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Research Article · Preprint Mathematics

Strong and energy-bounded permanent inequalities on bipartite 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 strong permanent contraction for every complex positive semidefinite matrix supported in K_(4,q), with any finite q, and in K_(5,5). The four-row proof is analytic; the five-by-five proof combines a retained third-minor budget with finite-column moment bounds and an exact tensor Bernstein certificate comprising 3,145,728 rational coefficients. Independently, a dimension-free energy criterion proves the ordinary self-Chollet inequality whenever the off-diagonal rectangle has squared row and column sums at most one and total squared energy at most 12+(201+16√(537))/988, including all positive semidefinite K_(12,q) supports. Strong five-row estimates for arbitrary finite column number are also obtained near row-isometries and under dispersed fourth moments. The structural core proofs, large-entry transfer and analytic reductions are given in full, with classical permanent norm, block and matching-polytope theorems credited. Arbitrary complex phases, missing edges and singular endpoints are included.

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

Scope of the work

Strong permanent comparison for every finite K₄,q and K₅,₅ support, an ordinary self-Chollet energy criterion including K₁₂,q, and additional stated arbitrary-column five-row strong regions.

Reproduction notes

The paper has 16 pages. The editable-source archive contains 7 files. Compile manuscript/main.tex. Run code/verify_bipartite_certificate.py with standard-library Python to reconstruct the exact tables and 3,145,728 Bernstein coefficients.

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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.

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Cite this edition

Scott Schubr. (2026). Strong and energy-bounded permanent inequalities on bipartite supports (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/strong-energy-bounded-permanent-inequalities-bipartite-supports/v/2

BibTeX
@techreport{luminary_strong_energy_bounded_permanent_inequalities_bipartite_supports_v2,
  title = {Strong and energy-bounded permanent inequalities on bipartite supports},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 2},
  url = {https://research.luminaryoddities.com/strong-energy-bounded-permanent-inequalities-bipartite-supports/v/2}
}

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