Spacetime Rigidity as a Unified Origin of Dark Matter and Dark Energy (v2)

Dark matter is reinterpreted as an *emergent manifestation of spacetime rigidity*. Within the Quantum Ultrametric Rigidity (QUR) framework, the effective number of gravitational degrees of freedom decreases when spacetime forms coherent ultrametric domains. This reduction contracts the effective dynamical volume (fewer distinguishable geometric configurations) while preserving the baryonic energy budget, producing an apparent gravitational overdensity that reproduces galactic rotation curves and lensing effects without introducing new particles or halo parameters. The same constitutive law Sext(L) · ρrig(L) ≈ MP⁴ links local curvature reinforcement (dark-matter-like behaviour) with the residual large-scale rigidity responsible for cosmic acceleration. Dark matter and dark energy thus emerge as two operational phases of a single geometric principle: the elastic response of spacetime to baryonic organization and the finite horizon-scale rigidity. Editorial Note —November 2025 This second edition clarifies notation, streamlines terminology, and improves coherence across the rigidity framework. The presentation is made lighter by avoiding specialised thermodynamic variables when not required, keeping the focus on the geometric role of rigidity. No physical results are modified: rotation curves, lensing interpretation, and the cosmological extension remain unchanged. The update aims only to enhance readability and ensure full internal consistency across the accompanying works.