This paper presents the mathematical foundation of the Discrete Gravitational Ontology (DGO)— a unified framework linking quantum mechanics, general relativity, and the arrow of timethrough a discrete relational substrate. Building on three prior works (Asplind, 2025a–c), thispaper formalizes DGO using coarse-graining, Gamma-convergence, and renormalization group(RG) flow, deriving Einstein’s equations and quantum behavior as emergent limits.Key insights:• Gravity is not a field in spacetime; it is the substrate of spacetime.• Spacetime, matter, and quantum behavior emerge from discrete gravitational relations.• Gamma-convergence demonstrates that general relativity is a macroscopic limit.• Probabilistic updates reproduce quantum interference and entanglement.• The effective Newton constant Geff(L) evolves with scale, yielding testable predictions.Testability: DGO predicts observable deviations from Newton’s law at sub-millimeterscales, potential gravitational wave anomalies, and cosmological implications from the runningof Geff(L).