Mathematics & Statistics

We attack a published ladder of fifteen covering-design cells C(v,k,t), 11<=v<=20 with (k,t) in {(4,3),(5,3),(5,4)}, each carrying its Schonheim lower bound L. We independently recompute all fifteen bounds; rebuild from scratch and machine-verify constructions attaining L on three classical cells (SQS(14), SQS(16) as the 2-flats of AG(4,2), and the small Witt design S(4,5,11)); and prove EXACT minimum sizes of coverings invariant under two named permutation groups - the regular cyclic group Z_v and the 1-rotational Z_{v-1} fixing a point - via branch-and-bound over orbit covers whose completeness and admissible bound we prove. Our exact cyclic values include C_Z11(11,4,3)=55, C_Z12(12,4,3)=60, C_Z13(13,4,3)=91, C_Z14(14,4,3)=98, C_Z15(15,4,3)=135, C_Z16(16,4,3)=144 and C_Z12(12,5,4)=132, each exceeding L, so no covering of these cells that attains the Schonheim bound admits the corresponding symmetry; in contrast C_Z11(11,5,4)=66=L (a cyclic model of S(4,5,11)), and on (16,4,3) the two groups disagree: cyclic optimality costs 4 extra blocks while an optimal 140-block 1-rotational design exists. Every exhibited object passes an independent exhaustive verifier implemented twice (TypeScript and Python); every optimality claim comes from a completed search run with reported node count. Objects, verifiers and solvers are attached.