Finite Activation, Phase Relaxation, and Charged-Lepton Mass Ratios

We study a finite effective model for the on-shell mass ratios of the charged leptons. Equal common and relative powers in the square-root mass vector define a circle. Four activation channels give fifteen nonempty states and the response \(\eta(t) = 4\sinh t/(11 + 4\cosh t)\). A three-depth carrier supplies \(q_0 = 2/9\) and \(\Omega = 2667\). On a selected principal-structural pair, a positive self-adjoint lift of the octahedral coupling realizes the prescribed residual stiffness and yields \(h = 1/(3\Omega^2)\). Its isotropic elimination leaves the angular selection to a separately specified shell-activation mean budget. Under this budget and the identification of the natural parameter with the mass angle, \(\theta_* = \eta^{-1}[\eta(q_0) - h]\) is the unique entropy-projection solution and the globally attracting equilibrium of a Fisher natural-gradient flow. The mass ratios remain \(m_\mu/m_e = 206.768285719792\ldots\) and \(m_\tau/m_e = 3477.441682622933\ldots\), with residuals of \(0.66\) and \(0.43\) standard deviations against the fixed CODATA and PDG references. A separate two-step history construction realizes the same score multiplicities and, with an explicitly prepared three-mode readout, the same mass amplitudes. These conditional realizations specify which operator, history, and readout choices a microscopic model must supply.