Chronological Residual Quotients and Constant-Coordinate Certificate Kernels in Finite Legal Reply Systems of Shogi

Abstract

We develop a chronological certificate theory for finite legal-reply systems arising from shogi. At each checkpoint, complete legal replies induce a finite full carrier relation together with mediator-wise projected witness sets. Future checkpoints act by synchronized intersection. Once a future action family and an observation contract are declared, standard behavioural residual equivalence yields the coarsest deterministic quotient preserving all declared future outputs. The quotient construction itself is classical; the new structure lies in the certificate system carried by legal-reply observations. We compare four contracts: Boolean exact-faithfulness verdicts, numerical defect spectra, labelled bad-set and collision-fiber certificates, and a lossless update state. Their future quotients form a strict refinement chain on a reproducible five-game family. We then isolate the primary abstract mechanism. In a projection system containing a constant coordinate, a labelled collision certificate records the entire full relation whenever that relation has at least two elements. Consequently every nontrivial certificate fiber is an optional-point fiber: its hidden full relation is empty or a singleton. We derive an exact projection-shadow update formula, characterize full future certificate equivalence, and prove generator-local completeness: equality after each individual future layer already implies equality after every intersection of future layers. The empty mediator in the shogi carrier system realizes the constant coordinate. The chronological contract matters. Full-cone and next-only action families induce different quotients, with a standard-game witness separating them. At the lossless contract no state merging occurs in the audited family, yet the transition representation still factors exactly into schedule control and shared payload actions. Across five games and both sides, the audit contains 3,208 chronological states, 114,511 direct full-cone transitions, and 7,078 factored transition records. We separate the abstract proofs, the shogi realization, and the finite computation, and give counterexamples showing that the constant coordinate, labelled fibers, current-certificate equality, and intersection updates are genuine hypotheses.