Time-Oscillation Dark Matter Model (TODMM): A Geometrical Field Formulation and Astrometric Validation Across Galactic Scales

We propose the Time-Oscillation Dark Matter Model (TODMM), an alternative theoretical framework designed to resolve the galactic missing mass problem without invoking hypothetical particulate dark matter relics. TODMM posits that observed dark matter dynamics emerge from large-scale local metric time perturbations, formulated via a coordinate shift $\tau(\vec{x}, t) = t + \frac{1}{\omega_0} \alpha(\vec{x})\sin(\omega_0 t + \phi)$. By constructing a covariant canonical scalar field action incorporating a non-linear self-interaction potential ($\frac{\lambda}{4}\Phi^4$) and a universal baryonic coupling term ($\beta \Phi \rho_b$), we derive the exact physical bridge linking the underlying scalar field $\Phi$ to the metric temporal component and the dimensionless amplitude $\alpha(\vec{x})$. To address physical singularities and cosmological constraints, we introduce a core-modified logarithmic amplitude profile paired with an exponential virial cut-off ($e^{-2r/R_{\text{vir}}}$) and a cosmological resonance frequency evolution $\omega(z)^2 = \omega_0^2(1+z)^{3/2}$. Numerical validation across six distinct astrophysical scales—ranging from ultra-faint dwarf galaxies (Segue 1, Draco dSph) and spiral galaxies (NGC 3198, LSB F568-6, Milky Way) to massive galaxy clusters (Coma Cluster)—demonstrates robust agreement with observed dynamical masses. Finally, we propose concrete falsifiable predictions to distinguish TODMM from $\Lambda\text{CDM}$, including ultra-precise optical atomic clock frequency drifts and pulsar timing array temporal modulations.