Toroidal Fractal Spacetime: A Unified Framework Connecting Toroidal Cosmology, Fractal Quantum Gravity, and Integrated Information

The Toroidal Fractal Spacetime (TFS) framework is a theoretical model proposing that the spacetime manifold has compact toroidal topology T⁴, within which the effective dimensionality of spacetime runs continuously with probed scale from 𝒟_s → 2 at ultraviolet (Planck) scales to 𝒟_s → 4 at infrared (cosmological) scales, consistent with predictions from Causal Dynamical Triangulations (CDT) and Asymptotic Safety Quantum Gravity (ASQG). The two independent toroidal coordinates correspond to position within a cosmic aeon (φ) and fractal depth (θ). The Big Bang is reinterpreted as a dimensional phase transition at minimum spectral dimension, naturally embedding the framework within cyclic cosmological models and removing the initial singularity. A generalized field equation is derived extending the Einstein equations by two additional stress tensors in the core framework: a fractal dimension term Tᵓʳᵃᶜμν encoding scale-dependent curvature (derived explicitly in Appendix B), and an optional information stress tensor Tᴵⁿᶠᵒμν [C, separable] coupling integrated information Φ to spacetime geometry (derived in Appendix C; setting λ = 0 recovers the core framework). A topological term S_topo, intended to capture boundary effects at the Big Bang phase transition, is retained in the action but its stress tensor Tᵐᵘᵖᵒμν vanishes identically in the smooth background (χ(T⁴) = 0 for an undeformed torus) and is deferred as a research direction rather than included in the field equation at this stage (Sec. III.A.2). The fractal and cosmological-constant terms reduce to standard General Relativity in the classical infrared limit. Three classes of testable predictions are derived, all independent of the information-geometry extension: a curved power-law rolloff in the CMB power spectrum at low multipoles (ℓ = 2–10); anomalous CMB angular power correlations at multipole pairs determined directly by the toroidal mode spectrum, with the dominant pairs and their amplitudes derived in Appendix D; and a periodically modulated primordial gravitational wave background. All speculative components are labeled throughout using a four-tier epistemic scheme.