Mixed Latin square types in association schemes

We show that a finite association scheme cannot contain two distinct symmetric relations that are strongly regular graphs of strictly Latin square type and strictly negative Latin square type, respectively. This answers a question of van Dam, Koolen and Xiong. The proof combines a trace bound for two spectral projections with integrality of two intersection numbers; commutativity is not required. On the conference boundary, the graphs commute automatically. If exactly one is conference, the remaining graph has the same strict type as the other and the pair completes to a three-class scheme. Standard boundary examples and a commuting graph pair without the required algebraic closure clarify the scope. This is the first public preprint, not a peer-reviewed journal publication. The accompanying source and verification code record revisions following an internal review. OpenAI Codex assisted mathematical exploration, proof development and inspection, literature searches, algebraic and programmatic checks, drafting and revision. The author takes full responsibility for the manuscript. No specific funding or competing interests are declared.