Fundamental Speed Theory: Hierarchical Tests on 171 SPARC Galaxies Without Dark Matter and a Unified Acceleration Scale

We present a mathematically rigorous formulation of the Fundamental Speed Theory (FST), a dimensionally consistent vector–tensor theory featuring a dimensionless vector field \( \nu^{\mu} \). We introduce characteristic scales \( L_0 = 10 \) kpc and \( M_0 = \hbar/(cL_0) \) and keep \( \hbar \) and \( c \) explicit throughout. In dimensionless form, the galactic field obeys

\( \frac{d^2\tilde{\nu}}{d\xi^2} + \frac{2}{\xi}\frac{d\tilde{\nu}}{d\xi} = \beta_{\mathrm{eff}}\tilde{\nu}^3 \),\( \qquad \beta_{\mathrm{eff}} \equiv -\frac{\lambda\nu_0^2}{6c_1} = 2.0\times 10^7 \ (\lambda<0). \)

We validate the theory on the SPARC sample using three primary hierarchical levels (Levels 1–3): Level 3 (zero free parameters) fits 65.7% of galaxies with mean \( \chi_{\nu}^{2}=0.809 \); Level 2 (estimated \( M, r_d \), no fitting) reaches 93.6% with mean \( \chi_{\nu}^{2}=0.347 \) for the 160 galaxies with \( \chi_{\nu}^2<3 \); and Level 1 (fitted \( M, r_d \)) fits all 171 galaxies with mean \( \chi_{\nu}^{2}=0.170 \) (91.2% with \( \chi_{\nu}^{2}<0.5 \)). We further report two derived formulations: Level 4 (coefficient-free) and Level 5 (unified), the latter showing that the field parameters unify into a single acceleration scale

\( A_0 = \frac{(c_1+c_3)\nu_0^2 c^2}{L_0} = 2.42\times 10^{-10}\ \mathrm{m/s^2}, \)

which reproduces the full formulation identically for all galaxies. A full three-dimensional numerical experiment with disk-like (anisotropic) boundary forcing confirms that the converged 3D field profiles and rotation-curve fits remain essentially unchanged relative to the 1D quasi-spherical approximation for the tested cases. We also perform an explicit sign-convention robustness check: running the full pipeline with the alternative (negative-sign) convention yields identical fits within numerical tolerance when implemented consistently. Solar System constraints are satisfied because the relevant acceleration arises from the galactic field gradient, giving a local FST acceleration at Earth of \( \sim 8\times 10^{-15} \) of the Newtonian value. All code is archived on Zenodo, and supplementary materials (including complete fit results and both sign-convention implementations) are provided. Extension of FST to cosmological scales is left as future work.