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 K₄,₇ satisfies per(|A|²) ≤ per(A) ∏ᵢ aᵢᵢ. 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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- Strong_Permanent_K47.pdf111.3 KiB · pdf
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04ce1c1e2f5cf6be5d1a7ccb24d24601339849f87388e6d2c52b14ac73fead5e - K47_Publication_Package.zip121.1 KiB · package
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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 inequality for arbitrary complex Hermitian positive semidefinite K₄,₇ supports, including phases, missing edges, singular and zero-diagonal endpoints. This is a separate attributed continuation of the published K₄,₄–K₄,₆ paper. Its published all-column heavy-entry lemma is an explicit input. Earlier Bernstein methods are credited.
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 seven-member publication package contains the eight-page paper, complete manuscript/main.tex, verify_k47_certificate.py, reader README, citation metadata, manifest and checksums. The standalone exact rational verifier regenerates 720 corner values and 38,416 Bernstein coefficients; normal and optimized Python runs and eight mutation failure controls were checked during preparation. An isolated source-only rebuild matched all eight pages of PDF text and raster output. Private assessments, internal reviews, gate records and ongoing K₄,₈ research are excluded.