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Occupation bounds for coupled permanent cofactors

Scott Schubr

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Abstract

We develop sharp all-order bounds for coupled permanent cofactors and occupation-resolved variance and curvature constraints for complex positive semidefinite Gram matrices. A uniform six-row theorem proves actual compensation whenever one selected occupation is at most 1/31. The proof retains the joint weighted coefficient budget and coherent root product; on positive loss within that band it gives G > P^2/20 + 2hE + R and reduces the remaining residual-diagonal bound to 30. Consequently an order-seven PSD matrix satisfies self-Chollet if one normalized off-diagonal squared modulus is at most 1/31; the two-factor consequence requires both factors to meet this condition. Aligned cofactor occupations give a sharp retained-curvature reserve, with coefficient 19/240 at six rows, uniformly optimal also on active rank-six and rank-five correlation bases. Exact projection overlap couples trace and occupation floors without double counting. Balanced coherent-column and covariance-deficit tests give further sufficient domains. Proofs retain complex phases, singular boundaries and kernel mixtures.

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Research details

Subjects
Mathematics · Linear Algebra and Matrix Inequalities
Keywords
permanent inequalities, positive semidefinite matrices, coupled cofactors, occupation bounds, Gram factors, adaptive certificates
Manuscript date
2026-10-23
Reader reuse
No additional permission granted

Scope

The uniform 1/31 domain and order-seven weak-pair theorem, actual cofactor coefficient constraints, sharp curvature reserve, projection-overlap bounds and balanced sufficient domains are proved with their stated hypotheses. Classical CLL, scaling and projection tools are attributed.

Disclosures

AI-assisted tools supported research exploration, literature review, manuscript preparation and internal verification. The author retains responsibility for the work. Internal AI-assisted and computational checks are distinct from external human peer review.

Reproduction

The complete 35-page manuscript and 21-member package contain eight editable LaTeX source files, eight standard-library Python checkers, exact rational fixture data, reader/build instructions, citation metadata and public checksums. Sixteen extracted normal/optimized checker runs passed. An isolated source rebuild reproduced all mathematical text and page geometry. The new exact checker verifies four complex rank-six and singular rank-five examples, including noncirculant cases. Finite computations corroborate identities and constants; universal claims rest on analytic proofs.

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