Characterization of the complementary order-five SNIEP region

Determine a finer description of the portion of the residual order-five symmetric nonnegative inverse eigenvalue problem region lying outside the newly identified impossibility region, either by deriving additional impossibility conditions or by constructing new symmetric nonnegative realizations.

Background

The paper studies the order-five symmetric nonnegative inverse eigenvalue problem (SNIEP) in the residual low-trace region of spectra not covered by previously known results. It identifies an explicit nonempty impossibility subregion consisting of spectra satisfying the defining residual-region conditions together with the inequality $4\lambda_1\leq 9\lambda_3-S$. No spectrum in that subregion admits a real symmetric entrywise-nonnegative realization.

The unresolved portion is the complementary region within the residual region, characterized by the strict inequality $4\lambda_1>9\lambda_3-S$. The paper leaves open whether further parts of this region can be ruled out by new necessary conditions or whether additional spectra in it can be realized through constructive methods.