We establish that a random -CNF formula with clauses having literals each, over a set of variables, is asymptotically almost surely renamable-Horn if . This lower bound on Horn non-renamability coincides with the upper bound established by Franco and Van Gelder in 2003 for a random -CNF formula to be q-Horn (Boros et al., 1994) or SLUR (Schlipf et al., 1995). Put together, these two results imply that the renamable-Horn, q-Horn and SLUR properties for random -CNFs share a common sharp threshold at . An immediate consequence is that the number of -CNF formulas which are q-Horn or SLUR, but not renamable-Horn, is asymptotically negligible as .