Emergent Gravity from a Discrete Coordinate Network with a Presentist Temporal Metric: A Self-Consistent Derivation

Abstract

This preprint introduces a self-consistent framework for quantum gravity where spacetime emerges from a discrete network at the Planck scale. Instead of treating spacetime as a fundamental continuum, the model describes space as a connected weighted graph and time evolution as an irreversible stochastic process. By doing so, it naturally eliminates closed timelike curves, avoids non-renormalizable ultraviolet divergences, and resolves major causality paradoxes.Key HighlightsEmergent Spacetime: Derives the edge-length–potential relation from a minimal action principle.General Relativity Limit: Shows how stochastic graph dynamics naturally generate the Regge action, yielding the Einstein field equations in the continuum limit.Standard Model Integration: Connects graph degrees of freedom to fermionic matter and gauge interactions using the braid group \(B_{3}\), explaining the three generations of fermions and the mass hierarchy.Black Hole Thermodynamics: Reproduces the Schwarzschild metric, Bekenstein–Hawking entropy, and Hawking temperature.Information Paradox Resolution: Solves the paradox via a finite number of horizon degrees of freedom embedded in a unitary global evolution.No Sign Problem: Solves the fermion sign problem for 1+1D numerical simulations by mapping onto an integrable quantum XXZ Hamiltonian.Validation & PredictionsThe analytical model is validated through a numerical simulation of a 1+1 dimensional radial graph, matching theoretical predictions with a relative error under \(10^{-3}\). The theory provides four testable experimental predictions for strong gravitational fields:Gravitational-wave echoes featuring discrete phase shifts determined by the Jones polynomial.Phase decoherence of light over cosmological distances.Planck-scale corrections to light deflection angles.A neutrino spectrum break in the 1.2–3.5 PeV range.