Poisson Spacetime Discreteness and Unimodular Gravity: A Consistency Analysis of the TSGT Framework

Abstract

Abstract Can Poisson discreteness of spacetime supply gravitational dynamics or a fluctuating cosmological term? We examine this question within the framework previously called Topological String-Gradient Theory (TSGT). We distinguish a random counting measure, its metric-dependent mean, and an independently prescribed volume form. The Poisson number-volume relation fixes neither the metric nor its dynamics; a two-potential weak-field counterexample makes this underdetermination explicit. An Einstein-Hilbert action with a prescribed volume form provides a consistent continuum completion, but its equations require a constant cosmological term when matter is separately conserved. A time-dependent term therefore needs additional dynamical structure. To make the statistical proposal precise without claiming such a completion, we analyze an explicitly assumed marked Poisson model on a fixed background. Its covariance is determined by the four-volume of overlap of the sampling regions, and its normalized diffusion limit has an exponential correlation in logarithmic volume. We also evaluate past-light-cone volumes in power-law cosmologies, exposing the geometric dependence hidden by the replacement of four-volume with the fourth power of the Hubble radius. These results yield reproducible consistency checks and a conditional statistical benchmark, rather than a new derivation of general relativity or a parameter-free dark-energy prediction. The required conservation law, backreaction, spatial perturbations, and statistical comparison with observations are specified as the remaining steps toward a testable model. About this revision This version substantially reframes the earlier TSGT manuscript as a consistency analysis. It corrects the distinction between Poisson counting and a prescribed volume constraint, the conservation condition for a cosmological term, and unsupported singularity-avoidance claims. It presents four propositions, an explicit overlap covariance, a Gaussian diffusion limit, and exact past-light-cone volumes for specified power-law cosmologies. The everpresent-Lambda motivation is attributed to the existing literature. These conditional results are not presented as a complete microscopic theory or an observational confirmation. Preparation and scope Author: Mustafa Karatum, Independent Researcher. This revision was prepared with OpenAI Codex assistance for mathematical checking, source checking, code preparation and editing, following an earlier Claude-assisted revision. AI assistance is not an independent peer review. No observational data were reanalyzed.