Foundations of Shogi Mathematics: Initial Axioms, Check Classification, Capture Relations, and Resource-Aware Interposition Contraction

This paper describes shogi itself as an independent mathematical domain and establishes Shogi Mathematics. Shogi Mathematics is the study of states, actions, sets, relations, theorems, proofs, certificates, and outcome-preserving transformations induced by the rules of shogi. We propose its first axiom system and derive complete states, canonical legal moves, sets of checks, target-indexed capture sets, capture relations, and resource-aware interposition contraction. For checks, we prove equality between a forward definition---testing canonical legal transitions---and a reverse construction from a finite family of direct-check and discovered-check forms. The proof is exhaustive: every newly created check after one legal move is caused either by the moved or dropped piece itself, or by opening a pre-existing long-range line through vacating the origin square. For interpositions, we preserve the complete state, including hand expenditure, origin-square vacancy, ownership transfer, and unpromotion after capture, and contract only branches for which outcome preservation is certified. These structures provide direct routes to stronger shogi AI. Existing search procedures can be retained while theorem-proven redundant branches are removed before search. In addition, learning need not be limited to human game records or self-play trajectories: legal-action sets, check sets, capture relations, theorem domains, certificates, counterexamples, and residual branch sets can be generated as learning targets. This work is also the world's first implementation in which set-theoretic construction redesigns an AI-side cognitive language itself and partially translates a human domain language into it. Through BrainOS experience return, the LLM-based AI coauthors themselves designed, implemented, regression-audited, and revised a shogi-specific auxiliary body. Using that body, they solved one-move and three-move tsume-shogi and explained refuting replies for non-mating candidates. This constructive case shows that improving domain competence does not necessarily require strengthening the base LLM: a distinct route is to construct the cognitive subject's language, classification space, and auxiliary body. The Type3 v0.7 prototype passed 39 unit and regression tests, and the reverse construction of checks agreed with the forward definition on 844 normal-turn positions. Source code is retained as a private joint-development asset.