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Strong Self-Permanent Inequalities for Complex Four-Row Bipartite Supports

Scott Schubr ORCID

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Abstract

For every finite column count, we prove the strong self-permanent inequality for complex positive semidefinite matrices whose off-diagonal support lies in a bipartite graph with one part of size four. The conclusion bounds the permanent of the entrywise squared modulus by the permanent of the original matrix times the product of its diagonal entries. The proof removes column-dependent moment restrictions: a matching-polytope bound controls row overlaps, paired endpoint deletion gives a column-independent scalar estimate, and a large-entry transfer completes the argument. Arbitrary complex phases, missing edges and singular matrices are included. A separate extension establishes the same inequality on five-by-five bipartite supports by combining the analytic reductions with an exact tensor Bernstein certificate. We also prove a dimension-free energy criterion for the ordinary self-Chollet inequality, covering all positive semidefinite bipartite supports with one part of size at most twelve, together with further five-row regimes. The earlier three-row core and the large-entry transfer are proved within the manuscript. The finite certificate is reconstructed by a portable exact-arithmetic supplement.

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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
2026-10-28
Reader reuse
none

Scope

Strong self-permanent comparison for complex positive semidefinite K₄,q supports with every finite q; a K₅,₅ exact-certificate extension; an ordinary self-Chollet energy criterion including K₁₂,q; and specified five-row regimes.

Disclosures

Generative AI tools substantially assisted mathematical exploration, proof development, drafting, verification-code development and internal critical checking. Scott Schubr directed these processes through his longitudinal N=1 AI governance research. OpenAI Codex (GPT-6.1 Sol; gpt-6.1-sol) assisted the journal-presentation revision. The author assumes responsibility for all content, mathematical arguments and attribution. The work was self-funded.

Reproduction

The complete manuscript has 17 pages. The six-member source package contains main.tex, the manuscript PDF, code/verify_bipartite_certificate.py, Read me.txt, Citation.bib and SHA256 manifest.txt. Compile main.tex. Run the Python checker normally and with -O to reconstruct 3,145,728 exact rational Bernstein coefficients.

References and related resources

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