Fit Details
Objective function, parameterization, and a reproducible calculator for the charged-lepton calibration.
Model
We evaluate $M_{\text{topo}}$ on eigenvalues $(N,w,T)$:
$$M_{\text{topo}}(N,w,T)=\Lambda_c\,e^{\lambda_c N}+\alpha_c\,w+\kappa_c\,T^2.$$
The charged-lepton labels used here are
$e:(1,1,1)$, $\mu:(3,0,2)$, $\tau:(5,-1,3)$.
Objective
We use a simple least-squares objective with unit weights:
$$J(\Lambda_c,\alpha_c,\kappa_c\mid\lambda_c)
=\sum_{\ell\in\{e,\mu,\tau\}}\big(M_{\text{topo}}(N_\ell,w_\ell,T_\ell)-m_\ell\big)^2.$$
At fixed $\lambda_c$, the best fit is obtained by minimizing $J$ over $(\Lambda_c,\alpha_c,\kappa_c)$.
| Lepton | (N, w, T) | Observed [MeV] | Predicted [MeV] | Residual [MeV] |
|---|---|---|---|---|
| electron (e) | (1,1,1) | 0.510999 | 0.510962 | -0.000037 |
| muon (μ) | (3,0,2) | 105.658376 | 105.659380 | 0.001004 |
| tau (τ) | (5,-1,3) | 1776.860000 | 1776.862423 | 0.002423 |
| Sum of squared errors | 0.000007 | |||
Hold-out check
A simple cross-check is to solve $(\Lambda_c,\alpha_c,\kappa_c)$ from $(e,\mu)$ only and predict $\tau$.
This UI does not solve the inverse problem, but you can verify sensitivity by perturbing the parameters and observing the $\tau$ residual.
For a full inverse solver, we can add a closed-form routine derived from Appendix F equations upon request.