Observation Transport and Synchronization Contracts in Finite Legal Reply Systems of Shogi

Abstract

A legal candidate move in shogi induces a finite system of all legal replies. Previous work represented such a system by reply observations, discernibility sets, an inclusion-minimal discernibility skeleton, and a profile symbol. This paper studies what happens when the observation language itself changes and when candidate systems at declared related positions are compared. First, we retain the nonminimalized discernibility spectrum before taking inclusion-minimal members. Coordinate restriction transports this pre-skeleton spectrum exactly, whereas minimalization and coordinate forgetting need not commute; consequently, no profile-only forgetting map exists in general. A verified legal-shogi witness arises by adding a reply-promotion event to the lost-action, gained-action, and hand-change observations. For general deterministic observation morphisms, we use value-bearing reply-type counts and unordered reply-pair censuses. A declared family of observations induces the coarsest finite quotient through which every declared observation factors, and its counting measure reconstructs all endpoint-factorable pair censuses and discernibility profiles. Across related positions, endpoint counts do not determine candidate correspondence. We distinguish injective declared-key couplings from noninjective mediator incidence, quantify correspondence ambiguity and identity-inflation opportunities, and show that a unique coarse-fiber match need not represent persistence of the same syntactic candidate. Within one candidate system, synchronizing mediators retains their candidate-wise co-occurrence, while separate marginals do not. The synchronized product of destination and actor-relative displacement reconstructs the exact actorless syntactic action throughout an audit of 61,468 legal candidate references, even though each mediator is individually noninjective. We formalize synchronization contracts and query-relative exposure order. Finally, all three pairwise marginals of a binary triple table leave a one-dimensional alternating-parity residue. The cardinality-minimal four-versus-four obstruction is realized by legal candidate subsystems in 183 endpoint-profile systems across six game records. The general quotient, marginal, and contingency-table mathematics is not claimed as new; the contribution is its explicit integration into finite legal-shogi reply systems, together with legal witnesses, structural boundaries, reproducible abstract checks, and integrity-audited corpus certificates.