Preprint Mathematics
A quantitative Chollet inequality for rank-four matrices of order seven
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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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Scott Schubr. (2026). A quantitative Chollet inequality for rank-four matrices of order seven (Version 1). Luminary Research Archive. https://luminaryoddities.com/researcharchive/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 = {Preprint},
note = {Version 1},
url = {https://luminaryoddities.com/researcharchive/a-quantitative-chollet-inequality-for-rank-four-matrices-of-order-seven/v/1}
}Reader correspondence
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