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Coupled permanent cofactors: sharp occupation bounds and compensation geometry

Scott Schubr

Luminary Oddities

  • Mathematics
  • Linear Algebra and Matrix Inequalities
Primary subject
Mathematics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
11 · Permanent link
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Overview

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.

Keywords permanent inequalities · positive semidefinite matrices · coupled cofactors · occupation bounds · Gram factors · adaptive certificates

Scope of the work

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.

Reproduction notes

63-page manuscript and 42 source-package members. Compile main.tex with the included fourteen TeX components. Standard-library Python runs the twenty-three verify*.py programs separately, normally and under -O, to reconstruct the coupled-cofactor, occupation, reserve, capacity, strip and restricted-family certificate controls described in the README.

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Author 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.

References and related publications

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Cite this edition

Scott Schubr. (2026). Coupled permanent cofactors: sharp occupation bounds and compensation geometry (Version 11). Luminary Research Archive. https://research.luminaryoddities.com/occupation-bounds-coupled-permanent-cofactors/v/11

BibTeX
@techreport{luminary_occupation_bounds_coupled_permanent_cofactors_v11,
  title = {Coupled permanent cofactors: sharp occupation bounds and compensation geometry},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 11},
  url = {https://research.luminaryoddities.com/occupation-bounds-coupled-permanent-cofactors/v/11}
}

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