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Strong and energy-bounded permanent inequalities on bipartite supports

Scott Schubr

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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 within the stated supports.

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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 inequalities for all finite K₄,q and full K₅,₅; ordinary energy bounds for all K₁₂,q; additional specified five-row regions. Replaces two unpublished standalone submissions while preserving all posted finite-column predecessors.

Disclosures

Author-posted preprint. AI-assisted internal research and verification were used.

Reproduction

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.

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).
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  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.