A Generating-Function Perspective on a Nonrealizable Trace-Zero Spectrum of Nonnegative 5 × 5 Matrices

We study a classical trace-zero spectrum that has played a central role in the analysis of the 5 × 5 nonnegative inverse eigenvalue problem. This spectrum is particularly illustrative because it fails to be realizable at the unperturbed parameter value, yet becomes realizable precisely once the symmetric perturbation exceeds a unique critical threshold. Using an exponential generating-function representation of power sums, we show that the refined Johnson-Loewy-London inequality is governed by a strictly increasing functional whose derivative is a polynomial in the perturbation parameter. This yields a transparent structural explanation of the sharp realizability threshold and recovers, in a unified way, earlier results of Salzmann, Friedland, and Laffey-Meehan. The method extends naturally to higher-order Johnson-Loewy-London inequalities and provides a convenient framework for analyzing parametrized families in the nonnegative inverse eigenvalue problem.