Sharp order-preserving integer models for short additive equalities

We ask how small an increasing integer model of a finite real set can be while preserving exactly the equalities between equal-length sums of at most q elements, with repetitions allowed. The signs of unequal comparisons may change. Let H_m(q) be the least diameter sufficient for every ordered real set of size m. For every integer q >= 2 we prove H_4(q) = q(q+1) and H_5(q) = q^2(q+1), and we determine H_6(2) = 24. The proofs use the linear space determined by the additive equalities and the inequalities prescribing the order of the elements. They also give bounds for larger sets, determine the first two asymptotic terms for each fixed m >= 4, and yield further families of sharp examples. An application bounds the integer alphabets needed to realize finite additive-square-free spectra. The supplementary data support the finite six-element classification. Preprint. Not externally peer reviewed. The main paper is accompanied by a required technical supplement and computational evidence archives.