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Strong and energy-bounded permanent inequalities on bipartite supports

Scott Schubr

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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.

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Edition files

Available editions

  1. Edition 3 · 2026-10-28 · public
  2. Edition 2 · 2026-10-27 · public
  3. Edition 1 · 2026-10-23 · public

Research details

Subjects
Mathematics · Linear Algebra and Matrix Inequalities
Keywords
permanent inequalities, positive semidefinite matrices, bipartite supports, Chollet conjecture, Bernstein certificates
Manuscript date
2026-10-23
Reader reuse
No additional permission granted

Scope

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.

Disclosures

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

Reproduction

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.

References and related resources

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  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.