The cubic threshold for coefficientwise log-concavity of multiple-Pochhammer series

Multiple-Pochhammer power series associated with a nonnegative log-concave sequence with no internal zeros were shown by Karp and Zhang to be coefficientwise log-concave at multiplicity two, while the cubic case remained open. We prove the cubic conjecture, in the equivalent fixed-sum form of coefficientwise Schur-concavity. Together with a degree-three obstruction for larger multiplicities, this yields the sharp classification: universal coefficientwise log-concavity holds precisely at multiplicities two and three. The proof reduces the coefficient inequalities to monotonicity of a symmetric residue kernel and shows that, in the cubic case, this kernel preserves the central ordering imposed by log-concavity.