We report the results of a lattice quantum chromodynamics calculation of $F_K/F_pi$ using M{o}bius domain-wall fermions computed on gradient-flowed $N_f=2+1+1$ highly-improved staggered quark (HISQ) ensembles. The calculation is performed with five values of the pion mass ranging from $130 lesssim m_pi lesssim 400$ MeV, four lattice spacings of $asim 0.15, 0.12, 0.09$ and $0.06$ fm and multiple values of the lattice volume. The interpolation/extrapolation to the physical pion and kaon mass point, the continuum, and infinite volume limits are performed with a variety of different extrapolation functions utilizing both the relevant mixed-action effective field theory expressions as well as discretization-enhanced continuum chiral perturbation theory formulas. We find that the $asim0.06$ fm ensemble is helpful, but not necessary to achieve a subpercent determination of $F_K/F_pi$. We also include an estimate of the strong isospin breaking corrections and arrive at a final result of $F_{K^pm}/F_{pi^pm} = 1.1942(45)$ with all sources of statistical and systematic uncertainty included. This is consistent with the Flavour Lattice Averaging Group average value, providing an important benchmark for our lattice action. Combining our result with experimental measurements of the pion and kaon leptonic decays leads to a determination of $|V_{us}|/|V_{ud}| = 0.2311(10)$.
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