The Renormalized Free Energy Framework: A Unified Resolution of Structural Bottlenecks through Gauge-Theoretic Reformulation

We present a unified mathematical framework for the resolution of structural bottlenecks across number theory (Riemann Hypothesis), cosmology (Curvature Relaxation), and fluid mechanics (Navier-Stokes). The framework distinguishes between three ontological layers: (1) Proven results, specifically the gauge-theoretic stability of arithmetic kernels (FST-RH) as a completed reference instantiation; (2) the Abstracted Renormalized Free-Energy Principle (RFEP) as a universal stability meta-rule; and (3) domain-specific Instantiations (e.g., Navier-Stokes regularity) as physical verifications. We demonstrate that the great unsolved problems are instances of a single structural challenge: Functional Positivity under Gauge Constraint (Pattern A). By identifying the universal "Null-Tail" property, we reduce infinite-dimensional stability problems to finite-dimensional certificates. The FST-RH instantiation constitutes a completed proof of the Riemann Hypothesis within this framework, serving as the reference implementation. Published in English and German versions.