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)−iτ(x), whose real component governs causal structure while the imaginary component encodes an internal temporal degree of freedom. Starting from a nonminimally coupled action ξR∣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 ξR∣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≃12(∇τ)2+V(τ), thereby modifying the Newtonian potential. Static, spherically symmetric configurations admit harmonic solutions τ(r)=Aln⁡(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=2πGA2 (or vflat2=A2/4 in units where 8πG=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.