A Geometric Scalar Field as a Unified Model of Dark Energy and Dark Matter

We present a minimal effective framework in which a single geometric scalar field, written as Z (x) = t (x) − i τ (x), accounts for both dark‑energy‑like behaviour in homogeneous cosmology and dark‑matter‑like phenomenology in galaxies. The real component t (x) is fixed by the cosmological slicing, while all dynamics reside in the internal scalar τ (x), which enters the action through a canonical kinetic term and a small nonminimal coupling ξ R τ². In FLRW backgrounds, the slow evolution of τ (t) generates an effective energy density and pressure that reproduce dark‑energy‑like behaviour. Allowing for spatial inhomogeneities, gradients of the same field produce an effective density ρₑff ≃ ½ (∇τ) ², modifying the Newtonian potential. In static disk geometries, the harmonic solution τ (r) = A ln (r/r₀) yields an r⁻² effective density and asymptotically flat rotation curves. The relation vflat² = 2πG A² matches SPARC data, while numerical solutions show that the same scalar field reproduces late‑time acceleration and galactic dynamics within a controlled effective description.