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Research Article · Preprint Mathematics

Row-dependent bounds for strong permanent contraction on four-by-eight 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 K₄,₈ satisfies per(|A|²) ≤ per(A) ∏ᵢ aᵢᵢ. The proof extends an established four-row framework by retaining a necessary lower bound on squared row moments and choosing a convex permanent bound as a polynomial of the row norms. An exact two-point obstruction shows why one fixed mixture in the inherited scalar framework is insufficient. A small-row reduction and a rational tensor Bernstein certificate cover the capped-entry region; a published all-column large-entry theorem covers its complement. The result is computer-assisted, includes arbitrary phases and singular endpoints, and does not establish all four-row supports or unrestricted Chollet.

Keywords permanent inequalities · Chollet conjecture · positive semidefinite matrices · bipartite supports · Bernstein certificates · computer-assisted proof

Scope of the work

Strong Q(B) ≤ d(B)P(B) for Hermitian positive semidefinite matrices supported in K₄,₈, including complex phases, missing edges, zero rows, and singular endpoints. A continuation of the published four-row results through six and seven columns. Does not establish K₄,₉, K₅,₅, all four-row supports, an arbitrary second factor, equality classification, or unrestricted Chollet.

Reproduction notes

The publication package contains the manuscript PDF, editable LaTeX source, a standalone Python exact-rational certificate verifier, reader instructions, citation metadata, and checksums. Run python -B verify_k48_certificate.py with Python 3.10 or newer. The verifier reconstructs 720 small-row corner/pairing values and 65,536 Bernstein coefficients across 256 boxes. The manuscript supplies the analytic reductions and the cited large-entry input.

The manuscript

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Author disclosures

AI tools assisted mathematical research, drafting, and verification. The author is responsible for the statements, proofs, attribution, and final text. Archive posting and internal checks do not constitute external peer review.

References and related publications

  1. L. B. Beasley. An inequality on permanents of Hadamard products. Bulletin of the Korean Mathematical Society 37 (2000), no. 3, 633–639.
  2. J. Chollet. Is there a permanental analogue to Oppenheim’s inequality? American Mathematical Monthly 89 (1982), 57–58.
  3. S. W. Drury. A counterexample to a question of Bapat and Sunder. Electronic Journal of Linear Algebra 31 (2016), 69–70.
  4. E. Carlen, E. H. Lieb and M. Loss. An inequality of Hadamard type for permanents. Methods and Applications of Analysis 13 (2006), no. 1.
  5. P. Pant and R. Singh. Structural Classes for Chollet’s Permanent Conjecture. arXiv:2604.24192v2 (2026).
  6. S. Sa-nguansin and K. Rodtes. Permanents of correlation matrices and the Chollet permanental conjecture. Linear and Multilinear Algebra 73 (2025), no. 18, 4084–4096.
  7. Z. Zeng, H. Liu and K. Ratnavelu. Sharp Self-Chollet Inequalities in Order Five: Rank Two and Structured Rank Three. Preprints.org, version 1 (2026), not peer-reviewed.
  8. S. Schubr. Strong permanent inequalities for three-row bipartite and triangle-free supports. Luminary Research Archive, first edition (2026).
  9. S. Schubr. Strong permanent contraction on four-row bipartite supports through six columns. Luminary Research Archive, first edition (2026).
  10. S. Schubr. Strong permanent contraction on four-by-seven bipartite supports. Luminary Research Archive, first edition (2026).

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Row-dependent bounds for strong permanent contraction on four-by-eight supports (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/row-dependent-bounds-strong-permanent-contraction-four-by-eight-supports/v/1

BibTeX
@techreport{luminary_row_dependent_bounds_strong_permanent_contraction_four_by_eight_supports_v1,
  title = {Row-dependent bounds for strong permanent contraction on four-by-eight supports},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/row-dependent-bounds-strong-permanent-contraction-four-by-eight-supports/v/1}
}

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