The Theory of Finite Systems: A Unified Framework for Fundamental Physics, Gravitation, and Fluid Dynamics under the Hernández-Valdivia Modular Paradigm

This paper establishes the Theory of Finite Systems as a definitive unified framework to resolve the non-renormalizable singularities and ultraviolet divergences of modern physics. We postulate that these anomalies are not physical failures but logical manifestations of the "Continuum Fallacy" and the "Gödel-Stokes Paradox". We introduce the Ramanujan-Hernández-Valdivia Limit (εHV ≈ 10⁻⁷⁰ m²) as a fundamental meta-axiom of discretization derived from Ramanujan's modular geometry. This limit acts as a universal regulator that: 1. Resolves the Navier-Stokes singularity through adaptive hyperviscosity and Hydrodynamic Cosmic Censorship. 2. Explains the Yang-Mills Mass Gap as a result of "topological friction" within the vacuum lattice. 3. Identifies Dark Energy as the emergent Bulk Viscosity (ζHV) of the superfluid vacuum. Finally, we present a falsifiable prediction for LHC Run 4: the existence of Landau-Ramanujan Oscillations in the vorticity of the Quark-Gluon Plasma, governed by the zeros of the Dedekind Eta function.