Non-Archimedean Stochastic Fixed-Point Theory Via Sheaf Methods

This work develops a novel framework for non-Archimedean stochastic fixed-point theory through a synthesis of nonlinear functional analysis with sheaf-theoretic probability, establishing new dynamical principles in exotic Banach spaces. By introducing Choquet-capacity-valued measures, we prove a random Borsuk-Ulam theorem for p-adic operators that reveals a fundamental connection between tropical convexity and quantum gravity via a stochastic holographic principle. Our main results include: (1) the constructive existence of forcing-measurable Nash equilibria, (2) complete resolution of non-separable ordinal games, and (3) the discovery of non-ergodic chaos phenomena in ℓ∞/c0 spaces where fixed points become ZFC-independent. The theory yields immediate applications across multiple disciplines: in probabilistic quantum computing, it provides p-adic optimization techniques for noise-resistant algorithms; for infinite-strategy game theory, it offers computable methods for high-dimensional equilibrium problems in economics and artificial intelligence; in turbulence modeling, it enables new approaches to unresolved scales through non-Archimedean stochastic PDEs; and for holographic quantum gravity, it establishes a numerical framework for emergent spacetime geometries. The framework incorporates constructive approximation schemes for fixed points, with supporting computational experiments that validate key conjectures in physically relevant low-dimensional cases. These advances collectively position nonlinear probability as a transformative paradigm bridging theoretical mathematics with computational physics.