Gravity as Unitary Reflection: A Rigorous Discrete Formulation from First Principles

We present a rigorous derivation of the Specular Lattice framework, a discrete and unitary formulation of gravity from first principles. Starting from the Einstein-Hilbert action and Regge calculus, we derive the effective refractive index of spacetime and demonstrate that the geodesic equation is equivalent to a sequence of 4 X 4 Householder reflections. We prove explicit error bounds for this discrete approximation, showing uniform convergence, and explicitly construct the lattice Hamiltonian, wherein the spatial curvature term becomes a sum over Householder reflections on edges. By computing the Hawking temperature directly from the lattice dynamics via Bogoliubov transformations, we show the explicit equivalence between the Householder reflection and the two-mode squeeze operator. Furthermore, we establish a combinatorial derivation of the Bekenstein-Hawking area law from the number of Bell pairs on the boundary, connecting the discrete framework to the ER=EPR conjecture. We address the restoration of diffeomorphism invariance in the continuum limit and the high-energy suppression of Lorentz invariance breaking. Crucially, the framework provides a unitary evolution perspective that resolves the black hole information paradox: the explicit construction of a unitary S-matrix demonstrates that the final radiation state remains pure, while tracing over interior degrees of freedom yields a thermal density matrix for an external observer. We validate this by numerically reproducing the Page curve. Finally, we integrate matter fields into the lattice and provide numerical validation for Schwarzschild and Kerr metrics, achieving convergence to machine precision.