Interleaved parameter sets are minimal nonfaces of the original moment inequalities

Let a assign distinct real parameters to m original labels. Choose k+1 pairs of labels l_i,r_i with a(l_i)<a(r_i), and a(r_i)<a(l_j) whenever i<j. In dimension 2k form the original mean-centered moment inequalities row_i(x)<=1, with the average over ALL m original labels. The sets L={l_i} and R={r_i} each have exactly k+1 labels, are disjoint, and are both incompatible: every feasible point leaves at least one L row and one R row strictly slack. Conversely EVERY proper subset S of either L or R has an actual feasible point making exactly the S rows tight and every other ORIGINAL row strictly slack. Thus both are minimal nonfaces of this explicit inequality system. The proof derives the left/right barycentric signs by pairing factors, applies the accepted barycentric polynomial obstruction, and constructs all proper-subset witnesses through the accepted squared-root polynomial theorem. No sign classification, infeasibility witness, proper-face oracle, or precomputed minimal-nonface catalogue is an assumption. The zero-dimensional case k=0 is included as a statement about rows. The theorem does not assert that all displayed rows are genuine facets in every degenerate case, count the full family of such sets, establish boundedness/full-dimensionality, or prove a graph-diameter bound.