Conflicting interpretations of experimental data preclude the understanding of the quantum magnetic state of spin-orbit coupled d$^2$ double perovskites. Whether the ground state is a Janh-Teller-distorted order of quadrupoles or the hitherto elusive octupolar order remains debated. We resolve this uncertainty through direct calculations of all-rank inter-site exchange interactions and inelastic neutron scattering (INS) cross-section for the d$^2$ double perovskite series Ba$_2M$OsO$_6$ ($M$= Ca, Mg, Zn). Using advanced many-body first principles methods we show that the ground state is formed by ferro-ordered octupoles coupled within the ground-stated $E_g$ doublet. Computed ordering temperature of the single second-order phase-transition and gapped excitation spectra are fully consistent with observations. Minuscule distortions of the parent cubic structure are shown to qualitatively modify the structure of magnetic excitations.
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