= Solution
The local <Fermi interaction> gives a three-body <leptonic decay> width of the form
$$
\Gamma(l\to a\bar\nu_a\nu_l)=\frac{G_F^2m_l^5}{192\pi^3}f(z),\qquad z=\frac{m_a^2}{m_l^2},\qquad f(z)=1-8z+8z^3-z^4-12z^2\log z,
$$
at tree level with negligible <neutrino masses> and universal weak <coupling constant (physics)> values. The fifth power follows from the dimension of $G_F$ and the three-body <phase space>; $f$ results from integrating the <V-A interaction> <matrix element> over the reduced <phase space> when the charged daughter has nonzero <mass>. It is more than a purely geometric volume correction because the spin-dependent <matrix element> is included.
As $z\to0$, $f(z)\to1$; increasing daughter <mass> reduces the available <energy> and suppresses the rate. The largest ratio is $m_\mu/m_\tau$, so the $\tau\to\mu$ correction is visibly below one. The <electron> is extremely light relative to either parent. Using the <lepton> <masses> gives
$$
\boxed{f_{\tau\mu}\simeq0.9725601,\quad f_{\mu e}\simeq0.99981295,\quad f_{\tau e}\simeq0.99999934}.
$$
These reproduce the supplied factors to their stated accuracy. Radiative and finite-<W boson> corrections are separate small corrections when a precision <lepton universality> test is performed.
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