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We report on a sub-percent scale determination using the omega baryon massand gradient-flow methods. The calculations are performed on 22 ensembles of$N_f=2+1+1$ highly improved, rooted staggered sea-quark configurationsgenerated by the MILC and CalLat Collaborations. The valence quark action usedis M\"obius Domain-Wall fermions solved on these configurations after agradient-flow smearing is applied with a flowtime of $t_{\rm gf}=1$ in latticeunits. The ensembles span four lattice spacings in the range $0.06 \lesssim a\lesssim 0.15$ fm, six pion masses in the range $130 \lesssim m_\pi \lesssim400$ MeV and multiple lattice volumes. On each ensemble, the gradient-flowscales $t_0/a^2$ and $w_0/a$ and the omega baryon mass $a m_\Omega$ arecomputed. The dimensionless product of these quantities is then extrapolated tothe continuum and infinite volume limits and interpolated to the physicallight, strange and charm quark mass point in the isospin limit, resulting inthe determination of $\sqrt{t_0}=0.1422(14)$ fm and $w_0 = 0.1709(11)$ fm withall sources of statistical and systematic uncertainty accounted for. Thedominant uncertainty in this result is the stochastic uncertainty, providing aclear path for a few-per-mille uncertainty, as recently obtained by theBudapest-Marseille-Wuppertal Collaboration.

physical review. d/physical review. d.

Scale setting the M\"obius Domain Wall Fermion on gradient-flowed HISQ  action using the omega baryon mass and the gradient-flow scales $t_0$ and  $w_0$