发表机构
University of Toronto Scarborough（多伦多大学士嘉堡校区）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文分类了所有满足 $g(T)<139$ 的四元素实数字母表，通过穷举搜索和 77 个显式单词证明截止值尖锐，并给出 $\{0,1,x,x+2\}$ 族中 167 的尖锐截止值。
AI 中文摘要
对于有限实数字母表 $T$，设 $g(T)$ 为 $T$ 上不含两个相邻且长度相等、和相等的非空块的单词长度的上确界。我们对所有满足 $g(T)<139$ 的四元素字母表进行了分类。在仿射等价意义下，这些字母表包括已知的端点求和族和七个孤立类。穷举有限搜索给出了孤立值，而 77 个显式单词证明了每个剩余实数字母表都存在长度为 139 的单词。后一论证使用了有限并集的直线的精确交集，因此它也涵盖了无理参数。该截止值是尖锐的，因为 $g(\{0,2,3,7\})=139$。我们还在族 $\{0,1,x,x+2\}$（$x>1$）中获得了 167 的尖锐截止值。所需的补充材料打印了单词证书并指定了完整搜索的证据。
英文摘要
For a finite real alphabet $T$, let $g(T)$ be the supremum of the lengths of words over $T$ containing no two adjacent nonempty blocks of equal length and equal sum. We classify all four-element alphabets with $g(T)<139$. Up to affine equivalence, these are the known endpoint-sum family and seven isolated classes. Exhaustive finite searches give the isolated values, while 77 explicit words prove that every remaining real alphabet admits a word of length 139. The latter argument uses exact intersections of finite unions of lines, so it covers irrational parameters as well. The cutoff is sharp because $g(\{0,2,3,7\})=139$. We also obtain a sharp cutoff of 167 within the family $\{0,1,x,x+2\}$, $x>1$. The required supplement prints the word certificates and specifies the complete-search evidence.
Comments14 pages, including the technical supplement. Computational data are available at https://doi.org/10.5281/zenodo.22559145
