= Solution
Write the leptonic partial <decay width> as
$$
\Gamma(\tau\to\ell\bar\nu_\ell\nu_\tau)=\frac{G_F^2m_\tau^5}{192\pi^3}\left|\frac{g_\tau g_\ell}{g_\mu g_e}\right|^2 f(r_\ell)(1+\delta_\ell),\quad
f(r)=1-8r+8r^3-r^4-12r^2\ln r,\quad r_\ell=\frac{m_\ell^2}{m_\tau^2}.
$$
Here $G_F$ is defined from <muon decay>, so the coupling denominator fixes the convention. Measure $\Gamma_\ell=B_\ell/\tau_\tau$. The ratio $B_\mu/B_e$ tests $|g_\mu/g_e|^2$ after the known <phase space> and radiative corrections. Comparing $B_e/\tau_\tau$ with the <muon> lifetime and the fifth power of the <mass> ratio tests $|g_\tau/g_\mu|^2$. Hadronic comparisons such as tau-to-pion versus pion-to-muon decay supply additional universality tests with their own hadronic and electromagnetic corrections.
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