Emergent Spacetime: Informational Metrics, Geometry and Causality

This paper establishes the geometric backbone of the Theory of Informational Selection (TSI), providing a rigorous informational foundation for emergent spacetime and General Relativity. The TSI describes spacetime, geometry, locality, and causal order as emergent phenomena arising from the correlational structure of the Informational Totality (IT). While Paper 1 established the mathematical ontology of the IT through correlational, variational, and functorial formulations, this second paper develops the emergence of spacetime from first principles. We introduce the fundamental informational distance dS, defined over the full Cartesian product of the IT, and construct the scale‑dependent informational metric tensor gμν∣λ through the granular operator Gλ. This operator maps discrete correlational neighborhoods into macroscopic points, generating emergent tangent spaces TpMλ. Geometric deviation from flatness is formalized via Ollivier–Ricci coarse curvature, which lifts naturally to a symmetric rank‑2 coarse Ricci tensor. Locality and causal order arise dynamically from anisotropic informational transitions, which break Euclidean symmetry and induce the physical Lorentzian signature (−,+,+,+). In the macroscopic limit, coarse curvature converges to the continuum Ricci tensor, and the Einstein field equations emerge as stability conditions coupling geometric curvature to the correlational density tensor Θμν.