A_Finite_Relational_Lattice_Framework

Abstract

Continuous field theories encounter ultraviolet divergences and metric singularities (r -> 0), necessitating ad-hoc renormalization counterterms and unphysical boundary cutoffs. Here, we present a finite relational lattice framework defined by an invariant minimum spatial separation a_0 = l_P and finite causal signal latency c. We demonstrate analytically and numerically that: (1) The gravitational potential and electrostatic field energy remain strictly regular at the origin, eliminating central divergences. (2) A relational Action Principle rigorously derives effective post-Newtonian equations of motion, accurately reproducing relativistic orbital precession and Lense-Thirring frame-dragging around Kerr sources. (3) Discretizing the radial Hamiltonian preserves the exact hydrogen Rydberg spectrum. (4) Modified Friedmann dynamics resolve cosmological singularities into a finite-density bounce. (5) Null-geodesic raytracing establishes the critical photon capture cross-section consistent with Event Horizon Telescope (EHT) observations, yielding a testable prediction for post-merger gravitational wave cavity echoes.