We perform a complete study of prompt double $J/��$ hadroproduction at leading order in the nonrelativistic-QCD factorization framework by including all possible pairings of the $c\bar{c}$ Fock states ${}^1\!S_0^{[8]}$, ${}^3\!S_1^{[1,8]}$, and ${}^3\!P_J^{[1,8]}$ with $J=0,1,2$. We find that the ${}^1\!S_0^{[8]}$ and ${}^3\!P_J^{[8]}$ channels of $J/��$ and $��^\prime$ production and the ${}^3\!P_J^{[1]}$ and ${}^3\!S_1^{[8]}$ channels of $��_{cJ}$ production, which have been overlooked so far, greatly dominate at large invariant masses and rapidity separations of the $J/��$ pair, and that their inclusion nearly fills the large gap between previous incomplete predictions within the color-singlet model and the recent measurement by the CMS Collaboration at the CERN LHC, leaving room for next-to-leading-order corrections of typical size.

5 pages, 4 figures, matches journal version