Sparse Query Compilation for Reverse-Cyclic Multiplayer Nim

Exact symbolic reasoning in multiplayer games must preserve more than a binary win/loss outcome. We study the phase-zero reverse-cyclic recurrence for Nim, representing a background position by its response to one additional heap. Let $m\ge 3$ be the number of cyclic outcome classes and let $q\ge 0$ bound the background mass above $2m$. For backgrounds with $p$ distinct heap heights below $m$ and binary-encoded multiplicities, we give a uniform deterministic algorithm for exact response queries. A shielding lemma, arithmetic descriptions of unblocked coordinates, and an existential rank-history identity compile each threshold query into $(p+q+2{)}^{O(q+1)}$ integer-linear systems, each with at most $2q+1$ variables. Fixed-dimensional integer programming yields polynomial bit time for every fixed $q$, with exact values, requested positive-label coordinates or absence, and preference-maximizing actions. The bound is XP in upper excess, not an FPT result in $q$ alone. The construction is supported by a sparse-response theorem and an explicit obstruction to composing response signatures. Definition-level tests and large-integer checks corroborate the interfaces; existing implementations cover restricted layers rather than the full uniform algorithm. Our results separate compact representation, composition, and exact inference in a multi-valued sequential problem.