Fundamental Kinematics in Time-Scalar Field Theory (TSFT): Reconstructing Velocity, Momentum, Inertia, and Kinetic Energy from Temporal Curvature

Time-Scalar Field Theory (TSFT) treats time as a scalar field Θ(x) on a compact 4-manifold with two boundaries. The theory’s domain of validity is the open set where the scalar density ρΘ = |∇Θ|2 > 0. We prove that fundamental kinematic quantities—velocity, momentum, inertia, and kinetic energy—emerge directly from Θ-geometry, recovering Newtonian mechanics in the flat-Θ limit while providing a unified basis for relativistic and gravitational effects. Five theorems and supporting lemmas establish these derivations rigorously from theten TSFT axioms [1] with no additional assumptions. Empirical ties to photon interference [4] and in-lab accelerometer data (Farrell, 2023–2025) confirm the scalar coupling α = 1.