Title Convex quotients of covering-process orders

Description A covering process grows a finite poset from a unique minimum by adding a new maximum of Hasse in-degree 1 or 2. Companion 32 characterises those posets by Hasse degrees, independently of a stored birth word. The present note asks what happens if, instead of adding a vertex, one identifies a nonempty subset to a point. The identification yields a leftover poset if and only if the subset is order-convex. Leftover Hasse degrees do not increase. The quotient remains covering if and only if the merged point still satisfies the covering bound. Two-element antichains and closed intervals are convex specializations; both overflow kinds inhabit on those specializations and on mixed convex sets. Adding a comparability between an incomparable pair keeps the ground set. Every covering poset of size at most 4 has only covering quotients and only covering added comparabilities. Overflow of merged out-degree, and of added-comparability out-degree, first occur at size 5; overflow of an already-born in-degree under added comparability first occurs at size 6; overflow of merged in-degree first occurs at size 7: all four bounds are sharp. Across the 170 unlabelled covering posets of size at most 7 there are 7,558 nonempty order-convex subsets, of which 6,285 are nontrivial, and 1,508 ordered incomparable pairs. None of these operations is a covering birth. No generation rule is constructed. Covering histories are finite posets of the mathematical type of finite causal sets. They are not identified with physical spacetime. Covering birth is not Rideout–Sorkin sequential growth. This note does not derive spacetime, a unique sequential growth law, or an interpolating continuum. It does not replace the parent programme manuscript, and it is not a new version of Companions 13–34.