Strong permanent contraction on four-row bipartite supports through six columns
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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.
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Research details
- Subjects
- Mathematics · Linear Algebra and Matrix Inequalities
- Keywords
- permanent inequalities, positive semidefinite matrices, bipartite supports, matrix contractions, scalar certificates, Chollet conjecture
- Manuscript date
- 2026-10-22
- Reader reuse
- No additional permission granted
Scope
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. The published three-row theorem is an explicit credited input. The inspected literature is credited in the manuscript.
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.
Reproduction
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.