From a discrete constraint model to general relativity: what survives

The Discrete Emergent Gravity (DEG) model postulates N “spacetime atoms”, each with a size variable and a momentum, bound by one global constraint (the total energy H = 0). In this form the atoms never interact. This companion paper checks the model step by step against general relativity (GR). (i) The bare postulate is a consistent, reparametrisation-invariant system; matter prescribed as a function of the evolution parameter is not.(ii) If each atom’s squared size is read as the volume of a cell, each atom is exactly a flat Friedmann–Lemaître–Robertson–Walker (FRW) patch with a cosmological constant and dust: the evolution parameter becomes cosmic time, and each atom’s energy is its dust energy.(iii) Connecting cells so that a mass attracts its neighbours works, and reproduces Newtonian gravity, the shell theorem and the turnaround radius exactly as in GR (Lemaître–Tolman–Bondi dust), but only with one local constraint per cell.(iv) Any version of the single global constraint forces the total dark (or total) mass to vanish, which is incompatible with the observed positive dark-matter density. No new physics is claimed beyond the explicit dictionary and these negative results. This paper accompanies, and corrects parts of, the core paper “Discrete Emergent Gravity: Time, Entropy, and Matter Coupling from a Single Constraint” (version 1.1).