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

Scott Schubr

Luminary Oddities

  • Mathematics
  • Linear Algebra and Matrix Inequalities
Primary subject
Mathematics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
2 · 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 ∘ 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.

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

Scope of the work

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.

Reproduction notes

The paper has 6 pages. The editable-source archive contains 8 files. Compile manuscript/main.tex or use code/build_pdf.py with Tectonic. Standard-library code/verify_certificate.py reconstructs the exact singular-five identity and positivity witnesses.

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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.

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Cite this edition

Scott Schubr. (2026). A quantitative Chollet inequality for rank-four matrices of order seven (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/a-quantitative-chollet-inequality-for-rank-four-matrices-of-order-seven/v/2

BibTeX
@techreport{luminary_a_quantitative_chollet_inequality_for_rank_four_matrices_of_order_seven_v2,
  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 2},
  url = {https://research.luminaryoddities.com/a-quantitative-chollet-inequality-for-rank-four-matrices-of-order-seven/v/2}
}

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