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Research Article · Preprint Mathematics

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

Scott Schubr

Luminary Oddities LLC

  • Mathematics
  • Linear Algebra and Matrix Inequalities
Primary subject
Mathematics
Subject subfield
Linear Algebra and Matrix Inequalities
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
1 · Permanent link
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Overview

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.

Keywords permanent · positive semidefinite matrix · Chollet inequality · rank four · sum of squares · computer-assisted proof

Scope of the work

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. No tight-frame, real-entry or spectral restriction is imposed within this class. The constants are not asserted optimal. This does not settle unrestricted Chollet, full rank-four order eight, general non-tight order nine, or a rank-unrestricted second factor.

Reproduction notes

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; no optimizer, discovery output or external certificate is needed. 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.

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Author 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, not Lavi’s strongest consequence. Internal AI-assisted review is not external human refereeing or journal acceptance. Novelty is supported relative to inspected primary sources; absolute priority remains provisional. The literature review was bounded, including an inaccessible relevant 2025 journal full text and incomplete inspection of the large prior supplement.

References and related publications

  1. J. Chollet, Is there a permanental analogue to Oppenheim’s inequality?, American Mathematical Monthly 89 (1982), 57–58.
  2. Q. Li, Q. Zhao and W. Luo, Chollet’s Permanent Conjecture Through Order Six via Border Induction, Preprints.org 202608.2196v1, August 31, 2026.
  3. Y. Lavi, A Conjecture on the Permanent of Positive Semidefinite Matrices, preprint dated September 26, 2026; Zenodo record posted September 27, 2026.
  4. Y. Lavi, Exact proof supplement to A Conjecture on the Permanent of Positive Semidefinite Matrices, version 4, September 26, 2026.
  5. 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.
  6. 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.

A stable scholarly record

Cite this edition

Scott Schubr. (2026). A quantitative Chollet inequality for rank-four matrices of order seven (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/a-quantitative-chollet-inequality-for-rank-four-matrices-of-order-seven/v/1

BibTeX
@techreport{luminary_a_quantitative_chollet_inequality_for_rank_four_matrices_of_order_seven_v1,
  title = {A quantitative Chollet inequality for rank-four matrices of order seven},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/a-quantitative-chollet-inequality-for-rank-four-matrices-of-order-seven/v/1}
}

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