An intrinsic kinematic relation $\boldsymbol{c=\frac{c_{0}}{H_{0}}\,\dot{a}}\,$ inferred from Type Ia supernovae

Based on the Hubble diagram of SNeIa, we present empirical evidence for a kinematic relation between the speed of light and the late-time cosmic expansion rate. To infer this relation, we employ the Dolgov-Barrow cosmology, described by Dolgov's power-law expansion $a=(t/t_0)^\mu$ and Barrow's varying-speed-of-light (VSL) $c=c_0\,a^{-\zeta}$. In this cosmology, light propagating through an expanding cosmic background undergoes an additional refraction induced by the variation of $c$ along its path, resulting in a modified Lema\^itre redshift relation $1+z=a^{-(1+\zeta)}$. The model yields a high-quality fit to the Pantheon SNeIa Hubble diagram and exhibits a remarkably tight posterior degeneracy along the locus $(1+\zeta)\,\mu=1$. In particular, the case ${\mu=2/3,\zeta=1/2}$, referred to as the VSL-Einstein-de Sitter case, is favored over flat $\Lambda$CDM by $\Delta\chi^2=2.8$, i.e. at 68% confidence level, despite having the same number of free parameters. The empirical relation $(1+\zeta)\,\mu=1$ entails that the speed of light is exactly proportional to the cosmic expansion rate, $c=c_0H_0^{-1}\,\dot a$, during late times, a synchronous behavior absent in the standard $\Lambda$CDM model. Although the empirical relation $(1+\zeta)\,\mu=1$ is inferred using the Dolgov-Barrow parameterization, the resulting expression $c=c_0H_0^{-1}\,\dot a$ is an intrinsic relation because it relates two physical quantities of the same dimension: speed of light and cosmic expansion rate. It is therefore independent of arbitrary choices of units or parameterization and encodes a purely kinematic correspondence between $c$ and $\dot a$. If confirmed by independent probes, this relation may point toward a more general kinematic principle governing late-time cosmic evolution. We discuss implications of this relation for late-time cosmology, including an alternative interpretation of cosmic acceleration.