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.

翻译：欧几里得空间中每一族有限紧凸体在无交指标集之间诱导一种分离关系：当相应并集的凸包无交时，两集合相互分离。本文研究以分离性作为原始关系建立的有限理论。其基本法则包括对称性、双边包含性和空性，等价于无环分离体的分离-极性形式。主要贡献在于具有均匀边界的高效有理实现定理及其所支持的精确推论理论。每个有限分离性分离体均可由有理多面体实现，其坐标以最大分离为索引。最大分离和最小拉德蒙划分可通过完整表格、生成元或成员关系预言机枚举；坐标值具有受控比特高度；每个坐标记录一个可读的最大分离证书。该实现使每一分离对的间隔至少为2，在任意小于1半径的外平行扩张下保持正确性，增厚后生成全维凸体。距离函数层通过利普希茨比较、包含单调性和外平行体记录标准凸分析稳定性。在逻辑层面，正蕴含恰为单前提包含性。欧几里得场景上的布尔推论具有完备性、可靠性及可判定性；可满足性为NP完全问题，有效性为coNP完全问题，正蕴含对于排序编码呈线性复杂度。分层定理表明布尔推理在分离体闭包之外不引入新的原子分离性。固定维度推论关系构成严格递减层级，当站点数为n时在n-1维处达到稳定。