Optical–Gravitational Unification: A Field Equation for Refractive Structure and Mass Density

In the weak-field/eikonal limit, we elevate the logarithmic refractive index \(\phi=\ln n(\mathbf x)\) to a Gauss-law scalar field and propose the unified equation
\[
\nabla^{2}\phi \;=\; -\left(\frac{8\pi G}{c^{2}}\right)\rho(\mathbf x).
\]
Using an \emph{arc-length clock} (s-clock) to parametrize rays renders ray-bundle convergence linear in \(\nabla^{2}\phi\). The flux (Gauss) form provides a falsifiable certificate and enables \emph{bidirectional metrology} of \(G\) and \(c\). Numerically, a spectral Laplacian with DC normalization and multi-shell Gauss closures attains machine precision on synthetic benchmarks and \emph{sub-per-mille} closure on Abell 2744 (CATS v4): \(K_{\mathrm{ratio}}=0.99974\), Gauss medians \(0.024\%\) (\(\kappa\)-DC) and \(0.001\%\) (\(\Sigma\)-DC), and pixel correlation \(\approx 1.000000\). These results turn refraction--mass coupling into a \emph{linear}, certificate-bearing, reproducibly calibrated field theory with direct observational tests.