From the Fundamental Speed Theory to a Unified Nuclear Radius Law: A Complete First-Principles Derivation with Empirical Validation

We present a first-principles-based derivation of a unified nuclear radius law within the Fundamental Speed Theory (FST). The central theoretical link used throughout is a local, high-density saturation relation between the FST speed field and matter density, \nu (\rho) = \nu_0(\rho /\rho_{\mathrm{gal}})^{1 / 4} (Appendix A), obtained under an explicit quasi-static saturation assumption that the field energy density tracks the local material energy density. Applied to nuclear matter, this yields a constant-density core and the classical A^{1 / 3} scaling. Small density variations with proton and neutron number generate a linear structure correction \Delta R \propto (5.18Z + 6.56N) (Appendix B). Scale dependence of the probing interaction yields a lepton-mass correction \Delta R \propto A^{1 / 3}(1 / m_e - 1 / m_e) ; the coefficient k_p is estimated from an RG-motivated flow argument and then fixed to the proton-scale normalization, with the remaining theoretical gap isolated for future work (Appendix C). Shell effects are modeled through the field response to shell-density oscillations (Appendix D). The resulting law is tested against the complete IAEA nuclear charge radii database (952 nuclei with A \geq 4 ), reducing the RMS error from 1.502 fm to 0.294 fm ( R^2 = 0.862 ). We also show that the lepton-mass term yields a proton-radius splitting of R_p^e - R_p^\mu \approx 0.034 fm at the adopted normalization. All appendices provide complete dimensional verification.