Luminary Research Archive
Luminary Research ArchiveIndependent scholarship
Luminary Portal
PublicEdition 1

Occupation bounds for coupled permanent cofactors

Scott Schubr

Original PDF · First-page preview

Loading original first page…

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.

Read the full paper in the original PDF.

Edition files

Available editions

  1. Edition 11 · 2026-10-27 · public
  2. Edition 10 · 2026-10-25 · public
  3. Edition 9 · 2026-10-24 · public
  4. Edition 8 · 2026-10-24 · public
  5. Edition 7 · 2026-10-24 · public
  6. Edition 6 · 2026-10-24 · public
  7. Edition 5 · 2026-10-24 · public
  8. Edition 4 · 2026-10-24 · public
  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
permanents, permanent cofactors, Gram matrices, occupation operators, matrix inequalities
Manuscript date
2026-10-22
Reader reuse
No additional permission granted

Scope

Sharp all-order coupled-cofactor Loewner bound and specified order-six refinements, including singular Gram kernels.

Disclosures

Author-posted preprint. AI-assisted internal research and verification were used.

Reproduction

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.

References and related resources

  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.