Volume XI (V): The Einsteinian Emergence — Numerical Verification of the C8 Gravitational Metric

A numerical proof of principle for Open Problem 2 (OP-2) of the Network Emergent Architecture (N.E.A.) framework: the emergence of Einstein gravity from the continuum limit of a discrete C8 cubic causal graph. Without presupposing any field equations or force laws, we perform three independent numerical experiments on a C8 lattice: (i) Laplacian relaxation around a central addressing-deficit source, (ii) ray-tracing of Stride-1 pulses through a rent-gradient field, and (iii) topological audit of the bipartite sublattice structure. The experiments yield:- The 1/r^2 gravitational force law with a pointwise slope of -2.0052 (error 0.26% at r=7, where the window r≈L/4 cancels near-field rigidity and far-field boundary compression across all tested lattice sizes).- Factor-of-2 light deflection angle from dual-protocol summation, matching the classic general-relativistic lensing prediction without any preset factor.- Topological origin of the 8π coupling constant from the double bookkeeping of bipartite sublattices, explaining why general relativity requires 8π as the necessary topological invariant of the C8 scaffold. Segmental slope auditing reveals the complete four-stage lifecycle of gravity emergence: near-field rigidity, catch-up compensation, the Einsteinian cruising regime, and boundary-induced horizon compression. OP-2 is numerically validated at the mechanism level. The remaining four analytic theorems are delineated as a well-posed 12–18 month formalisation programme whose boundary conditions, normalisation constants, and convergence targets are now empirically fixed by the spectroscopic audits reported in the companion paper (Volume X (Q)). All numerical code, output data, and reproduction instructions are publicly archived in the Lattice repository.