Research Article · Preprint Mathematics
Strong permanent contraction on four-row bipartite supports through six columns
- Primary subject
- Mathematics
- Subject subfield
- Linear Algebra and Matrix Inequalities
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · Permanent link
Overview
Abstract
We prove that every complex positive semidefinite matrix A supported in K₄,₆ satisfies per(|A|²) ≤ per(A) ∏ᵢ aᵢᵢ. The normalized off-diagonal block is a four-by-six operator contraction. For squared entries at most 7/10, the proof combines a four-vertex overlap estimate with a fixed mixture of rectangular permanent bounds. Separate affinity, deletion of inactive rows and piecewise separate concavity reduce the remaining inequality to fifteen rational corner values. A convex-border argument covers larger entries for every number of columns, using Lieb’s block inequality and the published three-row theorem. Arbitrary phases, missing edges and singular endpoints are included. The four-by-four and four-by-five conclusions follow by padding. No all-column four-row or unrestricted Chollet theorem is asserted.
Keywords permanent inequalities · positive semidefinite matrices · bipartite supports · matrix contractions · scalar certificates · Chollet conjecture
Scope of the work
Complex Hermitian positive semidefinite K₄,₆ supports, including arbitrary phases, missing edges and singular or zero-diagonal endpoints. K₄,₄ and K₄,₅ follow by zero padding and are consolidated here as one result family. The heavy-entry estimate has its stated all-column hypothesis; no full K₄,₇, K₅,₅, all bipartite order-ten, or unrestricted Chollet theorem is claimed. The published three-row theorem is an explicit credited input. Comparison with the unavailable Sa–Rodtes full text remains incomplete; no absolute priority claim is made.
Reproduction notes
The nine-member package contains the seven-page paper PDF, complete manuscript/main.tex, three exact Python supplements, reader README, citation metadata, manifest and checksums. Python 3.10 or newer and its standard library suffice for verify_scalar_budget.py, verify_pair_identity.py and verify_border_identity.py, including optimized Python execution. Compile with Tectonic or run pdflatex twice. The finite and symbolic certificates supplement the analytic proof. Private assessments, reviews and release records are excluded.
The manuscript
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Author disclosures
AI-assisted preparation and internal verification are disclosed in the paper. Internal checks are distinct from external human refereeing or formal proof-assistant verification. Classical ingredients and the published three-row predecessor are attributed; the paper states the limits of the available prior-work comparison.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Strong permanent contraction on four-row bipartite supports through six columns (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/strong-permanent-contraction-four-row-bipartite-supports-through-six-columns/v/1
BibTeX
@techreport{luminary_strong_permanent_contraction_four_row_bipartite_supports_through_six_columns_v1,
title = {Strong permanent contraction on four-row bipartite supports through six columns},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/strong-permanent-contraction-four-row-bipartite-supports-through-six-columns/v/1}
}Loading discussion…