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Research Article · Preprint Mathematics

Occupation bounds for coupled permanent cofactors

Scott Schubr

Luminary Oddities LLC

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

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. 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 occupation and frame-variance estimates and stated sufficient compensation domains retain their displayed hypotheses. Constants are not claimed optimal. General complementary self-Chollet assumptions remain where stated. The finite-forward identity does not settle the generic forward compensation budget; the pencil obstruction is not a counterexample to Chollet. Finite certificates corroborate the analytic arguments and do not establish a complete generic rank-six cover.

Reproduction notes

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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Author disclosures

AI tools assisted mathematical exploration, drafting and checking. The author is responsible for the statements and presentation. Internal computational and AI checks are not 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 4). Luminary Research Archive. https://research.luminaryoddities.com/occupation-bounds-coupled-permanent-cofactors/v/4

BibTeX
@techreport{luminary_occupation_bounds_coupled_permanent_cofactors_v4,
  title = {Occupation bounds for coupled permanent cofactors},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 4},
  url = {https://research.luminaryoddities.com/occupation-bounds-coupled-permanent-cofactors/v/4}
}

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