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The CKM angle $\gamma$ is measured for the first time from mixing-induced$CP$ violation between $B^0_s \rightarrow D_s^\mp K^\pm \pi^\pm \pi^\mp$ and$\bar{B}^0_s \rightarrow D_s^\pm K^\mp \pi^\mp \pi^\pm$ decays reconstructed inproton-proton collision data corresponding to an integrated luminosity of 9${\rm fb}^{-1}$ recorded with the LHCb detector. A time-dependent amplitudeanalysis is performed to extract the $CP$-violating weak phase$\gamma-2\beta_s$ and, subsequently, $\gamma$ by taking the$B^0_s$-$\bar{B}^0_s$ mixing phase $\beta_{s}$ as an external input. Themeasurement yields $\gamma = (44 \pm 12)^\circ$ modulo $180^\circ$, wherestatistical and systematic uncertainties are combined. An alternativemodel-independent measurement, integrating over the five-dimensional phasespace of the decay, yields $\gamma = (44^{\,+\,20}_{\,-\,13})^\circ$ modulo$180^\circ$. Moreover, the $B^0_s$-$\bar{B}^0_s$ oscillation frequency ismeasured from the flavour-specific control channel $B^0_s \rightarrow D_s^-\pi^+ \pi^+ \pi^-$ to be $\Delta m_s = (17.757 \pm 0.007 \,({\rm stat.}) \pm0.008 \,({\rm syst.})) \text{ps}^{-1}$, consistent with and more precise thanthe current world-average value.

Measurement of the CKM angle $\gamma$ and $B^0_s$-$\bar{B}^0_s$ mixing frequency with $B^0_s \rightarrow D_s^\mp h^\pm \pi^\pm \pi^\mp$ decays