The Geometry of Information: Quantum Ultrametric Rigidity and the Emergence of Continuum Physics

The long-standing conflict between General Relativity (GR) and Quantum Mechanics (QM) originates from their incompatible assumptions about spacetime: continuous curvature versus discrete probability. Quantum Ultrametric Rigidity (QUR) resolves this tension by introducing a finite hierarchical structure at the Planck scale, where spacetime consists of discrete patches carrying conjugate entropy–rigidity pairs (SI, χI). Their constitutive relation, S·ρrig ∼ MP⁴, expresses the equivalence between geometric tension and informational content, bounding both curvature and fluctuation within finite limits.Mathematically, QUR is grounded in Ultrametric Functional Analysis (UFA), developed in the Appendices (D–Z). Spectral convergence (Mosco limit), Rellich compactness, and Lieb–Robinson locality ensure that the discrete Hamiltonian reproduces the Wheeler–DeWitt equation in the continuum limit (N→∞, Lcore→0).Physically, QUR unifies geometry and information: time emerges from sequential coherence (“temporal stitching”), mass and gravity arise from rigidity gradients, and cosmic acceleration follows from ρrig ∝ MP²/L² without free parameters. At the experimental level, the theory predicts discrete loss of interference in the double-slit configuration and a quantized scaling of cosmological density — both providing direct falsifiability tests.QUR therefore completes rather than modifies existing physics: a finite, informationally coherent spacetime from which GR and QM emerge as effective limits. Future developments include applications to dark matter (QUR–DM), primordial black holes (QUR–PBH), and gauge coherence (QUR–SM).