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 K_(4,6) satisfies per(|A|^2)≤per(A)∏_i a_(ii). 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
Analytic strong permanent contraction through six columns on four-row complex bipartite supports, together with an arbitrary-column large-entry transfer based on the cited three-row theorem.
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 paper has 8 pages. The editable-source archive contains 9 files. Compile manuscript/main.tex. The three standard-library verifier scripts check the scalar budget, pair identity and border expansion.