The additive-square-free spectrum below 139

For a finite set T of real numbers, let g(T) be the largest possible length of a word over T containing no two adjacent nonempty blocks of equal length and equal sum; set g(T) = infinity if the lengths are unbounded. We classify all four-element sets with g(T) < 139. Up to affine changes of the alphabet, they consist of the known endpoint-sum family and seven isolated classes. The proof combines exhaustive searches for the isolated values with 77 explicit words that cover all remaining real parameters. The cutoff is sharp: 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 technical supplement contains the words and specifies the exact computations used in the proof. Preprint. Not externally peer reviewed. The main paper is accompanied by a required technical supplement and computational evidence archives.