Charged-Lepton Mass Hierarchy Law

We derive a closed no-target mass-ladder law for the charged leptons within an admitted three-shell Relator/Alpha source ledger and a scalar determinant-normalized Schur/Feshbach sector. In this framework, the electron, muon, and tau are not assigned different charged-current cores. They are the shell \(n=1,2,3\) realizations of the same Gaussian electron Relator core. The charged-current core and the path-dependent finite prefactors remain at the electron anchor. Changing the admitted shell changes only an internal Relator exponent-slot Alpha coordinate. It does not define a new QED running coupling and it is not inserted as an additive correction to a mass logarithm. The common electron readout has the form \(\displaystyle m_e=M_p\,C_p(\alpha)\exp\!\left[-\frac{\Theta_R}{\alpha}\right],\qquad \Theta_R=\frac{\pi}{8}.\) For an admitted charged shell \(n=2,3\), the same prefactor is kept at the electron anchor and only the exponent-slot coordinate changes, \(\displaystyle m_n=M_p\,C_p(\alpha)\exp\!\left[-\frac{\Theta_R}{\alpha_{\mathcal R,n}}\right].\) Thus the physical mass logarithm is read only after the shell source has been converted into a bare affine source load, Schur/Feshbach dressed, and mapped back to the Relator exponent slot. The shell ledger supplies a positive shell-admission ratio \(\Gamma_n\) and its bare readout \(\displaystyle T_n=\ln\Gamma_n,\) where \(T_n\) is not a physical mass logarithm. It is converted into the bare affine load \(\widehat{\Delta\Lambda}_{\lambda C,n}\), dressed into \(\widehat{\Delta\Lambda}_n\), and only then read as \(\displaystyle \mathscr L_{m,n}=\mathscr A_{\mathcal R}^{\mathrm{aff}}\!\left(\widehat{\Delta\Lambda}_n\right).\) The determinant sector is written uniformly for both branches as \(\displaystyle D_n=2+\mathfrak L^{\mathrm{light}}_n-\frac34\,\mathfrak H^{\mathrm{heavy}}_n+\varepsilon^{\mathrm{dyn}}_n.\) Here \(\mathfrak L^{\mathrm{light}}_n\) is the retained lower-sector determinant response, \(\mathfrak H^{\mathrm{heavy}}_n\) is the upper-heavy Schur/Stieltjes response, and \(\varepsilon^{\mathrm{dyn}}_n\) is the normal-collar determinant tail. In the admitted three-shell closure, the tau branch is terminal, while the muon upper response is the normalized rank-one terminal heavy return evaluated self-consistently at the solved output logarithms. With \(\displaystyle \mathsf A=\frac{\Theta_R}{\alpha}=\frac{\pi}{8\alpha},\qquad \rho_n=\frac{C_{\mathrm{UV}}^{(\mathrm{Gauss})}}{n+1}\ln D_n,\qquad C_{\mathrm{UV}}^{(\mathrm{Gauss})}=\frac12(\ln2+\gamma_{\mathrm E}),\) the resulting compact mass-log law is \(\displaystyle \mathscr L_{m,n}\equiv\ln\frac{m_n}{m_e}=\frac{\mathsf A\,T_n}{\mathsf A(1-\rho_n)+\rho_n T_n},\qquad n=2,3.\) The only supplied dimensionless theory input in the numerical audit is \(\alpha^{-1}\). Measured \(m_\mu\) and \(m_\tau\) enter only as external comparison values after the dimensionless ratios have been computed, and \(m_e\) is used only as the final MeV conversion scale. The no-target evaluation gives \(\displaystyle \mathscr L_{m,\mu}=5.33159877458397,\qquad \mathscr L_{m,\tau}=8.15399066072771,\) or \(\displaystyle \frac{m_\mu}{m_e}=206.768285919222,\qquad \frac{m_\tau}{m_e}=3477.22785082421.\) With \(m_e=0.51099895069\,\mathrm{MeV}\), these become \(\displaystyle m_\mu=105.658377140693\,\mathrm{MeV},\qquad m_\tau=1776.85978308121\,\mathrm{MeV},\) corresponding to \(+0.713\sigma\) and \(-0.780\sigma\) relative to the stated comparison values. No \(T_n\), \(\widehat{\Delta\Lambda}_{\lambda C,n}\), \(\widehat{\Delta\Lambda}_n\), \(D_n\), \(\rho_n\), \(\mathscr L_{m,n}\), or charged-lepton mass is fitted or supplied as an input.