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For $n\\ge2d+2$, the classical construction of Ahlswede and Khachatrian, later generalized by Mubayi and Zhao, gives \\[\n  \\mathsf{M}_d(n)\\ge \\binom{n-1}{d}+\\binom{n-4}{d-2}. \\] We introduce a two-cover lifting construction and prove the recursive lower bound \\[\n  \\mathsf{M}_d(n)\\ge\n  \\binom{n-1}{d}+\\binom{n-4}{d-2}+\\mathsf{M}_{d-3}(n-5) \\] for every $d\\ge 3$ and $n\\ge d+3$. 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