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Coupled permanent cofactors: sharp occupation bounds and compensation geometry

Scott Schubr

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Abstract

For complex positive semidefinite Gram matrices, permanent cofactors of the ordinary and entrywise squared Gram matrices determine a common residual operator. We prove its sharp all-order bound with optimal factor n-1 and a sharp occupation–variance inequality, retaining singular factors, complex phases and arbitrary ambient kernel components. These results express information that separate cofactor estimates discard. At order six, the same Gram geometry yields variance, inherited-defect and curvature reserves for the actual bordered permanent defect. A selected occupation at most 1/31 is sufficient for compensation and gives an order-seven self-conjugate domain. For the six-factor core F(t)=per(K+tJ), with P=F(1)+F'(1) and L=F(√(2))-P, we prove the sharp constraint P+3L≥7200 and classify its equality geometry. It yields actual compensation on 0≤ L≤4 without a diagonal cutoff. Weighted coherent projections and classical permanent bounds give further explicit capacity regions, including full-amplitude structured families and quantified higher-rank neighborhoods. A uniform creation-operator reserve proves strict diagonal monotonicity and convexity of the transformed deficit; exact counterexamples delimit extensions to unrestricted PSD directions and several fixed certificates. The sufficient regions retain their explicit occupation, loss, phase and neighborhood hypotheses. Classical inputs and the complete finite certificates supporting the restricted-family proofs are specified alongside the analytic arguments.

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  1. Edition 11 · 2026-10-27 · public
  2. Edition 10 · 2026-10-25 · 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, Gram matrix, coupled cofactors, occupation bounds, positive semidefinite matrices, Chollet inequality
Manuscript date
2026-10-23
Reader reuse
No additional permission granted

Scope

Sharp all-order coupled cofactor residual and occupation–variance bounds, with explicit six-factor compensation regions, a sharp occupation constraint, capacity domains and diagonal monotonicity/convexity.

Disclosures

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

Reproduction

The 64-page manuscript and 44-member source package include all fourteen TeX components, twenty-three exact Python checkers, certificate data, a fixed-timestamp Tectonic build helper and citation metadata. README.md gives compilation and normal/optimized checker commands; each checker’s proof or example role is specified in the manuscript.

References and related resources

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