t(11) = t(12) = 3, with the upper bounds proved self-containedly (sharpened fatness sum-bound: any 3-fat 4-edge MEIF has a pairwise intersection of size exactly 3, which is a global transversal; the m>=5 cases reduce to tau <= 2 or a triple intersection of size <= 3) and the lower bound H(r,7,5) machine-certified (21 edges, tau = 3 exact, local property verified exhaustively at the Y-level and, at r=11, independently on the literal 133-vertex hypergraph; the r=11 instance is tight: minimum empty-intersection span 31 against threshold 31).