= Solution
The <Cabibbo-Kobayashi-Maskawa matrix> is the unitary change of basis between down-type <quark> <mass eigenstates> and the combinations appearing in the weak <charged current>. In particular, $u\bar d$ and $u\bar s$ production carry $V_{ud}$ and $V_{us}$. Exclusive <tau lepton> decays into a charged <pseudoscalar meson> give
$$
\frac{B(\tau\to K\nu)}{B(\tau\to\pi\nu)}=\frac{f_K^2|V_{us}|^2}{f_\pi^2|V_{ud}|^2}\frac{(1-m_K^2/m_\tau^2)^2}{(1-m_\pi^2/m_\tau^2)^2},
$$
up to small radiative corrections. The <pseudoscalar decay constants> specify the vacuum-to-<meson> axial-current <matrix elements>, $\langle0|\bar q\gamma^\mu\gamma^5u|P(p)\rangle=if_Pp^\mu$.
The observed <branching fraction> ratio is $0.70/10.83=0.06464$, and the kinematic factor is approximately $0.8622$. Hence the directly determined combination is
$$
\boxed{\frac{f_K|V_{us}|}{f_\pi|V_{ud}|}\simeq0.2738}.
$$
With an \b[independent], representative lattice input $f_K/f_\pi\simeq1.193$, this becomes $|V_{us}/V_{ud}|\simeq0.2295$, close to the source's rounded expectation $0.224/0.975=0.2297$. Conversely that expectation and the hadronic input predict a <branching fraction> ratio near $0.0648$. Without the decay-constant input the table does not isolate a pure <Cabibbo-Kobayashi-Maskawa matrix> ratio. Setting $f_K=f_\pi$ would give approximately $0.0455$, demonstrating the importance of <flavor symmetry> breaking.<Leptonic Decays of Charged Pseudoscalar Mesons in the 2024 Review of Particle Physics>[Sections 72.1–72.2, axial-current definition, radiative corrections and lattice decay-constant ratios]{ref}
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