Deterministic Growth Dynamics of Causal Graphs: Topology and Dimension

We propose a deterministic variational principle — the Coverage Principle — as an alternative to the stochastic dynamics of classical sequential growth of causal sets. The Coverage Principle posits a single axiom of the theory: at each step, the causal graph grows so that the new node maximises coverage with minimal overlap. From this principle the following results are derived. Equilibrium configurations of the Coverage Principle are isostatic, which singles out D = 4 as the unique dimension in which the greedy growth algorithm attains the variational ideal without defect freezing (jamming). The sequence of maximal antichains converges to a smooth C1,α-manifold — the Frontier — by Allard’s regularity theorem. The Frontier is topologically isomorphic to S3, and the space of its possible continuations — the Extension manifold E — is isomorphic to B4. The natural symmetry group of E is Spin(4) ∼ = SU(2)L × SU(2)R. In the present work we do not prove the existence of a smooth limit in the strict sense, but identify the mechanism — the Coverage Principle and isostaticity — which apparently leads to the emergence of continuous geometry from the discrete structure of a graph. Since the dynamics of the theory is localised entirely on the Frontier, observable reality is three-dimensional. All results follow from the single axiom without additional postulates.