Invariant Speed of Light in a Physical Medium: A Three-Dimensional PTM Derivation of Local c Invariance, Clock Dilation, and Gravitational Redshift

This work presents a three-dimensional material-coordinate derivation of local invariant propagation speed c within the Physical Time Medium (PTM) framework and connects the same material deformation to gravitational clock dilation. In the retained PTM longitudinal sector, the reference-coordinate characteristic dX/dt = ±c/J transforms through the material map to dx/dt = v ± c. Relative to a materially embedded observer moving with the medium, the measured local characteristic is therefore ±c. The result is extended to arbitrary nonsingular local three-dimensional deformation using the full deformation gradient F, yielding a physical material-relative characteristic magnitude |c_phys| = c. The same framework gives a propagation-defined clock with frequency ν ∝ 1/J ∝ ρ. Combined with the PTM potential Φ = c² ln(ρ/ρ∞), this produces d ln ν/dr = g/c² and the weak gravitational clock relation Δν/ν ≈ ΔΦ/c² ≈ gΔh/c². The derivation does not assume a Lorentz transformation and does not claim a completed derivation of universal Lorentz covariance, all bound-matter clocks, or the strong-field clock law. These remain explicit tests of the broader PTM framework.