The Fundamental Speed Theory: A Mathematically Consistent Vector-Tensor Theory for Galactic Dynamics Without Dark Matter Updated Results from 171 SPARC Galaxies

\( \mathcal{V}^{\mu}\ \). 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 ℏ and c in all physical expressions. Galactic dynamics obey \( \frac{d^{2}\tilde{\mathcal{V}}}{d\xi^{2}} + \frac{2}{\xi}\frac{d\tilde{\mathcal{V}}}{d\xi} = \beta_{\mathrm{eff}}\tilde{\mathcal{V}}^{3} \) where \( \xi = r / L_{0} \) and \( \beta_{\mathrm{eff}} = \frac{\lambda}{6}\mathcal{V}_{0}^{2} = 2.0 \times 10^{7} \). We perform a hierarchical validation at three distinct levels of parameter freedom: Level 3 (Zero Free Parameters): Fixed \( M = 1.0\times10^{10}\,M_{\odot} \) and \( r_d = 3.0\,\mathrm{kpc} \) for all 175 galaxies. Even with no galaxy-specific parameters, FST correctly describes 65.7% of galaxies with mean \( \chi^{2}_{\nu} = 0.809 \). Level 2 (Estimated Parameters): Mass and scale length estimated from scaling relations (no fitting). Success rate rises to 94.9% with mean \( \chi^{2}_{\nu} = 0.283 \). Level 1 (Fully Fitted): Mass and scale length fitted per galaxy. Success rate reaches 100% with mean \( \chi^{2}_{\nu} = 0.170 \). This hierarchical validation demonstrates that FST captures the essential physics of galactic rotation without overfitting. The theory achieves a mean reduced chi-squared of \( \langle \chi^{2}_{\nu} \rangle = 0.170 \) across all 171 SPARC galaxies, with 91.2% of galaxies having \( \chi^{2}_{\nu} < 0.5 \) (excellent fit) and only 1.8% (three galaxies) having \( \chi^{2}_{\nu} > 1.0 \). The characteristic transition scale is \( \xi_c = \sqrt{2/\beta_{\mathrm{eff}}} = 3.16\times 10^{-4} \), corresponding to a fundamental scale \( r_c = \xi_c L_0 \approx 3.16 \) pc. Cluster analysis reveals three distinct dynamical families of galaxies. Solar System constraints are satisfied through a screening mechanism derived directly from the velocity field, with a characteristic screening length \( \lambda_{\mathrm{screen}} \sim 10^{13}\,\mathrm{m} \) (about 200 AU). Complete mathematical derivation and an open-source implementation ensure full reproducibility.