Contemporary physical theory is formulated predominantly in an equation-first manner, identifying physical law with differential equations and their solutions. This paradigm has proven extraordinarily successful in regimes where admissible states are weakly constrained, coefficients are uniform, and global solutions exist. However, it becomes structurally inadequate in regimes characterized by strong admissibility constraints, nonuniform topology, or histories that preclude global closed-form evolution. This work presents an operator-first physical framework in which admissibility precedes dynamics. Physical law is defined by constraints on allowable transformations acting on a coherence configuration space, and evolution is constructed as a product limit of admissible micro-transformations. Differential equations arise only as effective descriptions in regimes where admissibility constraints simplify and generators commute. The framework introduces a coherence-based admissible state space, topological sector structure, admissible generator families, and a monotone functional enforcing global directionality. Effective generators provide regime-dependent summaries of admissible evolution, while their resolvents encode accessibility and govern observable response. Measurement is interpreted as a probe of accessibility rather than of intrinsic state variables, yielding testable predictions including resonance structure, hysteresis, polarity inversion, and inertial modulation. The formulation is mathematically explicit and falsifiable. Conventional equation-based theories, including those of the Standard Model, emerge as special cases within the operator-first framework rather than as fundamental starting points. The result is a strictly more general procedural foundation for physical law.