发表机构
Oregon State University（俄勒冈州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该文为任意群构造至多32值的右不变度量，其等距群恰为保持所有右不变度量的置换，并据此完全分类自然群：阿贝尔群当且仅当2A=A或2A={0}，非阿贝尔群当且仅当非广义二环群，且布尔群界5为最优。
AI 中文摘要
对任意基数的每个群 $G$，我们构造了一个至多取 $32$ 个值的右不变度量，其等距群恰好是保持 $G$ 上所有右不变度量的置换。证明结合了子群入口秩和符号变差着色以及 Leemann 和 de la Salle 的短词刚性定理。他们的非交换定向刚性定理和直接的正则子群论证给出了右平移意义下自然群的完全分类：阿贝尔群 $A$ 是自然的当且仅当 $2A=A$ 或 $2A=\{0\}$，而非阿贝尔群是自然的当且仅当它不是广义二环群。特别地，每个域的可加群都是自然的。对于阿贝尔群，界改进为 $17$；对于布尔群，界改进为 $5$，且布尔界是精确的：$C_2^3$ 不允许少于五个值的此类度量。补充构造给出了一个可数值的包络度量，它同时在所有包含固定坐标标记的子群上精确实现仿射符号等距，以及对于奇素数 $p$，在 $\mathbb{F}_p$ 上至多取 $p+5$ 个值的符号基度量。没有声称其他界是最优的，也没有断言未着色的图正则表示。
英文摘要
Every group $G$, of arbitrary cardinality, admits a right-invariant metric with at most five values whose isometry group is exactly the group $Ξ(G)$ of permutations preserving every inverse-pair difference. Five is sharp. Writing $N(G)$ for the least number of values, including zero, we obtain $N(G)=1$ for the trivial group, $N(G)=2$ for $C_2$ and $C_3$, $N(G)=3$ for every other group outside an explicitly listed set $\mathcal{E}$ of 24 finite groups, $N(G)=4$ on $\mathcal{E}\setminus\{C_2^3, C_3^2\}$, and $N(C_2^3)=N(C_3^2)=5$. The exact result combines Cayley-graph realization theorems with verified finite certificates and hand proofs of the lower bounds. Separately, a uniform constructive 32-value bound uses subgroup-entry ranks, sign-variation conflict labels, and short-word rigidity, but no Cayley-index input or computation. In Knill's right-translation sense, a right-invariant metric on $G$ is naturalizing when every group law on its underlying set with isometric right translations is isomorphic to $G$; a group is natural when it admits such a metric. We prove that an abelian group $A$ is natural if and only if $2A=A$ or $2A=\{0\}$, and a nonabelian group is natural if and only if it is not generalized dicyclic. Every natural group admits a naturalizing metric with at most five values, and this bound is sharp.
Comments39 pages, 24 certificates in Appendix A; ancillary files include the certificates, verification programs, and logs. v3: exact five-value theorem (Theorem 1.1) added; expository revision; ancillary package updated
