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A quantitative Chollet inequality for rank-four matrices of order seven

Scott Schubr

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Abstract

For every complex Hermitian positive semidefinite matrix A of order seven and rank at most four, we prove per(A ∘ bar(A))≤ (192)/(245)(per A)^2. The inequality is strict when all diagonal entries are positive. The main input is the bound per(A ∘ bar(A))≤(3/5)(per A)^2 for every singular positive semidefinite matrix of order five. An exact rational principal-minor sum-of-squares certificate proves this singular-boundary inequality. A rank-preserving border comparison then gives the same constant in order six and the stated constant in order seven. The certificate and a standalone exact-integer verifier accompany the paper. Cauchy–Schwarz gives the corresponding two-factor inequalities when both factors have rank at most four.

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Edition files

Available editions

  1. Edition 2 · 2026-10-27 · public
  2. Edition 1 · 2026-10-20 · public

Research details

Subjects
Mathematics · Linear Algebra and Matrix Inequalities
Keywords
permanent, positive semidefinite matrix, Chollet inequality, rank four, sum of squares, computer-assisted proof
Manuscript date
2026-10-20
Reader reuse
No additional permission granted

Scope

An exact singular-five 3/5 self-Chollet certificate and rank-preserving transfer yield quantitative rank-four bounds through order seven, including 192/245 at order seven and paired consequences for two rank-four factors.

Disclosures

The author used AI-assisted tools for research, drafting and verification. Responsibility for the manuscript and its claims remains with the author. This Archive preprint has not undergone external journal peer review.

Reproduction

The eight-file source package includes the six-page manuscript, editable LaTeX source, exact rational certificate, standalone integer verifier, portable PDF build helper, reader instructions, citation metadata and SHA-256 manifest. Run python code/verify_certificate.py, also under -O. The certificate and full analytic argument are both essential.

References and related resources

  1. J. Chollet, Is there a permanental analogue to Oppenheim’s inequality?, American Mathematical Monthly 89 (1982), 57–58.
  2. S. Schubr, Chollet's inequality for positive semidefinite matrices of rank at most four, Luminary Research Archive, author-published edition1 (2026).
  3. Q. Li, Q. Zhao and W. Luo, Chollet’s Permanent Conjecture Through Order Six via Border Induction, Preprints.org 202608.2196v1, August 31, 2026.
  4. Y. Lavi, A Conjecture on the Permanent of Positive Semidefinite Matrices, preprint dated September 26, 2026; Zenodo record posted September 27, 2026.
  5. Y. Lavi, Exact proof supplement to A Conjecture on the Permanent of Positive Semidefinite Matrices, version 4, September 26, 2026.
  6. Y. Ma, Quantitative Chollet Inequalities for Matrices of Rank at Most Four: Spectral bounds and an order-nine tight-frame construction, arXiv:2609.37461v1, September 28, 2026.
  7. Artificial Hyperintelligence Eve, QX-NEPHRIEL / Seventh-String Closure: Quantitative Rank-Three Chollet Inequalities, version 1.0.0, research release prepared for Maciej Nowicki, October 3, 2026.