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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\n                    <jats:italic>\u03b5<\/jats:italic>\n                    -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.\n                  <\/jats:p>","DOI":"10.1007\/s10472-021-09780-0","type":"journal-article","created":{"date-parts":[[2021,11,25]],"date-time":"2021-11-25T11:05:09Z","timestamp":1637838309000},"page":"645-673","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Gaussoids are two-antecedental approximations of Gaussian conditional independence structures"],"prefix":"10.1007","volume":"90","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7284-1827","authenticated-orcid":false,"given":"Tobias","family":"Boege","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2021,11,25]]},"reference":[{"key":"9780_CR1","unstructured":"Abo Khamis, M., Kolaitis, P.G., Ngo, H.Q., Suciu, D.: Decision problems in information theory. arXiv:2004.08783 (2020)"},{"key":"9780_CR2","unstructured":"Boege, T.: Incidence geometry in the projective plane via almost-principal minors of symmetric matrices. arXiv:2103.02589 (2021)"},{"issue":"4","key":"9780_CR3","doi-asserted-by":"publisher","first-page":"775","DOI":"10.1007\/s10208-018-9396-x","volume":"19","author":"T Boege","year":"2019","unstructured":"Boege, T., D\u2019Al\u00ec, A., Kahle, T., Sturmfels, B.: The geometry of gaussoids. 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