Topological Renormalization of Spacetime: Extended f(R) Gravity, Bulk Viscosity, and the Dedekind Signature in Quark-Gluon Plasma

This paper proposes a definitive resolution to the historical obstruction in reconciling General Relativity with Quantum Mechanics by addressing the "continuum fallacy" (R³) and associated ultraviolet divergences. We introduce the Hernández-Valdivia Limit (ε_HV), not as a static constant, but as a dynamic topological stiffness coefficient within a quadratic Starobinsky-like gravity extension. Key findings include: • Asymptotic Safety: Information density saturates at the Ramanujan Packing Factor (Φ_Ram) at trans-Planckian scales, providing a natural ultraviolet regulator that prevents gravitational collapse. • Emergent Dark Energy: Dark Energy is derived as the emergent residual Bulk Viscosity (ζ_HV) arising from the geometric resistance of the vacuum lattice, naturally matching the observed Cosmological Constant. • Gödel-Stokes Paradox: Singularities in both Navier-Stokes and General Relativity are identified as logical manifestations of Gödel's Incompleteness Theorems applied to the non-physical assumption of infinite spatial divisibility (C∞). • Empirical Prediction: We present a falsifiable signature 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. The detection of this fine modular structure would constitute the first direct empirical evidence that spacetime operates as a discrete, finite information processing network.