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

This paper presents a three-dimensional material-coordinate derivation of local propagation-speed invariance for an observer physically embedded in a continuous physical substrate. Starting from a barotropic continuum description, mass conservation, and an exact finite-deformation longitudinal energy, the one-dimensional characteristic dX/dt=\pm c/J maps into physical coordinates as dx/dt=v\pm c, giving a material-relative characteristic speed of exactly c. The derivation is extended to three dimensions using the full deformation gradient F. Under arbitrary nonsingular local deformation, the material-coordinate characteristic maps back to physical space with magnitude c, demonstrating that the cancellation is tensorial rather than dependent on isotropic deformation. The same material description is used to construct a propagation-defined clock. Its frequency scales with local substrate density, and when combined with the retained density-defined gravitational potential, the model yields the weak gravitational clock-gradient relation d\ln\nu/dr=g/c^2, or \Delta\nu/\nu\approx\Delta\Phi/c^2\approx g\Delta h/c^2. Modern electromagnetic-isotropy and gravitational-redshift measurements are used as external benchmarks rather than inputs to the derivation. The paper also identifies explicit limits of the present result, including bound-matter clock composability, moving bound-system dynamics, and the universal strong-field clock response.