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Occupation bounds for coupled permanent cofactors

Scott Schubr

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Abstract

We prove 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 compensation whenever one selected occupation is at most 1/31, giving an order-seven weak-pair consequence and a bounded residual chart. Weighted coherent projection and its occupation resolution give actual phase-dependent lower certificates for the ordinary derivative. The occupation-projection defect is bounded by (364373/5) eta^4 for eta at most one, without conditioning or residual-scale cutoffs. On their stated nonnegative ordinary-loss domains, unequal opposite-pair and Fourier families admit strict occupation-capacity comparisons. On nonnegative ordinary loss, for the full family of two third-root orbits, arbitrary unequal norms and complex overlap phases are allowed at every amplitude: the actual occupation projection satisfies Psi^2 > 2 Uopt, while a separate explicit floor satisfies B0^2 > (3/2) Uopt. Bounded adaptive duals and fixed feasible capacity weights extend this control to an entrywise 2^-100 neighborhood of bounded orbit references, with Psi^2-Uopt > 2^21, including complex noncirculant rank-six and rank-five strata. Complete rational covers, analytic unbounded tails and portable exact readers accompany the proofs.

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  1. Edition 11 · 2026-10-27 · public
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  10. Edition 2 · 2026-10-23 · public
  11. Edition 1 · 2026-10-23 · public

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 paper proves the stated all-order cofactor bounds, occupation and curvature reserves, 1/31 sufficient domain, uniform Newton projection-defect bound, specified structured-family comparisons and explicit higher-rank neighborhoods. The full two-third-root-orbit factor-two and B0 lower-floor theorems require nonnegative ordinary loss; their amplitudes and overlap phases are unrestricted. The adaptive tube requires PSD actual/reference cores with diagonal at most 30, nonnegative reference loss and entry distance at most 2^-100. Classical projection, balancing, CLL, Jensen and complete-search 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 57-page manuscript and 38-member package contain twelve editable LaTeX sources, twenty-one Python standard-library checkers, necessary exact fixture data, reader/build instructions, citation metadata and a complete public checksum manifest. Forty-two extracted normal/optimized reader executions passed. An isolated source rebuild reproduced all mathematical text and page geometry. The orbit reader verifies complete rational 48/30 and 586/455 covers with analytic unbounded tails; the adaptive-tube reader checks low/high overlap constants and exact complex active-rank-six/rank-five examples. Shared local arithmetic dependencies and the distinction between analytic proofs, complete finite covers and finite nonemptiness checks are explicit. Generic coverage is not inferred from examples.

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