Temporal Dynamics as Time-First General Relativity: k = 4G/c2 Linking Local Time Offsets, Curvature, and Cosmic Acceleration

We present Temporal Dynamics (TD) as a time-first reparameterization of General Relativity (GR). In TD, mass sources a local slow-time fraction Δ T(x); keeping the universal converter UC=1/c exact implies that the straight-path delay over a segment of length D corresponds to a curve-length increment Δ L=Δ T̄\,D. Fixing the single TD mass constant to k=4G/c² (m\,kg⁻¹) makes the TD↔GR bridge exact: Δ Lrad=rₛ, Δ Ldiam=2rₛ, and the TD horizon rule DBH=kM. The TD potential φ=-(c²/2)Δ T yields the general motion law a=(c²/2)∇Δ T, reproducing the Newtonian limit aᵣ=-GM/r². Taking Δ T(r)=rₛ/r reconstructs the Schwarzschild line element, so the PPN parameters match GR (β=γ=1) along with classic tests (gravitational redshift, light bending, Shapiro delay, perihelion precession). A variational formulation identifies Δ T with the ADM lapse via N²=1-Δ T; varying the action yields the Einstein equations with no extra propagating degrees of freedom (GR-equivalent dynamics, including gravitational waves with speed c and GR polarizations/flux). Cosmologically, a residual micro-push ε per clock unit defines κ=ε/c and recovers the Hubble law v=κ D; constant κ maps to Λ=3κ²/c², while a slow drift κ(z) corresponds to w(z)≠ -1, reframing the H₀ tension as differing ε₀ estimates. TD thus organizes tested GR physics with a single constant k and provides a micro→macro interpretation of dark energy via ε, without modifying GR’s empirical content.