The additive-square-free spectrum below 139

Abstract

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.

arXiv comment14 pages, including the technical supplement. Computational data are available at https://doi.org/10.5281/zenodo.22559145