Dark Matter and Dark Energy from Geometric Relaxation of Spacetime

We propose a geometric extension framework in which dark-matter and dark-energy phenomenology can be captured through a single dynamical field contribution: the vector trace mode of spacetime torsion. Starting from a Riemann-Cartan geometry, we construct an effective field theory (EFT) where torsion acquires dynamics through radiative corrections from fermionic vacuum polarization. The resulting massive vector field exhibits a characteristic relaxation time τ ≈ 30 Myr, arising from the infrared scale of gravitational systems. This relaxation dynamics generates: Dark Matter-like effects through spatial gradients (galactic rigidity). Dark Energy-like behavior through temporal evolution (effective cosmological constant) in the late-time slow-roll regime. Observational Tests: We present comprehensive observational tests using multiple independent probes: DESI Galaxy Survey: Detection of an environmental dependence of the local expansion rate $H(z)$, with a differential shift Δ H ≈ +1.26 km s⁻¹ Mpc⁻¹ between low- and high-density regions (>30σ statistical significance). Cosmological MCMC: Joint analysis of Cosmic Chronometers, BAO, and Type Ia Supernovae (Pantheon+) shows that the TIDE framework provides an improved fit relative to ΛCDM with Δ χ² = 39.40. Gravitational Lensing: Analysis of the El Gordo cluster and other high-speed mergers supports a geometric origin for the dark matter-baryon offset, consistent with the relaxation time τ ≈ 30 Myr. Universal Scaling: The timescale τ ≈ 30 Myr emerges consistently across galactic rotation curves and cluster collision dynamics.