Minimal Sparse Pairwise Completion Makes Every Higher-Order Payoff Layer Globally Outcome-Flexible: Sharp Edge Counts and Exact Evolutionary-Stress Certificates

This paper establishes sharp sparse-completion theorems for evolutionary games with arbitrary finite higher-order interactions. It asks whether a prescribed and highly sparse layer of pairwise payoffs can determine global evolutionary behavior even when the higher-order payoff structure is fixed in advance. For every number of strategies at least three and every prescribed interior equilibrium, the paper constructs a pairwise payoff completion using exactly n minus one adjustable coefficients. This minimally sparse completion can produce globally attracting, permanent evolutionary dynamics and a hyperbolic evolutionarily stable state. Remarkably, the same signed support pattern can also produce a globally attracting or permanent hyperbolic state that is strictly not evolutionarily stable. Thus, interaction topology and signs alone do not determine the relationship between ecological attraction and evolutionary stability. A rank obstruction proves that fewer than n minus one raw pairwise coefficients cannot provide this universal flexibility. At the minimum support size, the paper classifies the possible unsigned support architectures in the pure affine setting. Only two support types can produce a strict evolutionarily stable state, and only the broom support can also support the contrasting non-ESS construction. Additional obstruction theorems show that a prescribed tree skeleton requires at least n minus two self-loops and that a loopless tree cannot support the required strict evolutionary stability. A superpattern result extends the construction to larger supports containing the signed broom configuration. These results show that an arbitrarily complicated finite higher-order evolutionary environment can be globally redirected by a linearly small and provably minimal pairwise layer. They also identify precise limitations on what can be inferred from sparse interaction networks without quantitative payoff information. The paper is accompanied by exact rational certificates, independent verification code, automated corruption tests, unit tests, and a complete reproducibility workflow.