An elementary presentation of abstract Cuntz semigroups

We give a countable continuous first-order theory whose complete models are precisely the expanded path spaces of abstract Cuntz semigroups. The underlying space is the generalized Thomsen space, equipped with its interleaving metric, a directed comparison predicate, addition, and rational piecewise-linear functional calculus. We prove a converse reconstruction theorem for every complete model, without a separability or realization hypothesis. Function-preserving maps that decrease the comparison predicate correspond exactly to Cuntz-semigroup morphisms, and language embeddings correspond exactly to order embeddings. With homomorphisms decreasing atomic formulas, this resolves the category-axiomatizability problem recorded by Farah in 2026. The equivalence also identifies metric ultraproducts of the presentations with the presentations of categorical Cuntz-semigroup ultraproducts, for every ultrafilter. The endpoint reconstruction is external. A closed substructure of the scalar path space shows that quantified gluing cannot be replaced by a universal theory in the given language. GitHub repository