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Research Article · Preprint Mathematics

Strong permanent contraction on four-by-seven 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 that every complex positive semidefinite matrix A supported in K_(4,7) satisfies per(|A|^2)≤per(A)∏_i a_(ii). The normalized off-diagonal block is a four-by-seven operator contraction. The new argument retains third-order minor contributions and controls their coupled phase cancellations through the row Gram matrix. A split according to the smallest squared row norm separates an analytic corner reduction from a balanced-row polynomial inequality. The latter is verified by an exact rational tensor Bernstein certificate on 256 boxes. An established all-column large-entry theorem covers the complementary region. The computer-assisted result includes arbitrary phases, missing edges and singular endpoints.

Keywords permanent inequalities · positive semidefinite matrices · bipartite supports · matrix contractions · Bernstein certificates · computer-assisted proof

Scope of the work

Computer-assisted strong permanent contraction on every complex PSD K₄,₇ support, including singular and missing-edge cases, via retained third-minor budgets and exact balanced-region certificates.

Reproduction notes

The paper has 8 pages. The editable-source archive contains 7 files. Compile manuscript/main.tex. The standard-library verify_k47_certificate.py reconstructs the small-row and 256-box rational certificate calculations.

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Author disclosures

The paper identifies its main theorem as computer-assisted. Independent internal AI-assisted whole-source analytic, attribution and code reviews are distinct from external human refereeing or formal proof-assistant verification. Classical permanent bounds, prior Bernstein techniques and the published six-column predecessor are attributed. The author retains responsibility for the manuscript; originality is qualified by the inspected literature.

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Cite this edition

Scott Schubr. (2026). Strong permanent contraction on four-by-seven bipartite supports (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/strong-permanent-contraction-four-by-seven-bipartite-supports/v/2

BibTeX
@techreport{luminary_strong_permanent_contraction_four_by_seven_bipartite_supports_v2,
  title = {Strong permanent contraction on four-by-seven bipartite supports},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 2},
  url = {https://research.luminaryoddities.com/strong-permanent-contraction-four-by-seven-bipartite-supports/v/2}
}

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