343阶Heisenberg群的小Davenport常数
The small Davenport constant of the Heisenberg group of order 343
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中文总结 AI 辅助
该研究针对343阶Heisenberg群，证明了其小Davenport常数为18，采用White的框架并开发$p=7$特定分层方法，结合SAT程序与穷尽计算完成上界验证，填补了该领域的研究空白。
中文摘要 AI 辅助
对于有限群$G$，记$\boldsymbol{\textsf{d}}(G)$为不存在非空子序列可排序后乘积为1的序列的最大长度。对于奇素数$p$，记$H_{p^3}=\boldsymbol{\text{UT}_3}(\boldsymbol{\text{F}}_p)$。Godara与Sarkar证明了$\boldsymbol{\textsf{d}}(H_{27})=6$，并猜想$\boldsymbol{\textsf{d}}(H_{p^3})=3p-3$；近期预印本中，White证明了下一个情况$\boldsymbol{\textsf{d}}(H_{125})=12$，并给出$\boldsymbol{\textsf{d}}(H_{343})$的范围为$18 \boldsymbol{\textsf{d}}(H_{343})=18$。我们采用White的乘积一判据与扩展框架，开发了针对$p=7$的特定方向分层方法。一个显式无乘积一的序列给出了下界；对于上界，我们按中心项数量及其商多重集占据的射影方向对长度为19的假设无乘积一序列进行分层。通过理论论证排除了最多两个方向的支撑，其有限辅助陈述经穷尽检验；三方向情形与5个中心项情形通过精确有限计算解决；剩余30个层由反例引导的SAT程序编码，独立实现的检查器验证了全部9,920,815个种子割与全部27,207个学习割，每个最终不可满足实例均附带经检查的LRAT证书，另有独立实现级审计验证了主编码、证明档案与下界见证。
英文摘要
For a finite group $G$, let $\mathsf{d}(G)$ denote the maximum length of a sequence having no nonempty subsequence whose terms can be ordered to have product one. For an odd prime $p$, let $H_{p^3}=\operatorname{UT}*3(\mathbb{F}*p)$. Godara and Sarkar proved $\mathsf{d}(H*{27})=6$ and conjectured $\mathsf{d}(H*{p^3})=3p-3$; in a recent preprint, White proved the next case $\mathsf{d}(H_{125})=12$ and left $18\leq\mathsf{d}(H_{343})\leq24$. We prove $\mathsf{d}(H_{343})=18$. We adopt White's product-one criterion and spread framework and develop a $p=7$-specific direction stratification. An explicit product-one-free sequence gives the lower bound. For the upper bound, we stratify a hypothetical product-one-free sequence of length $19$ by the number of central terms and by the occupied projective directions of its quotient multiset. Supports on at most two directions are excluded by a theoretical argument whose finite auxiliary statements are exhaustively checked; the three-direction case and the case of five central terms are settled by exact finite computations. The remaining thirty strata are encoded by a counterexample-guided SAT procedure. A separately implemented checker verifies all $9{,}920{,}815$ seed cuts and all $27{,}207$ learned cuts, and each final unsatisfiable instance is accompanied by a checked LRAT certificate. A separate implementation-level audit verifies the master encoding, the proof archives, and the lower-bound witness.
