CDUFD Supplementary Material IV: Emergent Spacetime, Gravity, and Cosmology

This is the fourth supplementary material of the CDUFD qualitative framework. Based on axioms A1–A5, we formalise the emergence of spacetime, gravity, time, and cosmology. From the discrete kernel of A1 we construct an intrinsic distance and, via Gromov–Hausdorff convergence, obtain a smooth Riemannian manifold. Assuming a dynamic exponent z=1 for the spacetime‑emergence critical point (Lorentz invariance), a Wick rotation and the Osterwalder–Schrader theorem yield a (3+1) -dimensional Lorentzian manifold. The continuum limit of the Regge action gives the Einstein–Hilbert action. Physical time emerges from critical slowing down, and the arrow of time originates from entropy production together with causality. The CDUFD axioms provide the necessary conditions for the Verlinde and Jacobson derivations, leading to Newton’s law and the Einstein field equations. The cosmological constant is identified with the critical vacuum energy of an infrared critical point; dark matter candidates are heavy gauge bosons protected by topological charges. The paper is fully consistent with the qualitative framework and lists all open problems.