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Research Article · Preprint Mathematics

Strong permanent contraction on four-by-seven bipartite supports

Scott Schubr

Luminary Oddities LLC

  • Mathematics
  • Linear Algebra and Matrix Inequalities
Primary subject
Mathematics
Subject subfield
Linear Algebra and Matrix Inequalities
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
1 · Permanent link
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Overview

Abstract

We prove that every complex positive semidefinite matrix A supported in K₄,₇ satisfies per(|A|²) ≤ per(A) ∏ᵢ aᵢᵢ. 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. It does not prove the all-column four-row case, the K₅,₅ case, or unrestricted Chollet.

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

Scope of the work

Computer-assisted strong permanent inequality for arbitrary complex Hermitian positive semidefinite K₄,₇ supports, including phases, missing edges, singular and zero-diagonal endpoints. This is a separate attributed continuation of the published K₄,₄–K₄,₆ paper. Its published all-column heavy-entry lemma is an explicit input. No K₄,₈, K₅,₅, all bipartite order-ten or order-eleven, or unrestricted Chollet theorem is asserted. Earlier Bernstein methods are credited; comparison with the unavailable Sa–Rodtes full text remains incomplete and worldwide priority is provisional.

Reproduction notes

The seven-member publication package contains the eight-page paper, complete manuscript/main.tex, verify_k47_certificate.py, reader README, citation metadata, manifest and checksums. The standalone exact rational verifier regenerates 720 corner values and 38,416 Bernstein coefficients; normal and optimized Python runs and eight mutation failure controls were checked during preparation. An isolated source-only rebuild matched all eight pages of PDF text and raster output. Private assessments, internal reviews, gate records and ongoing K₄,₈ research are excluded.

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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.

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Strong permanent contraction on four-by-seven bipartite supports (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/strong-permanent-contraction-four-by-seven-bipartite-supports/v/1

BibTeX
@techreport{luminary_strong_permanent_contraction_four_by_seven_bipartite_supports_v1,
  title = {Strong permanent contraction on four-by-seven bipartite supports},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/strong-permanent-contraction-four-by-seven-bipartite-supports/v/1}
}

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