Sigmazeta Infinity Omega Protocol Maximum-Strength Closure of the Apex Dominance Criterion

We develop a maximum-strength mathematical closure of the Apex Dominance Criterionwithin Strategic Systems Theory. The framework formalizes structural dominance in multiagent environments through five principal components: operational power, multiplicativestacking, information asymmetry, layered arena coupling, and systemic resilience. We introduce the canonical dominance functional, dominance-gap geometry, perturbation envelopes,spectral strategic operators, minimax adversarial closure, entropy concentration, Lyapunovreinforcement, and mean-field limits. These culminate in the Sigmazeta Infinity OmegaProtocol: a unified hierarchy of dominance conditions whose top level, the Diamond ApexClosure, provides sufficient conditions for unique, robust, spectral, adversarially stable, andasymptotically persistent apex control within the model. The resulting architecture yieldsa domain-agnostic formalism for strategic competition across organizational, technological,economic, and institutional systems