inference Complete proof in proofs/proof_t6_t7.md (Theorem 1), via Lemma 5 (rigidity): under L(6) every minimal empty-intersection family is the rigid m=4 gadget with span exactly 16, every two of whose edges meet in exactly 2 vertices; any edge avoiding such a 2-set would give three edges with no common vertex, contradicting the 3-wise intersection property that L(6) forces (any minimal witness of size 2 or 3 spans <= 3r-3 = 15). Rigidity classification machine-verified by exhaustive enumeration of all MEIF type-vectors (code/classify_minimal.py: unique survivor at r=6); key step verified exhaustively over all 74613 one-edge extensions of the gadget on 22 vertices (code/gadget_check.py: all 489 L(6)-compatible extensions meet E1 ∩ E2); randomized falsification found no L(6) family with tau >= 3 (code/random_maximal_search.py). Lower bound gadget verified against the literal definition over all 58650 vertex subsets.