Residual-Checker Hypergraphs and Distinguished-Checker Decomposition in Finite Legal Evasion Systems of Shogi

We study one-ply legal evasion systems in shogi after a checking move has already been fixed. For every geometrically admissible defensive candidate, we retain more than a Boolean legal/illegal verdict: if the candidate leaves the defender's king attacked, we attach the complete nonempty set of surviving checker squares. If the king is safe but the candidate is an immediate mating pawn drop, we attach a separate terminal colour \(\), not a checker square. These typed candidate-wise residues distinguish a geometrically empty mate from a mate in which a nonempty candidate domain is completely obstructed. For a position with a unique primary checker \(c\), the self-check residues form a finite hypergraph. Shogi rules force an ancestry asymmetry before any hypergraph dualization is performed: capturing \(c\), or strictly interposing on its checking ray, removes \(c\) from the residual checker set, whereas a pure king move may leave \(c\) active at the new king square. We exploit this partition to give an exact decomposition of inclusion-minimal residual obstruction cores into those containing \(c\) and those avoiding it. The result yields three mutually exclusive certificate modes: primary-only, secondary-only, and bifurcating. We give both a direct minimal-transversal proof and an equivalent monotone-Boolean proof. A singleton terminal-colour factorization then separates pawn-drop-mate obstruction from surviving-attacker ancestry. The theory is accompanied by legal-shogi witnesses and finite audits. A 24-ply legal sequence from the standard initial position ends in a mate whose geometric evasion domain is empty. A separate ordinary-rule construction exhibits a pawn-drop interposition that neutralizes the original rook check but is illegal by the pawn-drop-mate rule because the checking rook is absolutely pinned from recapturing the pawn. A deterministic hirate-reachable family contains 393 unique-checker all-obstructed mates: 259 primary-only, 88 secondary-only, and 46 bifurcating. An independent nine-record corpus contributes 18 further mates, and exhaustive abstract verification checks 131,474 distinguished-vertex decompositions and 32,906 terminal-colour factorizations on all hypergraphs with at most four checker vertices.