Catalyst-rank limits for second-order entanglement concentration

Exact catalysts can improve the second-order yield of pure-state entanglement concentration, but the optimal yield does not specify how the catalyst must grow. We obtain a necessary rank condition for this improvement. Let the source have a fixed nonuniform finite squared Schmidt vector, with entropy $H$ and information variance $V$ defined using base-two logarithms, and fix a required squared fidelity $q\in(0,1)$. For pure outputs of $n$ source copies, with a pure catalyst returned exactly and uncorrelated, suppose that the target contains $nH+b\sqrt n+o(\sqrt n)$ Bell pairs. If the catalyst rank $K_n$ satisfies $\limsup \log_2K_n/\sqrt n\le c<\infty$, we give an explicit necessary fidelity bound strictly below $e^{-b^2/(2V)}$ for every $b>0$. Consequently, attaining the known unrestricted coefficient $\sqrt{2V\ln(1/q)}$ requires $\log_2K_n/\sqrt n\to\infty$. Catalysts with $\log_2K_n=o(\sqrt n)$ cannot improve the unassisted coefficient. The proof combines a fidelity concentration estimate, a Gaussian limit of the majorization inequalities, and a positive tail term retained by the rank constraint. The record contains a standalone PDF preprint and an accompanying source archive with the LaTeX manuscript, figure files, reproducible Python figure source and build instructions. No experimental dataset is used. The preprint has not undergone journal peer review. AI assistance is disclosed in the manuscript.