Energy‐Constrained Kinematics of Earthquake Rupture on Extended Faults: A Physical–Mathematical Framework

Earthquake rupture is a spatially extended, time‐dependent process governed by the interplay between rupture kinematics and energy partitioning along the fault. Although kinematic source models and energetic formulations are widely used, their local physical consistency is typically not enforced explicitly. Here, a physical–mathematical framework is introduced that links rupture geometry, kinematic fields, and local energy balance within a unified spatial–temporal–energetic description of extended seismic sources. Rupture propagation is described through a spatially defined rupture‐time field, from which rupture‐front velocity is derived as the inverse gradient of rupture time. Local energy consistency is imposed through a balance between energy release rate, fracture energy, and radiated energy, leading to a formulation in which rupture velocity and radiative efficiency are locally constrained by a common energy balance. Within this framework, kinematic rupture models can be interpreted as implicitly defining a local energy budget. A synthetic diagnostic example demonstrates that not all kinematically admissible rupture‐time fields correspond to energetically plausible configurations, and that strongly heterogeneous rupture‐velocity patterns may imply violations of local energy‐balance conditions. The proposed formulation does not provide direct constraints for inversion, but establishes necessary conditions for physical admissibility that are independent of specific constitutive laws or numerical implementations. By making explicit the energetic assumptions embedded in kinematic descriptions, this framework provides a physically grounded basis for interpreting rupture heterogeneity, radiated energy variability, and source‐process observations.