The gaussoid axioms are conditional independence inference rules which characterize regular Gaussian CI structures over a three-element ground set. It is known that no finite set of inference rules completely describes regular Gaussian CI as the ground set grows. In this article we show that the gaussoid axioms logically imply every inference rule of at most two antecedents which is valid for regular Gaussians over any ground set. The proof is accomplished by exhibiting for each inclusion-minimal gaussoid extension of at most two CI statements a regular Gaussian realization. Moreover we prove that all those gaussoids have rational positive-definite realizations inside every $\varepsilon$-ball around the identity matrix. For the proof we introduce the concept of algebraic Gaussians over arbitrary fields and of positive Gaussians over ordered fields and obtain the same two-antecedental completeness of the gaussoid axioms for algebraic and positive Gaussians over all fields of characteristic zero as a byproduct.

翻译：Gaussoid axioms 是有条件的独立推断规则, 以三元素基底为固定的 Gaussian CI 结构特征。 已知没有一套有限的推断规则完全描述正常的 Gaussian CI 。 在本篇文章中, 我们显示 Gaussoid axioms 逻辑上意味着最多对正常的高斯人有效、 对任何一组地的常数两个序数的每个推论规则。 证据通过展示最多对两部CI 声明的包含- 最小缩数扩展, 一个常规的Gaussian 实现。 此外, 我们证明所有这些类在身份矩阵周围每个 $\ varepsilon- balls 中都有合理的正- 确定值。 为了证明我们引入了任意字段的平方位高官和对定地的正数高官的概念, 并获得对正值零特性的所有领域的正位平方位平方位平方位平方位平方位平方位平方位平。