A quantitative Chollet inequality for rank-four matrices of order seven
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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 ∘ conjugate(A)) ≤ (192/245)(per A)². The inequality is strict when all diagonal entries are positive. The main input is the bound per(A ∘ conjugate(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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- Rank_Four_Chollet_Order_Seven.pdf80.9 KiB · pdf
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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
Complex Hermitian positive semidefinite matrices of rank at most four, of orders five, six and seven. With P(A) = per A and Q(A) = per(A ∘ conjugate(A)), the bounds are Q(A) ≤ (3/5)P(A)² in orders five and six and Q(A) ≤ (192/245)P(A)² in order seven. They are strict for positive diagonal; a zero diagonal gives P(A) = Q(A) = 0. The paired estimates require BOTH factors to have rank at most four. The class includes tight and non-tight frames, complex entries and arbitrary spectra.
Disclosures
AI tools assisted drafting and verification; the author is responsible for the content. The principal-minor feature cone and symmetry framework are credited to Lavi; the border mechanism is credited to Li–Zhao–Luo. The 119/120 comparison is identified as a direct dimension-counting consequence.
Reproduction
Manuscript revision 1.1, October 20, 2026; intended first public Archive edition 1. The eight-file package contains the complete six-page PDF, LaTeX source, essential exact rational certificate, standalone exact-integer verifier, portable PDF build helper, README, citation and seven-entry checksum manifest. Python 3.9 or later and its standard library suffice for certificate verification, including optimized mode. Rebuilding the PDF requires Tectonic and its LaTeX resources, or a compatible LaTeX installation. The README documents an optional existing cache and offline mode; availability of the required cached resources is separate from package integrity. Rebuilt PDF bytes may differ by compiler resources or metadata.
References and related resources
- J. Chollet, Is there a permanental analogue to Oppenheim’s inequality?, American Mathematical Monthly 89 (1982), 57–58.
- Q. Li, Q. Zhao and W. Luo, Chollet’s Permanent Conjecture Through Order Six via Border Induction, Preprints.org 202608.2196v1, August 31, 2026.
- Y. Lavi, A Conjecture on the Permanent of Positive Semidefinite Matrices, preprint dated September 26, 2026; Zenodo record posted September 27, 2026.
- Y. Lavi, Exact proof supplement to A Conjecture on the Permanent of Positive Semidefinite Matrices, version 4, September 26, 2026.
- 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.
- 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.