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Strong permanent inequalities on three chorded seven-cycle supports

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

We prove per(|A|^2)≤per(A) for complex positive semidefinite correlation matrices supported on a seven-cycle with chords 03,15,26. More precisely, the difference dominates one fiftieth of the matching-partition loss. The proof combines elementary cycle estimates with an explicit nonnegative integer polynomial identity, supplied with a standard-library verification program. The two-chord subgraph with chords 03,15 inherits the same bound. For the crossing placement 03,14 we give a separate analytic proof with constant 1/12. All results include missing edges and singular matrices. The normalized arguments require only vertex-star magnitude bounds, and we describe equality and diagonal scaling.

Keywords permanent inequality · positive semidefinite matrix · Chollet conjecture · seven-cycle support · graph matching · exact integer certificate

Scope of the work

Quantitative strong permanent inequalities on the three specified chorded seven-cycle supports and their subgraphs, with matching-loss factors 1/50 or 1/12 under the stated vertex-star magnitude bounds.

Reproduction notes

The paper has 7 pages. The editable-source archive contains 6 files. Compile manuscript/main.tex. The standard-library verify_certificate.py reconstructs the 323-term integer identity and graph budgets.

The manuscript

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Author disclosures

AI tools assisted drafting and verification, as disclosed in the manuscript; the author is responsible for the final statements, proofs and attribution. Internal analytic and exact-code reviews are not external human refereeing. Originality remains provisional. The preceding cycle/single-chord paper, structural graph literature, cycle expansion and polynomial-positivity framework are credited.

References and related publications

  1. J. Chollet, Is there a permanental analogue to Oppenheim’s inequality?, American Mathematical Monthly 89 (1982), 57–58
  2. S. W. Drury, A counterexample to a question of Bapat and Sunder (2016)
  3. P. Pant and R. Singh, Structural Classes for Chollet’s Permanent Conjecture, arXiv:2604.24192v2
  4. S. Schubr, Strong permanent inequalities for complex positive semidefinite cycle and single-chord supports, edition 1 (2026)
  5. D. Handelman, Representing polynomials by positive linear functions on compact convex polyhedra (1988)
  6. R. Mehatari, M. R. Kannan and A. Samanta, On the adjacency matrix of a complex unit gain graph (2022)

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Strong permanent inequalities on three chorded seven-cycle supports (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/strong-permanent-inequalities-complex-two-three-chord-seven-cycle-supports/v/2

BibTeX
@techreport{luminary_strong_permanent_inequalities_complex_two_three_chord_seven_cycle_supports_v2,
  title = {Strong permanent inequalities on three chorded seven-cycle supports},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 2},
  url = {https://research.luminaryoddities.com/strong-permanent-inequalities-complex-two-three-chord-seven-cycle-supports/v/2}
}

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