Galactic Rotation Curves and Spiral Persistence from Scalar-Time Geometry Without Dark Matter

We present a galactic-scale dynamical analysis within Time-Scalar Field Theory (TSFT), a framework in which time is treated as a physical scalar field governing the rate of all physical evolution. In this formulation, gravitational and inertial phenomena emerge from spatial gradients in the scalar proper-time rate field Θ(x). We show that radial structure in Θ naturally produces asymptotically flat galactic rotation curves and stabilizes spiral arms against differential winding, without invoking non-baryonic dark matter or modified inertia. Starting from the canonical TSFT definition of proper-time geometry, we derive the effective acceleration experienced by matter and light as a function of ∇ln Θ. In disk galaxies, baryonic structure induces a persistent radial Θ-gradient extending beyond the stellar disk, yielding constant circular velocities at large radius. Applying this framework to the Andromeda galaxy (M31) using a scale-fixed nonlinearity parameter tied to disk structure, we reproduce the observed rotation curve from ∼ 5–125 kpc with reduced χ2 ≈ 1.2. We further show that azimuthal perturbations of Θ couple coherently to surface density waves, producing phase-locked spiral structure that suppresses winding without requiring massive halos. Observational consequences include a natural explanation of the radial acceleration relation across both kinematic and lensing probes, and testable deviations from dark matter and MOND-based predictions. The results establish TSFT as a falsifiable geometric alternative framework for galactic dynamics.