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Abstract

We prove joint log-concavity of the raw lattice-simplex sum ∑_(i+j+k≤ r)(A-i)_+^2(B-j)_+^2(C-k)_+^2 for every integer radius r≥0 and arbitrary positive real endpoints. The proof combines exhaustive cell coverage with exact positive-coefficient certificates over unbounded parameter orthants. Independent real exponents can all be at least two, and this coordinatewise threshold is sharp. In every dimension we also prove the corresponding theorem with up to two unrestricted occupancies and arbitrarily many binary matroid companions, including activities in [0,1]. This uses degree-elevated homogenization, the classical complete-log-concavity criterion and compensated composition.