The concept of nimbers - a.k.a. Grundy-values or nim-values - is fundamental to combinatorial game theory. Beyond the winnability, nimbers provide a complete characterization of strategic interactions among impartial games in disjunctive sums. In this paper, we consider nimber-preserving reductions among impartial games, which enhance the winnability-preserving reductions in traditional computational characterizations of combinatorial games. We prove that Generalized Geography is complete for the natural class, ℐ^P, of polynomially-short impartial rulesets, under polynomial-time nimber-preserving reductions. We refer to this notion of completeness as Sprague-Grundy-completeness. In contrast, we also show that not every PSPACE-complete ruleset in ℐ^P is Sprague-Grundy-complete for ℐ^P.

By viewing every impartial game as an encoding of its nimber - a succinct game secret richer than its winnability alone - our technical result establishes the following striking cryptography-inspired homomorphic theorem: Despite the PSPACE-completeness of nimber computation for ℐ^P, there exists a polynomial-time algorithm to construct, for any pair of games G₁, G₂ in ℐ^P, a Generalized Geography game G satisfying: nimber(G) = nimber(G₁) ⊕ nimber(G₂).