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

The Renormalized Free-Energy Principle (RFEP) establishes the mathematical foundation of Functional Stability Theory (FST). It formalizes how renormalized energy functionals detect and resolve structural bottlenecks across number theory (Riemann Hypothesis), cosmology (Dark Energy), fluid mechanics (Navier–Stokes), and complexity theory (P vs NP). The framework identifies a single structural challenge underlying all these problems: Functional Positivity under Gauge Constraint (Pattern A). Three ontological layers are distinguished: (1) the abstract RFEP as a universal stability meta-rule; (2) the Dissipative Selection Principle (DS1–DS3) as the mechanism by which physical evolution selects the unique dissipation-minimising attractor; and (3) domain-specific instantiations as concrete verifications. Domain Proof Papers — Current Status (March 2026) PaperVersionStatusOpen Problem Turbulencev1.3Journal-readyDFC1 empirical (only input) Dark Energyv1.6Framework NoteHu-Sawicki parameters quantitatively open Yang-Millsv2.1ConditionalAnalytical proof of λ < 0 Navier-Stokesv2.1ConditionalAssumption G₂ (projection regularity) NS Log-Distancev1.3Proof of Life ✓TLL for 3D NS analytically open BSDv1.1ReformulationHigher Gross-Zagier (rank ≥ 2) Hodgev1.1No-Go Theorem= Deligne's question (1982) P vs NPv1.2ReformulationUniformity Bridge Proof Architecture Framework (Pattern A) ├── DS1-DS3 (Dissipative Selection) └── Second-Order Resolvent Dominance │ ┌──────────┼──────────┐ │ │ │ PROVEN CONDITIONAL OPEN (rigorous) (reduced to (= open threshold) research) │ │ │ TU: YM: BSD: DFC⇒NL' Doeblin Rank ≥ 2 (journal) α(β)>0 Gross-Zagier │ DE: NS: Hodge: Screening G₂ (proj.) Deligne's (valid.) G₃ (Gronw.) question │ NS-LDI: PvNP: TLL+LDI Uniformity (Lorenz✓) Bridge Published in English and German versions. Source code: functional-stability-theory (framework/ directory). Changes in Version 1.6 (March 2026) Major: Pattern A falsifiability criterion (PA1–PA3): explicit falsification via first-order resolution Changes in Version 1.5 (March 2026) Major: Dissipative Selection Principle (DS1–DS3) as formal meta-theorem Major: Pattern B (Flow Selection) for Dark Energy; DS3 failure examples Note: This record contains legacy files (FST-GP, FST-Meta, FST-UnifiedFramework) from earlier versions. The current paper is FST_Unified_Renormalized_Energy_Framework (EN+DE).