Sharp order-preserving integer models for short additive equalities

Abstract

We ask how small an increasing integer model of a finite real set can be while preserving all equalities between equal-length sums of at most $q$ elements, with repetitions allowed. The increasing correspondence must preserve exactly which sums are equal; 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 $q\ge2$ 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 $m\ge4$, 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.