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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 subfield
Linear Algebra and Matrix Inequalities
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
1 · Permanent link
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Overview

Abstract

For a complex positive semidefinite Gram matrix B = VV* of order n ≥ 2, this paper couples the permanent-cofactor contractions of B and |B|². With a = per B, c = per |B|² and the transposed cofactor contractions T and U, it proves the sharp bound 0 ⪯ U − aT/n! ⪯ (n − 1)(c − a²/n!)I. The difference is represented as a one-mode occupation expectation in the orthogonal difference of two nested invariant subspaces. At order six, a central-transposition statistic gives a computable refinement involving fifteen mixed permanents. Applications retain all Gram kernel directions and include complex rank-five and rank-six examples with an improved sufficient certificate. Residual occupation polynomials, exact kernel thinning, zero-safe top-coefficient cancellation and complex examples without real-rooted occupation polynomials are also established. No unrestricted Chollet inequality is proved.

Keywords permanents · permanent cofactors · Gram matrices · occupation operators · matrix inequalities

Scope of the work

Sharp all-order coupled-cofactor Loewner bound and specified order-six refinements, including singular Gram kernels. Concrete improved sufficient certificates do not resolve the generic rank-six compensation gap or unrestricted Chollet.

Reproduction notes

The six-member package includes complete LaTeX and verify.py. Python 3.10 or later with the standard library reconstructs the complex examples, verifies ranks, full support, phase triangles and 378 exact principal minors. Normal and optimized runs retain explicit checks. The sharp all-order cofactor inequality is proved analytically in the manuscript.

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Author disclosures

Author-posted preprint. AI-assisted internal research and verification do not constitute external human peer review or proof-assistant formalization.

References and related publications

  1. J. Chollet, Problem 6482, Amer. Math. Monthly 89 (1982), 57–58.
  2. K. Rodtes, Chollet’s permanent conjecture for 4 × 4 matrices, Linear Multilinear Algebra 72 (2024), 2633–2638, arXiv:2211.02839.
  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.
  4. V . E. Paksoy, A permanent inequality for positive semidefinite matrices,Electron. J. Linear Algebra 39 (2023), 71–77, publisher’s article.
  5. A. W. Harrow, The church of the symmetric subspace (2013), arXiv:1308.6595.
  6. J. M. Zhang, H. F. Song and Y . Liu, Permanent variational wave functions for bosons (2021), arXiv:2106.14679.
  7. S. Geller and E. Knill, Measuring multiparticle indistinguishability with the generalized bunching probability, arXiv:2509.04550v2 (2025).
  8. C. Ryba, Stable centres of wreath products, Algebraic Combinatorics 6 (2023), 413–455, doi:10.5802/alco.264.
  9. A. Prékopa and G. Alexe, Dual methods for the numerical solution of the univariate power moment problem, RUTCOR Research Report 14-2003, report.
  10. W. Tang and F. Tang, The Poisson binomial distribution—old & new, arXiv:1908.10024 (2019).
  11. S. Schubr, Chollet’s inequality for positive semidefinite matrices of rank at most four (2026), authorpublished preprint, edition 1.
  12. S. Schubr, Sharp limitations of two permanental cofactor comparisons (2026), author-published preprint, edition 1.

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Occupation bounds for coupled permanent cofactors (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/occupation-bounds-coupled-permanent-cofactors/v/1

BibTeX
@techreport{luminary_occupation_bounds_coupled_permanent_cofactors_v1,
  title = {Occupation bounds for coupled permanent cofactors},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/occupation-bounds-coupled-permanent-cofactors/v/1}
}

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