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Research Article · Preprint Mathematics

Strong and energy-bounded permanent inequalities on bipartite 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

Strong permanent contraction is proved for every complex positive semidefinite matrix supported in K₄,q, for any finite q, and in K₅,₅. The four-row proof is analytic; the five-by-five proof combines a retained third-minor budget, finite-column moment bounds and an exact tensor Bernstein certificate with 3,145,728 rational coefficients. Independently, a dimension-free energy criterion proves the ordinary self-Chollet inequality under squared row and column bounds of one and total squared energy at most 12 + (201 + 16√537)/988, including all positive semidefinite K₁₂,q supports. Strong five-row estimates with arbitrary finite column count are also obtained near row-isometries and under dispersed fourth moments. Structural proofs, large-entry transfer and analytic reductions are included, with classical inputs credited. Complex phases, missing edges and singular endpoints are covered; full strong contraction for arbitrary K₅,q is not claimed.

Keywords permanent inequalities · positive semidefinite matrices · bipartite supports · Chollet conjecture · Bernstein certificates

Scope of the work

Strong inequalities for all finite K₄,q and full K₅,₅; ordinary energy bounds for all K₁₂,q; additional specified five-row regions. No unrestricted K₅,₆, all-K₅,q, generic rank-five or unrestricted Chollet theorem. Replaces two unpublished standalone submissions while preserving all posted finite-column predecessors.

Reproduction notes

The seven-member package includes the manuscript, complete LaTeX source and code/verify_bipartite_certificate.py. Python 3.10 or later with its standard library reconstructs 3,145,728 exact Bernstein coefficients on 3,072 boxes and checks 3,072 independent midpoint evaluations. Both normal and optimized Python retain the explicit acceptance checks. The manuscript supplies the continuum reductions.

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Author disclosures

Author-posted preprint. AI-assisted internal research and verification do not constitute external human peer review or proof-assistant formalization.

References and related publications

  1. L. B. Beasley, An inequality on permanents of Hadamard products, Bulletin of the Korean Mathematical Society 37 (2000), no. 3, 633–639.
  2. J. Chollet, Is there a permanental analogue to Oppenheim’s inequality?, American Mathematical Monthly 89 (1982), 57–58.
  3. S. W. Drury, A counterexample to a question of Bapat and Sunder, Electronic Journal of Linear Algebra 31 (2016), 69–70. https://journals.uwyo.edu/index.php/ela/article/view/1631.
  4. E. Carlen, E. H. Lieb and M. Loss, An inequality of Hadamard type for permanents, Methods and Applications of Analysis 13 (2006), no. 1. https://doi.org/10.4310/MAA.2006.v13.n1.a1. Square form: arXiv:math/0508096v1, Theorem 1.1, https://arxiv.org/html/math/0508096v1.
  5. E. H. Lieb, Proofs of some conjectures on permanents, Journal of Mathematics and Mechanics 16 (1966), 127–134.
  6. J. Edmonds, Maximum matching and a polyhedron with 0,1-vertices, Journal of Research of the National Bureau of Standards, Section B 69B (1965), 125–130. https://doi.org/10.6028/jres.069B.013.
  7. P. Pant and R. Singh, Structural Classes for Chollet’s Permanent Conjecture, arXiv:2604.24192v2 (2026). https://arxiv.org/abs/2604.24192v2.
  8. S. Sa-nguansin and K. Rodtes, Permanents of correlation matrices and the Chollet permanental conjecture, Linear and Multilinear Algebra 73 (2025), no. 18, 4084–4096. https://doi.org/10.1080/03081087.2025.2588584.
  9. S. Schubr, Strong permanent inequalities for three-row bipartite and triangle-free supports, Luminary Research Archive, first edition (2026).
  10. S. Schubr, Strong permanent contraction on four-row bipartite supports through six columns, Luminary Research Archive, first edition (2026).
  11. S. Schubr, Strong permanent contraction on four-by-seven bipartite supports, Luminary Research Archive, first edition (2026).
  12. S. Schubr, Row-dependent bounds for strong permanent contraction on four-by-eight supports, Luminary Research Archive, first edition (2026).
  13. Z. Zeng, H. Liu and K. Ratnavelu, Sharp Self-Chollet Inequalities in Order Five: Rank Two and Structured Rank Three, Preprints.org (2026), version 1, posted 18 August 2026, Section 6. https://www.preprints.org/manuscript/202608.1240/v1.

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Strong and energy-bounded permanent inequalities on bipartite supports (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/strong-energy-bounded-permanent-inequalities-bipartite-supports/v/1

BibTeX
@techreport{luminary_strong_energy_bounded_permanent_inequalities_bipartite_supports_v1,
  title = {Strong and energy-bounded permanent inequalities on bipartite supports},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/strong-energy-bounded-permanent-inequalities-bipartite-supports/v/1}
}

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