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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. At six rows, the shared occupations, inherited defect and retained curvature give explicit sufficient compensation domains, including the selected-occupation 1/31 theorem and structured phase-dependent capacity comparisons with higher-rank neighborhoods. For the actual core F(t)=per(K+tJ), P=F(1)+F′(1), L=F(√2)−P, we prove the sharp constraint P+3L≥7200 and classify equality by a fourth-root orbit plus two zero residual factors. It yields actual compensation for 0≤L≤4 without an initial diagonal cutoff. A separate 1/32 entrywise fourth-root tube has a strict capacity margin. The intrinsic criterion L≥0, P+3L−7200≤2^−160 derives bounded diagonals and two small residual rows and gives another explicit capacity margin. The tube and intrinsic regions also satisfy the strict actual three-quarter self-bound. The proofs retain complex phases, singular limits and rank-five kernel mixtures. Classical inputs and the attributed small-order induction are identified.

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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 paper proves its stated all-order cofactor, variance/occupation and curvature results, the 1/31 domain, structured-family comparisons and explicit neighborhoods. New Section29 proves P+3L≥7200 with exact equality geometry, cutoff-free actual compensation on 0≤L≤4, a 1/32 fourth-root capacity tube, a two-small-row capacity region and intrinsic localization at D≤2^−160 under L≥0. The strict three-quarter corollary applies to the stated capacity regions. All row-gauge factors, nonnegative-loss hypotheses and rank/kernel endpoints are retained.

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 60-page manuscript and 40-member package contain thirteen editable TeX sources, twenty-two Python standard-library checkers, exact adaptive fixture data, reader/build instructions, citation metadata and a complete 39-entry checksum manifest. All 44 extracted normal/optimized checker executions passed, with mathematical outputs agreeing after the explicitly reported optimization flag is accounted for. An isolated source rebuild matches every page’s extracted mathematical text and geometry. The new Fraction-only checker verifies the strict capacity and three-quarter constants; universal occupation, localization and tube statements have analytic proofs. Complete rational covers certify their stated structured families. Shared local arithmetic dependencies and internal verification limits are disclosed.

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