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Occupation bounds for coupled permanent cofactors

Scott Schubr

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Abstract

For a complex positive semidefinite Gram matrix B = VV*, this paper couples the permanent-cofactor contractions of B and |B|² and proves the sharp all-order bound 0 ⪯ U − aT/n! ⪯ (n − 1)(c − a²/n!)I, where a = per(B) and c = per(|B|²). Orthogonal trace sectors give a joint occupation and frame-variance bound for the actual bordered permanent defect: for six unit Gram factors, E ≥ v²/270 + 7A/1620; at rank at most five, E ≥ 6(v² + A)/625. The all-order trace-one positive-operator formulation includes mixed factors with their actual operator variance, arbitrary complex phases, singular limits and kernel mixtures. The quadratic power is sharp. Quantitative base-defect reserves, rank-one secant compensation domains, frame-variance occupation-box certificates and conditional multiplicative compensation domains are established. A finite-forward identity separates a nonnegative curvature remainder from degree-six ordinary occupation loss, while an explicit rank-one pencil obstructs uniform positive multiplicative transfer to a prescribed forward point. Earlier stability domains and the coherence-1/15 certificate are retained.

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  1. Edition 11 · 2026-10-27 · public
  2. Edition 10 · 2026-10-25 · public
  3. Edition 9 · 2026-10-24 · public
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  9. Edition 3 · 2026-10-23 · public
  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 occupation and frame-variance estimates and stated sufficient compensation domains retain their displayed hypotheses. General complementary self-Chollet assumptions remain where stated. The finite-forward identity and pencil obstruction concern the stated auxiliary comparison. Finite certificates corroborate the analytic arguments.

Disclosures

AI tools assisted mathematical exploration, drafting and checking. The author is responsible for the statements and presentation.

Reproduction

The fifteen-member package contains the 24-page PDF, five linked LaTeX sources, five standard-library Python checkers, exact adaptive certificates, citation metadata, reader instructions and a checksum manifest. Follow README.md with Python 3.10 or later. Run verify.py, verify_adaptive.py adaptive_certificates.json --self-test, verify_uniform.py, verify_coupling.py and verify_forward.py in normal and optimized modes. Keep all included TeX files together. Exact finite checks support identities and examples; universal statements rely on the displayed analytic proofs.

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