Instant TravelLuminary Oddities — Enter the Portal
Research Archive

Research Article · Preprint Mathematics

Sharp limitations of two permanental cofactor comparisons

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
Read / download PDF Publication filesCite this edition

Overview

Abstract

We study two auxiliary cofactor comparisons motivated by permanent Hadamard inequalities. For a complex positive definite correlation matrix B, let R be its transposed permanental cofactor matrix and L that of its entrywise squared modulus. We compare zᵀLz and zᵀ|R|◦²z, where zᵢ = |xᵢ|², with (per B)(x*B⁻¹x)(x*Rx). Both comparisons have sharp universal constant one in orders one and two. At every fixed order at least three, neither admits any finite universal constant, even among dense irreducible positive definite correlation matrices with nonreal triangle phases. An explicit roots-of-unity family gives quantitative divergence. The cancellation mechanism is classical; the result concerns these auxiliary comparisons and does not disprove Chollet’s inequality.

Keywords permanent · permanental cofactors · matrix inequalities · correlation matrices · roots of unity · sharp constants

Scope of the work

The note classifies two specific auxiliary cofactor comparisons: sharp constant one in orders one and two, and no finite universal constant at each fixed complex order at least three. It does not disprove Chollet’s conjecture, determine the sign of the full compensated border expression, or decide the separate cofactor matrix comparison. Classical roots-of-unity cancellation is credited.

Reproduction notes

The four-file package contains the complete four-page PDF, editable LaTeX source, reader instructions, and a SHA-256 manifest. The proof is analytic and does not require numerical data or computational certificates. Compile main.tex with Tectonic or a standard LaTeX installation.

The manuscript

Read the paper

Open PDF

The complete manuscript, as published in this edition.

If the reader is unavailable on your device, open or download the PDF directly.

Author disclosures

AI-assisted derivation, exact symbolic checks and separate computational checks were used in developing the note. The theorem is analytic and its proof does not depend on numerical search or finite-sample inference. Internal scientific and reader checks do not constitute external human refereeing. The author is Scott Schubr, Luminary Oddities LLC.

References and related publications

  1. B. Reznick. An inequality for products of polynomials. Proceedings of the American Mathematical Society 117 (1993), 1063–1073.
  2. B. Seron, L. Novo and N. J. Cerf. Boson bunching is not maximized by indistinguishable particles. Nature Photonics 17 (2023), 702–709.
  3. L. Pioge, K. K. Pietrasz, B. Seron, L. Novo and N. J. Cerf. A logical implication between two conjectures on matrix permanents. Linear Algebra and its Applications 725 (2025), 309–318.

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Sharp limitations of two permanental cofactor comparisons (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/sharp-limitations-two-permanental-cofactor-comparisons/v/1

BibTeX
@techreport{luminary_sharp_limitations_two_permanental_cofactor_comparisons_v1,
  title = {Sharp limitations of two permanental cofactor comparisons},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/sharp-limitations-two-permanental-cofactor-comparisons/v/1}
}

Loading discussion…