Topological String-Gradient Theory: Metric Replication of Space-Time and Cosmological Evolution

Abstract

We present the Topological String-Gradient Theory (TSGT), a framework in which space-time is discrete at the Planck scale and gravity is the gradient of the spacing between discreteelements (“pixels”). Two kinematic postulates—a spatial pixel strain set by the Newtonianpotential and an equal temporal strain—reproduce the weak-field Schwarzschild metric, includ-ing the full light deflection ∆θ = 4GM/c2b; this reproduction is by construction. The discretesubstrate is a random Poisson sprinkling of density L−4p , which preserves Lorentz invariancestatistically and removes the preferred-frame problems of lattice formulations; lattice-derivedpredictions of earlier versions are retracted. If cosmic expansion creates new pixels ratherthan stretching existing ones—the only option compatible with an invariant Lp—then Poissonfluctuations in the pixel count fix the cosmological term to ρΛ ≃ 2.8 × 10−27 kg m−3, withina factor of two of the observed value; this argument is Sorkin’s heuristic, following herefrom TSGT’s own postulates. Written locally, the number–volume correspondence is theunimodular constraint; enforcing it in an Einstein–Hilbert action yields the trace-free Einsteinequations, making TSGT provably equivalent to General Relativity at the classical level.The framework’s independent content lies in the integer character of the pixel count and inthe fluctuating dark-energy term it implies (w̸ = −1). Unresolved problems, including theabsence of a dynamical equation for the pixel density, are listed explicitly