Query-Relative Soundness of Legal-Reply Compression in Shogi : Quotient Criteria, Canonical Repair, and Action-Edge Counterexamples

Abstract

Compressing a finite set of legal Shogi replies into coarse classes can reduce the amount of downstream checking, but such compression is meaningful only relative to what is being asked about those replies. We formalize this point as a query-relative quotient condition. For a position $P$, a legal candidate move $m$, the legal reply set $R_m$, a partition $\Pi$ of $R_m$, and a declared query $Q:R_m\to Y$, we call $\Pi$ sound for $Q$ when $Q$ is constant on every reply class. Equivalently, $Q$ factors uniquely through the quotient map $R_m\to R_m/\Pi$, and representative-only evaluation is independent of which representative is chosen. We then define the canonical repair of an unsound partition by splitting each old class only by its $Q$-values, and prove that this is the coarsest refinement on which $Q$ becomes well-defined. For several queries, repair is obtained by their joint value tuple and is order independent. We connect this elementary quotient structure to the reply-by-action-edge incidence formalism introduced in our preceding work. For an action edge $e$, the query $Q_e(r)$ asks whether $e$ survives reply $r$. A reply partition is sound for a family of such queries exactly when the corresponding restricted incidence rows are constant inside every class. Two standard-start reachable Shogi witnesses show why the criterion is not vacuous. In one position, the natural capture/non-capture partition is sound for one edge-survival query but unsound for another; the same third query is sound under capture/non-capture but unsound under king/non-king. A second reachable position gives an independent failure inside a king-move class. These examples are reproduced by a standalone legal-move verifier. We do not claim novelty for quotient well-definedness or partition refinement themselves; the contribution is their explicit query-relative realization for legal Shogi replies and their connection to Shogi action-edge survival.