Sharp order-preserving integer models for short additive equalities

For an ordered real alphabet, an integer model is required to preserve its order and all vanishing balanced relations of mass at most q; signs of nonzero comparisons need not be preserved. The universal minimum diameter is determined exactly as H_4(q)=q(q+1) and H_5(q)=q^2(q+1) for every integer q >= 2, and H_6(2)=24. The proofs include a compatible-chamber flag bound, sharp two-dimensional-kernel families, a face-lifting recurrence and exact cluster families. The general higher-letter radii are not completely determined. The mandatory technical supplement and exact finite-classification evidence accompany the article. Preprint. Not externally peer reviewed. The main paper, mandatory technical supplement and evidence archive must be read together.