A Unified dark sector from the geometry of complex time

We develop a unified cosmological framework in which physical time is promoted to a complex scalar field T (x) = t (x) − i τ (x), whose real component determines causal structure while the imaginary component encodes an internal temporal degree of freedom. Starting from a nonminimally coupled action ξ R |T|², we show that the coupling simultaneously generates an effective potential V (τ) and controls the backreaction of τ on the geometry. In homogeneous cosmology, the evolution of the internal component τ (t) contributes an effective energy density and pressure that naturally reproduce dark‑energy phenomenology; in particular, the modified Friedmann equations admit regimes in which cosmic acceleration is driven directly by the ξ R |T|² term rather than by additional matter fields. Allowing for spatial inhomogeneities τ = τ (x), the same complex‑time field produces an effective gravitational density ρₑff ≃ ½ (∇τ) ² + V (τ), thereby modifying the Newtonian potential. Static, cylindrically symmetric configurations admit harmonic solutions τ (r) = A ln (r/r₀), whose gradient energy scales as r⁻², yielding asymptotically flat rotation curves and halo‑like mass profiles without particle dark matter. The model predicts the falsifiable scaling relation vflat² = π G A² (or equivalently vflat = A/2 in units where 4πG = 1), which matches the observed amplitudes of SPARC galaxy rotation curves. Numerical simulations confirm that the complex‑time field simultaneously accounts for late‑time acceleration, realistic galactic dynamics, and observationally consistent energy scales. Altogether, the geometry of complex time provides a single covariant mechanism linking dark energy, dark‑matter phenomenology, and quantum attenuation, suggesting that cosmic expansion and gravitational structure may arise from an internal temporal degree of freedom intrinsic to spacetime itself.