Radial Normalization and the No-Fit Criterion for Scale-Invariant Lepton Mass Relations: A Conditional Application to the Quantized Dimensional Ledger

Empirical relations among particle masses may reveal nontrivial structure without determining absolute numerical scales. This paper examines that distinction using the charged-lepton Koide relation and formulates a general no-fit criterion for scale-invariant mass relations. The charged-lepton Koide relation constrains the electron, muon, and tau masses through a homogeneous dimensionless ratio. In mass-root space, it can be represented geometrically as a cone of fixed angle relative to the democratic direction. This geometry constrains the shape of the charged-lepton mass vector but leaves its overall radial scale undetermined. The central theorem states that if a relation F(m1, m2, m3) is homogeneous of degree zero under common positive rescaling, F(lambda m1, lambda m2, lambda m3) = F(m1, m2, m3), then it cannot determine the absolute mass scale S = m1 + m2 + m3. If one solution satisfies the relation, every commonly rescaled solution satisfies the same relation while possessing a different radial scale. Consequently, a proposed derivation that combines a scale-invariant relation with the empirical mass sum, fitted mass ratios, or any algebraically equivalent target-mass input has not derived the absolute mass spectrum. It has normalized a scale-invariant structure using observed data. The paper distinguishes three levels of claim: Koide cone: structural constraint. Koide cone plusass input has not derived the absolute mass spectrum. It has normalized a scale-invariant structure using observed data empirical radius: reconstruction of observed data. Koide cone plus independently derived radius: candidate absolute mass derivation. Using the sum-scale convention, S_l = m_e + m_mu + m_tau, the numerical audit gives S_l c^2 = 1883.03(12) MeV. The corresponding Planck-normalized empirical target is R_l,obs = S_l/m_P = (1.54234 +/- 0.00010) x 10^-19. This value is not presented as a theoretical prediction or derivation. It is the empirical target that any future absolute charged-lepton scale theory must reproduce without using the charged-lepton masses, their sum, their ratios, or fitted equivalent parameters as inputs. The no-fit criterion is broader than the Koide relation. It applies to any homogeneous degree-zero mass relation and provides a method for distinguishing dimensional consistency, structural representation, normalized reconstruction, partial derivation, and genuine absolute derivation. The paper also explains why a physically complete charged-lepton mass theory must ultimately connect its scale-setting rule to electroweak symmetry breaking, charged-lepton Yukawa couplings, chirality, renormalization conventions, and the scale at which any proposed boundary condition is defined. The final section applies the criterion conditionally to the Quantized Dimensional Ledger. The no-fit theorem does not depend on QDL. Within the QDL case study, a future Quantized Dimensional Cell recurrence theory would need to derive a dimensionless factor R_l,QDC satisfying S_l = m_P R_l,QDC without using charged-lepton mass data as input. The paper does not derive the charged-lepton masses, Yukawa couplings, electroweak scale, Standard Model gauge structure, or the QDL recurrence factor. Its contribution is to isolate the missing radial-normalization problem and provide a precise, non-circular standard by which future mass-spectrum claims can be evaluated. This record forms part of the QDL/QDC Completion Series and defines the radial-scale and no-fit gate for the charged-lepton mass program.