Mathematical Formulation and Lagrangian Structure of the Distributed Density Equilibrium Theory

This paper presents the full mathematical formulation and Lagrangian structure of the Distributed Density Equilibrium Theory (DDET), a unifying field theory based on the internal-to-structural energy ratio R = ρ / γ. Building upon prior containment and vessel-based models, the framework is extended to include formal field equations, gradient-derived disequilibrium terms, and a scaleinvariant action integral. The paper derives Lagrangian densities for scalar fields, vector redistributions, and coupling terms with curvature and memory dynamics. Additional tests include path integral quantization, anomaly cancellation under gauge and gravitational transformations, and a cross-scale consistency analysis spanning atomic, biological, planetary, and cosmological systems. A universal collapse threshold at R ≈ 0.1 is confirmed across simulations and domainspecific applications. Numerical modules and empirical protocols reinforce the theory’s predictive capacity, renormalizability, and falsifiability. By preserving structural continuity without requiring dualities or hidden dimensions, DDET emerges as a mathematically complete, observationally testable, and physically coherent alternative to current unification models.