  We present a lattice determination of the inclusive decay rate of the process
$\tau\mapsto X_{us} \nu_\tau$ in which the $\tau$ lepton decays into a generic
hadronic state $X_{us}$ with $\bar u s$ flavour quantum numbers. Our results
have been obtained in $n_f=2+1+1$ iso-symmetric QCD with full non-perturbative
accuracy, without any OPE approximation and, except for the presently missing
long-distance isospin-breaking corrections, include a solid estimate of all
sources of theoretical uncertainties. This has been possible by using the
Hansen-Lupo-Tantalo method [1] that we have already successfully applied in [2]
to compute the inclusive decay rate of the process $\tau\mapsto X_{ud}
\nu_\tau$ in the $\bar u d$ flavour channel. By combining our first-principles
theoretical results with the presently-available experimental data we extract
the CKM matrix element $\vert V_{us}\vert$, the Cabibbo angle, with a $0.9$\%
accuracy, dominated by the experimental error.
