的 Davenport 常数的值——是标准的；贡献在于它们针对新的非阿贝尔目标的组合以及完成证明的有限验证。
英文摘要
The small Davenport constant $\mathsf{d}(G)$ of a finite group $G$ is the maximal length of a product-one-free sequence over $G$. For the exponent-$p$ Heisenberg group $H_{p^3}$ of order $p^3$, Godara and Sarkar proved $\mathsf{d}(H_{27})=6$ and posed $\mathsf{d}(H_{p^3})=3p-3$ for every odd prime $p$, leaving $p\ge5$ open. We settle the first open case: $\mathsf{d}(H_{125})=12$. The lower bound is the explicit product-one-free sequence $x^4y^4v^4$. For the upper bound we record a product-one criterion that reduces the non-commutative problem to additive combinatorics over $\mathbb{F}_5^2$, and then reduce "every length-13 sequence has a product-one subsequence" to a single finite statement -- a spread bound on quotient multisets -- which we verify by an exhaustive, memory-flat search in C, its verdict independently reproduced by a second search with a different pruning strategy. Every auxiliary lemma is machine-checked. The argument is genuinely $p$-specific: we identify the exact step that fails for $p\ge7$ (a Chevalley-Warning shortcut whose forced block need not be wide), exhibit the obstructing multiset for $p=7$, and leave only $18\le\mathsf{d}(H_{343})\le24$. The techniques -- the Cauchy-Davenport theorem, Chevalley-Warning, and Olson's value of the Davenport constant of $C_p^2$ -- are standard; the contribution is their assembly against a new non-abelian target and the finite verification that closes it.
Comments8 pages. The load-bearing spread bound is verified by two independent exhaustive searches (C and Python). Verification code: https://github.com/pw/heisenberg-davenport-125
