In DGDCG I the Coverage Principle was introduced as a deterministic growth algorithm for a graph embedded in RD. It was shown there that the limiting configuration — the Frontier — is a smooth C1,α-manifold topologically isomorphic to S3, that the space of its continuations is isomorphic to B4 with natural symmetry group Spin(4) ∼= SU(2)L × SU(2)R, and that the dynamics is localised on the Frontier, which accounts for the three-dimensionality of observable reality. The present paper proves that RD is not a postulate but the unique geometry compatible with the existence of a non-trivial limiting object — the Frontier. The limit is understood in the sense of weak convergence of integer-rectifiable varifolds; its existence on RD was established in DGDCG I. We show that a varifold limit of positive infinite mass, invariant under the isometries of the arena, exists if and only if the arena is isometric to RD. The proof eliminates spherical geometry by compactness, hyperbolic geometry by the absence of an intrinsic scale in the isoperimetric balance, and arenas with handles by the failure of transitivity of the isometry group. The conclusion follows from the Hadamard–Cartan theorem.
Andrei Okhremenko (Thu,) studied this question.