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Research Article · Preprint Mathematics

Strong permanent inequalities for complex two- and three-chord seven-cycle supports

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

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. These structural results do not establish the unrestricted Chollet conjecture.

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

Scope of the work

The strong bound applies to the specified three-chord seven-cycle support and its subgraphs, including the two-chord 03,15 placement, with matching-loss constant 1/50. The crossing two-chord 03,14 placement has a separate analytic 1/12 bound. Arbitrary complex phases, missing edges and singular PSD endpoints are included. The normalized proof holds on the stated Hermitian vertex-star magnitude domain. It does not establish all triangle-free order-seven supports, general rank-six/order-seven matrices, arbitrary second PSD factors or unrestricted Chollet.

Reproduction notes

The six-member package contains the complete seven-page PDF, editable manuscript/main.tex, certificate.json, verify_certificate.py, reader instructions and MANIFEST.sha256. The Python standard-library verifier reconstructs 53 matchings, 15 cycles and the exact 323-term nonnegative integer identity with zero residual. Run python -B verify_certificate.py, optionally with -O. The analytic reduction, endpoint arguments and crossing-class proof are in the manuscript; no solver or private dependencies are needed.

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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 for complex two- and three-chord seven-cycle supports (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/strong-permanent-inequalities-complex-two-three-chord-seven-cycle-supports/v/1

BibTeX
@techreport{luminary_strong_permanent_inequalities_complex_two_three_chord_seven_cycle_supports_v1,
  title = {Strong permanent inequalities for complex two- and three-chord seven-cycle supports},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/strong-permanent-inequalities-complex-two-three-chord-seven-cycle-supports/v/1}
}

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