Quantum Fundamental Speed Theory (QFST): A Unified Framework for Galactic Dynamics, Cosmic Structure, and Dark Energy

We present a complete quantum formulation of the Fundamental Speed Theory (QFST), built upon the classical FST Lagrangian and the previously established emergence of spacetime geometry from the speed field \nu^{\mu}. The theory demonstrates that a single field can account for three distinct physical phenomena: (i) Galactic dynamics: The classical field equation \frac{d^2\tilde{\nu}}{d\xi^2} +\frac{2}{c}\frac{d\tilde{\nu}}{d\xi} = \beta_{\mathrm{eff}}\tilde{\nu}^3 with \beta_{\mathrm{eff}} = 2.0\times 10^{7} explains galactic rotation curves without requiring dark matter, achieving \chi_{\nu}^{2} = 0.170 for 171 SPARC galaxies. (ii) Cosmic dark matter: Quantization of the \nu field yields particles of mass m = \frac{\hbar}{c\lambda_{\mathrm{screen}}} = 3.88\times 10^{-24}\mathrm{eV} where \lambda_{\mathrm{screen}} = 1.647 pc is the screening length. On cosmological scales, these quanta form large-scale structure via the Schrödinger-Poisson system. (iii) Dark energy: The logarithmic potential V(\tilde{\nu}) = -V_{0}\ln \left(1 + \frac{\tilde{\nu}^{2}}{\nu_{0}^{2}}\right) produces a cosmological constant. Matching the observed dark energy density yields the prediction \tilde{\nu}_{\mathrm{now}} = 8.9\times 10^{-12} for the cosmic vacuum field value. A rigorous calculation shows that quantum corrections to the classical field equation are suppressed by (\ell_{\mathrm{Pl}} / \lambda_{\mathrm{screen}})^{2} = 1.01\times 10^{-103} on galactic scales. Consequently, classical FST dominates where it is required, while quantum effects become relevant only on cosmological scales. This scale separation resolves any apparent tension between the classical and quantum descriptions: the quanta are not needed to explain galactic rotation curves, yet they naturally account for cosmic structure formation. All derivations are presented with complete dimensional analysis in SI units. This work is theoretical; experimental verification is left for future research.