A New Impossibility Region for the $5\times5$ Symmetric Nonnegative Inverse Eigenvalue Problem

Jiashun Jin, Zheng Tracy Ke, Bingcheng Sui

arXiv:2608.19435v1 Primary: math.RA Secondary: math.SP

Abstract

We present a new impossibility region for the $5\times5$ symmetric nonnegative inverse eigenvalue problem. The region lies in the low-trace regime and, to the best of our knowledge, has not been identified previously. The proof uses a suitable diagonal shift to transform the problem to a critical high-trace boundary and then reduces a complementary commuting matrix to a weighted five-cycle. The characteristic polynomial and spectral identities of this five-cycle provide the main tools for deriving the resulting contradiction.

Disclosure

“the connection in Section 3 motivates further study of the symmetric doubly stochastic inverse eigenvalue problem (SDIEP), including the special spectral families arising from network modeling. Acknowledgements The authors used OpenAI’s GPT-5.6 in preparing this manuscript. All mathematical arguments were independently verified by the authors. 6 Appendix This section contains the proofs of Lemmas 4.1–4.5. The only external theorem used in these proofs is the following high-”
PDF page 23

Classification: Drafting limited passages
Multiplier: 5
Verified: 2026-08-22T18:15:39.749530Z

Structural counts

Pages 42 pdf

Theorems 2 source

Lemmas 5 source

Propositions 0 source

Corollaries 0 source

Definitions 0 source

Displayed equations 222 source

Bibliography entries 33 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file main-arXiv.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.