The additive-square-free spectrum below 139

For a finite real alphabet T, let g(T) be the supremum of lengths of words with no adjacent equal-length blocks having equal sum. We classify all four-element real alphabets with g(T) < 139. The eleven finite values arise from the endpoint-sum family classified by Freedman and Brown and seven isolated affine classes. A certificate of 77 fixed words of length 139 covers the complement in the full real parameter plane through exact rational intersections. Complete finite searches provide the isolated values and the sharp example g({0,2,3,7})=139. The same method gives sharp cutoff 167 on the real line {0,1,x,x+2}, x>1. The required technical supplement contains the certificate words and the computational evidence. This paper contains no five-element model theory. Preprint. Not externally peer reviewed. The main paper must be read with the mandatory technical supplement and the accompanying evidence archives.