Physical Law From Structural Coherence

Modern physics relies on several highly successful mathematical frameworks — quantum theory, gauge theory, general relativity, and thermodynamics. These theories describe nature extremely well, yet their mathematical forms are usually introduced through independent postulates. This work asks a prior question: why do these mathematical structures appear at all? The framework developed here begins from two minimal structural requirements for persistence:coherent return (self-consistency under transformation) and triadic closure (the minimal structure required for such verification). Within clearly stated domain restrictions — independent composability, smooth relational locality, and finite resolution — these postulates impose strong constraints on how physical systems can be represented. From these structural constraints, several familiar elements of modern physics arise as minimal admissible forms: • complex Hilbert space and unitary quantum evolution• quadratic weighting (Born rule)• relational geometry and Einstein-type gravitational response• irreversible temporal ordering from non-invertible return• entropy and the Second Law from indistinguishability growth• compact unitary gauge structure The framework does not attempt to predict parameter values or replace existing physical theories. Instead, it explains why the mathematical forms used by those theories are structurally admissible once coherent persistence under finite resolution is required. As a numerical application, the Koide relation for charged-lepton masses is shown to arise as a geometric consequence of triadic equilibrium in quadratic mass-amplitude space, conditional on the charged-lepton family realizing a coherent triadic structure. The result is a structural program for physics: rather than proposing new dynamical laws, it constrains the mathematical forms that viable physical laws may take.