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 \( m_e=M_p\,C_p(α)exp\![-Θ_R/α], Θ_R=π/8.\) For an admitted charged shell \(n=2,3\), the same prefactor is kept at the electron anchor and only the exponent-slot coordinate changes, \( m_n=M_p\,C_p(α)exp\![-{Θ_R}{αR,n}].\) 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 \(Γ_n\) and its bare readout \( T_n=lnΓ_n,\) where \(T_n\) is not a physical mass logarithm. It is converted into the bare affine load \({ΔΛ}λ C,n\), dressed into \({ΔΛ}_n\), and only then read as \( Lm,n= ARᵃᶠᶠ\!({ΔΛ}_n).\) The determinant sector is written uniformly for both branches as \( D_n=2+ Lˡⁱᵍʰᵗ_n-34\, Hʰᵉᵃᵛʸ_n+εᵈʸⁿ_n.\) Here \( Lˡⁱᵍʰᵗ_n\) is the retained lower-sector determinant response, \( Hʰᵉᵃᵛʸ_n\) is the upper-heavy Schur/Stieltjes response, and \(εᵈʸⁿ_n\) 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 \( A=Θ_R/α=π/8α, ρ_n={CUV(Gauss)}{n+1}ln D_n, CUV(Gauss)=12(ln2+γE),\) the resulting compact mass-log law is \( Lm,n≡_n/m_e= A\,T_n/ A(1-ρ_n)+ρ_n T_n, n=2,3.\) The only supplied dimensionless theory input in the numerical audit is \(α⁻¹\). Measured \(m_μ\) and \(m_τ\) 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 \( Lm,μ=5.33159877458397, Lm,τ=8.15399066072771,\) or \( m_μ/m_e=206.768285919222, m_τ/m_e=3477.22785082421.\) With \(m_e=0.51099895069\,MeV\), these become \( m_μ=105.658377140693\,MeV, m_τ=1776.85978308121\,MeV,\) corresponding to \(+0.713σ\) and \(-0.780σ\) relative to the stated comparison values. No \(T_n\), \({ΔΛ}λ C,n\), \({ΔΛ}_n\), \(D_n\), \(ρ_n\), \( Lm,n\), or charged-lepton mass is fitted or supplied as an input.