Chollet's inequality for positive semidefinite matrices of rank at most four
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Abstract
We prove Chollet's inequality in every matrix order for pairs of complex Hermitian positive semidefinite matrices when both factors have rank at most four. A controlled-loss extension of the occupation-border comparison of Li, Zhao and Luo propagates the strict reserve in an established singular-five bound with factor 3/5. We optimize the scalar balancing parameter, sharpen the resulting factors in orders seven through nine, and obtain an exact factor below 0.959 in order ten. The all-order conclusion combines this continuation with Rodtes' order-four theorem and Ma's large-order theorem. We include the complete interpretation and positivity proof of the inherited principal-minor sum-of-squares certificate, together with its unchanged data and standalone verifier. The coefficient-one inequality is strict for positive diagonal in orders at least five. The certificate and quantitative base are prior inputs; the continuation optimizes two explicitly defined coefficientwise losses.
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Research details
- Subjects
- Mathematics · Linear Algebra and Matrix Inequalities
- Keywords
- permanent, positive semidefinite matrices, Chollet inequality, rank four, Hadamard product, border comparison
- Manuscript date
- 2026-10-22
- Reader reuse
- none
Scope
We prove Chollet's inequality in every matrix order for pairs of complex Hermitian positive semidefinite matrices when both factors have rank at most four.
Disclosures
ChatGPT Sol6.1 assisted research, proof development, drafting and verification through workflows developed in the author's AI governance research. Scott Schubr directed and reviewed the work and assumes responsibility for all content.
Reproduction
Complete searchable manuscript and 9-member source package with editable sources and necessary reader-facing verification material. Build instructions and computational scope are stated in the package; proofs are given in the manuscript.
References and related resources
- J. Chollet, Is there a permanental analogue to Oppenheim’s inequality?, American Mathematical Monthly 89 (1982), 57–58. doi:10.1080/00029890.1982.11995380.
- S. Schubr, A quantitative Chollet inequality for rank-four matrices of order seven, research release, Archive edition 1, 20 October 2026. Luminary Research Archive, edition 1.
- K. Rodtes, Chollet’s permanent conjecture for 4 × 4 matrices, Linear and Multilinear Algebra 72 (2024), 2633–2638. doi:10.1080/03081087.2023.2279150.
- Y. Ma, Quantitative Chollet Inequalities for Matrices of Rank at Most Four: Spectral bounds and an order-nine tight-frame construction, arXiv:2609.37461v1, 28 September 2026. arXiv:2609.37461v1.
- Q. Li, Q. Zhao and W. Luo, Chollet’s Permanent Conjecture Through Order Six via Border Induction, Preprints.org 202608.2196v1, 31 August 2026. doi:10.20944/preprints202608.2196.v1.
- Y. Lavi, A Conjecture on the Permanent of Positive Semidefinite Matrices, preprint dated 26 September 2026, Zenodo record posted 27 September 2026. doi:10.5281/zenodo.23001182.
- Y. Lavi, Exact proof supplement to A Conjecture on the Permanent of Positive Semidefinite Matrices, version 4, 26 September 2026. doi:10.5281/zenodo.22976216.