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Occupation bounds for coupled permanent cofactors

Scott Schubr

Luminary Oddities LLC

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

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Overview

Abstract

We develop 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 actual compensation whenever one selected occupation is at most 1/31. The proof retains the joint weighted coefficient budget and coherent root product; on positive loss within that band it gives G > P^2/20 + 2hE + R and reduces the remaining residual-diagonal bound to 30. Consequently an order-seven PSD matrix satisfies self-Chollet if one normalized off-diagonal squared modulus is at most 1/31; the two-factor consequence requires both factors to meet this condition. Aligned cofactor occupations give a sharp retained-curvature reserve, with coefficient 19/240 at six rows, uniformly optimal also on active rank-six and rank-five correlation bases. Exact projection overlap couples trace and occupation floors without double counting. Balanced coherent-column and covariance-deficit tests give further sufficient domains. Proofs retain complex phases, singular boundaries and kernel mixtures. The generic rank-six compensation inequality and unrestricted Chollet conjecture remain open.

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

Scope of the work

The uniform 1/31 domain and order-seven weak-pair theorem, actual cofactor coefficient constraints, sharp curvature reserve, projection-overlap bounds and balanced sufficient domains are proved with their stated hypotheses. The generic rank-six compensation problem remains open. The 2025 Sa-nguansin–Rodtes full proof was inaccessible for comparison; no exhaustive containment or first-ever priority is claimed. Classical CLL, scaling and projection tools are attributed. Internal AI-assisted review is not external human refereeing.

Reproduction notes

The complete 36-page manuscript and 21-member package contain eight editable LaTeX source files, eight standard-library Python checkers, exact rational fixture data, reader/build instructions, citation metadata and public checksums. Sixteen extracted normal/optimized checker runs passed. An isolated source rebuild reproduced all mathematical text and page geometry. The new exact checker verifies four complex rank-six and singular rank-five examples, including noncirculant cases. Finite computations corroborate identities and constants; universal claims rest on analytic proofs.

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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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  3. Q. Li, Q. Zhao and W. Luo, Chollet’s permanent conjecture through order six via border induction, Preprints.org 202608.2196v1 (31 August 2026), doi:10.20944/preprints202608.2196.v1.
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A stable scholarly record

Cite this edition

Scott Schubr. (2026). Occupation bounds for coupled permanent cofactors (Version 7). Luminary Research Archive. https://research.luminaryoddities.com/occupation-bounds-coupled-permanent-cofactors/v/7

BibTeX
@techreport{luminary_occupation_bounds_coupled_permanent_cofactors_v7,
  title = {Occupation bounds for coupled permanent cofactors},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 7},
  url = {https://research.luminaryoddities.com/occupation-bounds-coupled-permanent-cofactors/v/7}
}

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