A Calculus of Apartness over Separoids: Effective Convex Representation, Stratified Conservativity, and the Complexity of Entailment

Abstract

Every finite family of compact convex bodies in Euclidean space induces an apartness relation between disjoint index sets: two sets are apart when the convex hulls of the corresponding unions are disjoint. This paper studies the finite theory obtained by taking apartness as the primitive relation. Its basic laws are symmetry, bilateral subsumption, and vacuity, equivalently the separation-polarity form of acyclic separoids. The main contribution is an effective rational realization theorem with uniform margins and the exact consequence theory it supports. Every finite apartness separoid is realized by rational polytopes whose coordinates are indexed by maximal separations. Maximal separations and minimal Radon partitions can be enumerated from a full table, generators, or a membership oracle; the coordinate values have controlled bit height; and each coordinate records a readable certificate of one maximal separation. The realization separates every apart pair with clearance at least 2, remains correct under outer parallel enlargement by any radius below 1, and yields full-dimensional convex bodies after thickening. The distance-function layer records standard convex-analytic stability through Lipschitz comparison, monotonicity under inclusion, and outer parallel bodies. On the logical side, positive entailment is exactly one-premise subsumption. Boolean consequence over Euclidean scenes is sound, complete, and decidable; satisfiability is NP-complete, validity is coNP-complete, and positive entailment is linear for sorted encodings. A stratification theorem shows that Boolean reasoning introduces no new atomic apartness beyond separoid closure. Fixed-dimensional consequence relations form a strictly decreasing hierarchy that stabilizes in dimension n minus 1 for n sites.