Query-Relative Quotients and Certificate-Carrying Observation Reducts

Abstract

We present a framework for reducing a finite set of legal moves or legal replies in a fixed shogi position while preserving exactly the distinctions required by declared queries. Our preceding work first formalized complete positions and canonical legal actions as foundational objects of shogi mathematics, then introduced an incidence geometry of action-edge deletion under replies, and subsequently established query-relative soundness and canonical repair for a given reply partition. The canonical quotient and joint query tuple used here reorganize that repair construction into a form independent of an initial partition. Building on this foundation, we study realizations of the canonical quotient by finitely many atomic query generators, products of independent query modules, and dependencies among observation components in a fixed position. We characterize when a subset of components $S\subseteq J$ preserves the same quotient as the full observation in three equivalent ways: reconstruction of deleted components, finite functional dependency, and a hitting-set condition for a family of discernibility sets. We call an inclusion-minimal sufficient component set an \emph{observation reduct}, and the intersection of all reducts the \emph{position-relative observation core}. Deletability admits a finite reconstruction-table certificate, whereas non-deletability admits a witnessing pair of legal moves. In a constructed position with action-edge loss, action-edge gain, and hand-resource difference as components, exhaustive enumeration of 161 legal moves yields exactly two reducts, with the gained-edge component alone forming the core. We do not claim novelty for quotients, functional dependencies, rough-set reducts, or minimal hitting sets as general mathematical notions. The principal contribution is to extend earlier query-relative compression in shogi mathematics to observation dependency, reducts, cores, and success/failure certificates, and to connect these structures with shogi-specific action and resource observations under a proof-carrying plugin boundary.