The Covariant N-Max Framework: Riemannian Geometry and the Asymmetric Knuth–Moore Isomorphism

Multi-agent extensive-form games have historically defied efficient pruning due to the speculative nature of unconstrained, general-sum opponents. While static Euclidean algorithms like Paranoid establish lower bounds, they inherently distort the N-player Nash Equilibrium. In this paper, we resolve this topological limit by introducing Covariant Hypermax. By formally mapping the discrete search tree onto a κ-weighted Riemannian manifold, the multi-player zero-sum invariant induces a continuous geometric contraction of the evaluation space. As material mass (κ) decreases, the speculative shadow naturally decays until the geometry dictates a strictly bipartite Alpha-Beta isomorphism, mathematically anchoring the search complexity to an O(b^(d*(N-1)/N)) asymptotic limit. We prove that bounding opponents via this covariant metric allows the algorithm to safely operate at the absolute precipice of the geometric shadow, yielding an approximate pruning without violating the authentic N-player Nash Equilibrium. For example, empirical validation in 3-player chess demonstrated that against the classical max^N baseline, the Covariant framework evaluated 79.2x fewer nodes at depth d = 6, while systematically maintaining an 84.8% strategic win rate.