Strong permanent contraction on four-by-seven bipartite supports
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Abstract
We prove that every complex positive semidefinite matrix A supported in K_(4,7) satisfies per(|A|^2)≤per(A)∏_i a_(ii). 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.
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9637a5e2314fbcb7e729b8c014fe3c330a01ff433bacf4ab4ba5572643267465
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Research details
- Subjects
- Mathematics · Linear Algebra and Matrix Inequalities
- Keywords
- permanent inequalities, positive semidefinite matrices, bipartite supports, matrix contractions, Bernstein certificates, computer-assisted proof
- Manuscript date
- 2026-10-22
- Reader reuse
- No additional permission granted
Scope
Computer-assisted strong permanent contraction on every complex PSD K₄,₇ support, including singular and missing-edge cases, via retained third-minor budgets and exact balanced-region certificates.
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.
Reproduction
The paper has 8 pages. The editable-source archive contains 7 files. Compile manuscript/main.tex. The standard-library verify_k47_certificate.py reconstructs the small-row and 256-box rational certificate calculations.