inference Complete proof in proofs/proof_t6_t7.md (Theorem 2), via Lemma 6 (near-rigidity): surviving MEIFs at r=7 are (m,span) in {(4,19),(4,20),(5,19),(5,20)}, rigid up to at most one extra vertex of a 2-element type. Case A (an m=4 MEIF exists): some pair of its edges meets in exactly 2 vertices; the 3-wise intersecting pinch gives tau <= 2. Case B (no m=4 MEIF): every 4 edges share a vertex; the m=5 MEIF has every triple intersection equal to the complementary pair of distinguished vertices, so every edge of G contains >= 4 of the 5 distinguished vertices, and any 2 of them form a transversal. Classification machine-verified exhaustively (code/classify_minimal.py: exactly 7 surviving labeled type-vectors at m=4 and 11 at m=5, with the intersection properties used); both lower-bound witnesses verified against the literal definition (all 988095 vertex subsets for the gadget; planted-search sanity check for the m=5 witness); randomized falsification (gadget-planted, m=5-planted, unplanted) found no L(7) family with tau >= 3.