Poisson Spacetime Discreteness and Unimodular Gravity: Consistency Conditions and Exact Past-Cone Statistics of a Fluctuating Cosmological Term in the TSGT Framework

Abstract

Abstract Can Poisson discreteness of spacetime supply gravitational dynamics or a fluctuating cosmological term? We examine this question for the framework previously called Topological String-Gradient Theory (TSGT) and then make its statistical proposal quantitative. A random counting measure, its metric-dependent mean and an independently prescribed volume form are distinct objects: the number–volume relation fixes neither the metric nor its dynamics, as a two-potential weak-field counterexample shows. An Einstein–Hilbert action with a prescribed volume form is a consistent completion, but with separately conserved matter its cosmological term is constant. We therefore analyse an explicitly assumed marked Poisson model of the everpresent-Λ type on fixed backgrounds. Besides the overlap covariance, we prove that the term is a reverse martingale along nested regions for any mark distribution, and that in the diffusion limit its probability of keeping one sign follows Lévy's arcsine law, independently of the amplitude. For the flat ΛCDM benchmark the past light cone has four-volume 0.1230 (c/H0)^4, so the rms dark-energy density is 0.950 α times the critical density and the observed value is a 0.721/α standard-deviation draw. Conditional on that value, the spread at redshift z = 1 is 4.93 α times the critical density; present typicality (α ≳ 0.36) and approximate constancy to z ≈ 1 (α ≲ 0.014) are therefore incompatible on this background. For the equal-time spatial correlation generated by overlapping past cones we derive a closed form for power-law expansion, the universal small-separation law 1 − Corr = r/(ct) + O(r³) and its ΛCDM analogue. The resulting structure function is fixed by geometry alone, independently of the amplitude α; thus no choice of α makes a local realization both typical today and spatially smooth. All results are checked numerically, including genuine (3+1)-dimensional sprinklings. They are conditional benchmarks, not a derivation of general relativity or a dark-energy prediction, and they specify what a viable completion must change. About this revision This version keeps the consistency analysis of the previous version and adds quantitative results for the marked-Poisson (everpresent-Λ type) cosmological term on fixed backgrounds: the reverse-martingale property, the arcsine sign-persistence law, the past-light-cone coefficient of the flat ΛCDM benchmark, the conditional history and the resulting constancy tension, and a closed-form equal-time spatial correlation. The earlier statement that a fluctuating Λ yields no well-defined observed w(z) is corrected. Prior work is attributed explicitly (Sorkin; Ahmed, Dodelson, Greene and Sorkin; Barrow; Zuntz; Zwane, Afshordi and Sorkin; Das, Nasiri and Yazdi; Josset, Perez and Sudarsky), bibliographic data were updated, and strong-field statements of earlier versions are collected in an appendix. Files main.pdf: manuscript. main.tex: single-file LaTeX source (figures are drawn from coordinates in the source). verify.py: reproduces every quoted number and plotted coordinate and fails if the manuscript and the computation disagree (Python 3.10+, NumPy, SciPy, fixed seed). verification_report.txt and verification_results.json: actual outputs (40 checks passed). requirements.txt: Python dependencies. Preparation and scope Author: Mustafa Karatum, Independent Researcher. This revision was prepared with AI assistance (Anthropic Claude) for mathematical checking, literature and source checking, verification code, and editing, building on earlier Claude- and Codex-assisted revisions. AI assistance is not an independent peer review. No observational data were reanalysed.