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. The new full-domain cases are orders eight and nine, where we obtain self-conjugate factors 243/320 and 1215/1352. A controlled-loss extension of the occupation-border comparison of Li, Zhao and Luo spends the strict reserve in an established singular-five bound with factor 3/5. We include the complete interpretation and positivity proof of its exact principal-minor sum-of-squares certificate, together with the unchanged data and standalone verifier. Rodtes’ order-four theorem and Ma’s theorem for orders at least ten complete the all-order conclusion. The coefficient-one inequality is strict for positive diagonal in orders at least five. The certificate and quantitative base are prior inputs; the new continuation balances two multiplicative losses.
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- Rank_Four_Chollet_All_Orders.pdf89.2 KiB · pdf
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77471ce16a45f09029455500461d61f98ff1ab2dc3b1b46ff8c4aa2bf8e33e22 - Revised_publication_package.zip179.7 KiB · package
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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
All-order Chollet inequality for complex Hermitian PSD pairs when BOTH factors have rank at most four; controlled-loss continuation closes full-domain orders eight and nine. Complete inherited singular-five certificate proof is assembled with unchanged data and checker. Orders at least ten use Ma’s cited theorem; orders at most four use Rodtes with identity padding.
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
The seven-page manuscript gives the complete analytic continuation and all-order consequence in the main text, with the full singular-five exact certificate proof in Appendix A. The nine-member source package contains the unchanged certificate, exact integer verifier, scalar checks, editable LaTeX source, portable build helper, reader instructions, citation metadata and member checksums.
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.