The Fundamental Speed Theory: A Mathematically Consistent Vector-Tensor Theory for Galactic Dynamics

We present a mathematically rigorous formulation of the Fundamental Speed Theory (FST), a vector-tensor theory of gravity featuring a dimensionless vector field \( \mathcal{V}^{\mu} \). The theory introduces characteristic scales \( M_{0} = \hbar /(cL_{0}) \) and \( L_{0} = 10 \mathrm{kpc} \) to ensure complete dimensional consistency, with explicit inclusion of \( \hbar \) and \( c \) in all physical expressions. The dimensionless Lagrangian density is \( \mathcal{L}_{V} = M_{0}^{4}[-\frac{c_1}{2}(L_{0}^{2}\nabla_{\mu}\mathcal{V}_{\nu})(\nabla^{\mu}\mathcal{V}^{\nu}) - \frac{\lambda}{4!}(\mathcal{V}_{\mu}\mathcal{V}^{\mu})^{2}] \). Galactic dynamics obey \( \frac{d^{2}\mathcal{V}}{d\xi^{2}} + \frac{2}{\xi}\frac{d\mathcal{V}}{d\xi} = \beta_{\mathrm{eff}}\mathcal{V}^{3} \) where \( \xi = r / L_{0} \) and \( \beta_{\mathrm{eff}} = \lambda \mathcal{V}_{0}^{2} / 6 = 2.0 \times 10^{7} \). FST achieves \( \chi^{2} / \mathrm{dof} = 0.189 \) across 137 SPARC galaxies using universal parameters \( c_{1} = 0.51 \), \( c_{2} = - 0.07 \), \( c_{3} = 0.32 \), \( \lambda = 1.2 \times 10^{14} \), \( \mathcal{V}_{0} = 1.0 \times 10^{- 3} \), \( \Upsilon_{\star} = 1.0 \). Solar System constraints are satisfied through a screening mechanism with \( \lambda_{\mathrm{screen}} = \hbar /(m_{\mathrm{eff}}c) \approx 2.5 \mathrm{~nm} \). Complete mathematical derivation and open-source implementation ensure full reproducibility.