Time-Scalar Field Theory and Quantum Mechanics: Schrodinger, Dirac, and Klein-Gordon Equations from Scalar-Time Continuity

We derive the Schr¨odinger, Dirac, and Klein-Gordon equations from the scalar-time continuity law of Time-Scalar Field Theory (TSFT). By promoting evolution to the scalar-time parameter Θ and introducing a local time-rate field α(x), classical Hamiltonian and quantum operator dynamics unify under the relation dO
dΘ = α{O,H} (classical) and dO dΘ = i¯hα[ ˆH ,O] (quantum). Schr¨odinger’s equation follows from ∂Θ = α∂t; the Dirac equation emerges from a Θ-weighted tetrad with a spin-connection term ∝ ∂μ ln α; and the Klein-Gordon equation arises as the relativistic scalar field equation in the effective metric induced by Θ. Standard
quantum mechanics is recovered when ∇α = 0, while TSFT predicts falsifiable gradient corrections accessible to precision matter-wave and spinor interferometry. We supply quantitative estimates (e.g. δϕ ∼ 10−3 rad for Q = 109 cavities) and cross-link TSFT’s microphysics to its cosmological continuity, emphasizing multi-scale  coherence and experiment-first falsifiability.