  Using HISQ $N_f=2+1+1$ MILC ensembles with five different values of the
lattice spacing, including four ensembles with physical quark masses, we have
performed the most precise computation to date of the $K\to\pi\ell\nu$ vector
form factor at zero momentum transfer,
$f_+^{K^0\pi^-}(0)=0.9696(15)_\text{stat}(12)_\text{syst}$. This is the first
calculation that includes the dominant finite-volume effects, as calculated in
chiral perturbation theory at next-to-leading order. Our result for the form
factor provides a direct determination of the Cabibbo-Kobayashi-Maskawa matrix
element $|V_{us}|=0.22333(44)_{f_+(0)}(42)_\text{exp}$, with a theory error
that is, for the first time, at the same level as the experimental error. The
uncertainty of the semileptonic determination is now similar to that from
leptonic decays and the ratio $f_{K^+}/f_{\pi^+}$, which uses $|V_{ud}|$ as
input. Our value of $|V_{us}|$ is in tension at the 2--$2.6\sigma$ level both
with the determinations from leptonic decays and with the unitarity of the CKM
matrix. In the test of CKM unitarity in the first row, the current limiting
factor is the error in $|V_{ud}|$, although a recent determination of the
nucleus-independent radiative corrections to superallowed nuclear $\beta$
decays could reduce the $|V_{ud}|^2$ uncertainty nearly to that of
$|V_{us}|^2$. Alternative unitarity tests using only kaon decays, for which
improvements in the theory and experimental inputs are likely in the next few
years, reveal similar tensions. As part of our analysis, we calculated the
correction to $f_+^{K\pi}(0)$ due to nonequilibrated topological charge at
leading order in chiral perturbation theory, for both the full-QCD and the
partially-quenched cases. We also obtain the combination of low-energy
constants in the chiral effective Lagrangian
$[C_{12}^r+C_{34}^r-(L_5^r)^2](M_\rho)=(2.92\pm0.31)\cdot10^{-6}$.
