Rigorous Derivation of the Einstein Field Equations from a Discrete Information Network Substrate

Abstract:

This document provides a fully explicit, non-circular, line-by-line mathematical derivation of the continuous Einstein Field Equations (EFE) starting from a discrete, undirected graph substrate. Addressing long-standing challenges in discrete spacetime models, this derivation successfully bridges the gap between quantum information theory and macroscopic General Relativity without relying on pre-existing background manifolds or continuous coordinates.

Methodology & Key Derivations:

The framework models the universe as a dynamic network of finite-capacity nodes. By defining "mass" as localized Information Debt (the Kullback-Leibler divergence between actual and ideal packet distributions) and routing information via optimal transport (Wasserstein-1 distance), the derivation proceeds through the following rigorous steps:

1. Emergent Metric & Signature: The local metric is extracted via the symmetric anti-commutator trace of a $U(N)$ matrix algebra. The physical Lorentzian signature $(-, +, +, +)$ is natively recovered through the algebraic structure of the temporal anti-Hermitian generators, bypassing the need for ad hoc Wick rotations.

2. Discrete-to-Continuum Limit: The discrete Ollivier-Ricci curvature is expanded in the Gromov-Hausdorff limit to map the discrete graph Laplacian to the continuous Laplace-Beltrami operator, meticulously tracking the volume deviation of geodesic balls to scale information density.

3. Variational Principle: The macroscopic dynamics are governed by the minimization of the Graph Dirichlet Energy. The explicit Taylor expansion of the entropy term organically yields the Einstein-Hilbert action and the cosmological constant.

4. The Stress-Energy Tensor: By applying standard metric variation to the emergent matter Lagrangian, the optimal transport of Information Debt is mathematically proven to generate a stress-energy tensor identical to that of a classical massless scalar field.

Context:

This derivation serves as a foundational mathematical proof for the Graph-cMERA Unified Ontology (Null Orientation Framework). It demonstrates that the kinematics of General Relativity are the inevitable, emergent consequence of a discrete network minimizing its structural information debt.