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We here focus on Nelson-Oppen and polite theory combinations, leading to a total of five model-theoretic properties to be considered: stable infiniteness, smoothness, finite witnessability, strong finite witnessability, and convexity. Our first result is an improvement on polite theory combination, showing that it is possible when only assuming stable infiniteness and strong finite witnessability, and thus implying smoothness is not a prerequisite for this method. Second, we provide examples of Boolean combinations of the aforementioned 5 properties whenever they are possible (e.g., a theory that admits all the properties, a theory that admits none, etc.), sharp in the sense that no theories within simpler signatures may exhibit the exact same properties, and prove which combinations cannot occur. Among these examples, the most surprising one is that of a polite yet not strongly polite theory in one sort, a combination whose previous example in the literature was two-sorted.<\/jats:p>","DOI":"10.1007\/s10817-025-09746-5","type":"journal-article","created":{"date-parts":[[2025,12,23]],"date-time":"2025-12-23T14:47:26Z","timestamp":1766501246000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Combining Combination Properties, Part I: Nelson-Oppen and Politeness"],"prefix":"10.1007","volume":"70","author":[{"given":"Guilherme V.","family":"Toledo","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yoni","family":"Zohar","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Clark","family":"Barrett","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,12,23]]},"reference":[{"key":"9746_CR1","doi-asserted-by":"publisher","unstructured":"Toledo, G.V., Zohar, Y., Barrett, C.W.: Combining combination properties: An analysis of stable infiniteness, convexity, and politeness. In: Pientka, B., Tinelli, C. (eds.) Automated Deduction - CADE 29 - 29th International Conference on Automated Deduction, Rome, Italy, July 1-4, 2023, Proceedings. Lecture Notes in Computer Science, vol. 14132, pp. 522\u2013541. Springer, Cham (2023). https:\/\/doi.org\/10.1007\/978-3-031-38499-8_30","DOI":"10.1007\/978-3-031-38499-8_30"},{"key":"9746_CR2","doi-asserted-by":"crossref","unstructured":"Barrett, C., Sebastiani, R., Seshia, S.A., Tinelli, C.: In: Biere, A., Heule, M., Maaren, H., Walsh, T. (eds.) Frontiers in Artificial Intelligence and Applications. Chapter 33. satisfiability modulo theories. Frontiers in artificial intelligence and applications. IOS Press, Amsterdam (2021)","DOI":"10.3233\/FAIA201017"},{"issue":"2","key":"9746_CR3","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1145\/357073.357079","volume":"1","author":"G Nelson","year":"1979","unstructured":"Nelson, G., Oppen, D.C.: Simplification by cooperating decision procedures. ACM Trans. Program. 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