Variational Structure of Hierarchical Interaction Theory: Lagrangian Framework for Modified Gravitational Dynamics

Following the fixation of the functional form g(r) for gravitational acceleration (DOI: 10.5281/zenodo.19824773), we present the variational structure from which this expression is derived. The framework is based on a scalar field Φ with a non-linear kinetic term, governed by a Lagrangian density ℒ_field. We provide the complete Lagrangian, the resulting field equation, and the particle Lagrangian that yields equations of motion consistent with solar-system tests. No dark matter or cosmological constant is introduced. This note serves to establish priority for the variational formulation.