Research Archive

Luminary Research ArchiveIndependent scholarship

Luminary Portal

Research Article · Preprint Mathematics

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
9 · Permanent link

You are reading a preserved edition. Read the current edition (v10).

Read / download PDF Publication filesCite this edition

Overview

Abstract

We prove sharp all-order bounds for coupled permanent cofactors and occupation-resolved variance and curvature constraints for complex positive semidefinite Gram matrices. At six rows, the shared occupations, inherited defect and retained curvature give explicit sufficient compensation domains, including the selected-occupation 1/31 theorem and structured phase-dependent capacity comparisons with higher-rank neighborhoods. For the actual core F(t)=per(K+tJ), P=F(1)+F′(1), L=F(√2)−P, we prove the sharp constraint P+3L≥7200 and classify equality by a fourth-root orbit plus two zero residual factors. It yields actual compensation for 0≤L≤4 without an initial diagonal cutoff. A separate 1/32 entrywise fourth-root tube has a strict capacity margin. The intrinsic criterion L≥0, P+3L−7200≤2^−160 derives bounded diagonals and two small residual rows and gives another explicit capacity margin. The tube and intrinsic regions also satisfy the strict actual three-quarter self-bound. The proofs retain complex phases, singular limits and rank-five kernel mixtures. Classical inputs and the attributed small-order induction are identified. The generic rank-six compensation inequality, the generic three-quarter self-bound and unrestricted Chollet remain open.

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

Scope of the work

The paper proves its stated all-order cofactor, variance/occupation and curvature results, the 1/31 domain, structured-family comparisons and explicit neighborhoods. New Section29 proves P+3L≥7200 with exact equality geometry, cutoff-free actual compensation on 0≤L≤4, a 1/32 fourth-root capacity tube, a two-small-row capacity region and intrinsic localization at D≤2^−160 under L≥0. The strict three-quarter corollary applies only to the stated capacity regions; it is not asserted on the whole L≤4 strip or generically. All row-gauge factors, nonnegative-loss hypotheses and rank/kernel endpoints are retained. No global localization, optimal threshold, generic rank-six solution or new general Marcus/CLL/creation theorem is claimed. The complete 2025 Sa-nguansin–Rodtes proof was inaccessible, so external novelty and containment comparisons remain qualified. Internal AI-assisted checks are not external human refereeing.

Reproduction notes

The complete 61-page manuscript and 40-member package contain thirteen editable TeX sources, twenty-two Python standard-library checkers, exact adaptive fixture data, reader/build instructions, citation metadata and a complete 39-entry checksum manifest. All 44 extracted normal/optimized checker executions passed, with mathematical outputs agreeing after the explicitly reported optimization flag is accounted for. An isolated source rebuild matches every page’s extracted mathematical text and geometry. The new Fraction-only checker verifies the strict capacity and three-quarter constants; universal occupation, localization and tube statements have analytic proofs. Complete rational covers certify only their stated structured families; finite examples do not establish generic coverage. Shared local arithmetic dependencies and internal verification limits are disclosed.

The manuscript

Read the paper

Open PDF

The complete manuscript, as published in this edition.

If the reader is unavailable on your device, open or download the PDF directly.

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

  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.
  13. R. Bhatia, P . Grover and T. Jain, Derivatives of tensor powers and their norms, arXiv:1102.2414v3 (2015).
  14. H. Terao, Algebras generated by reciprocals of linear forms, J. Algebra 250 (2002), 549–558, doi:10.1006/jabr.2001.9121.
  15. M. Marcus, The permanent analogue of the Hadamard determinant theorem, Bull. Amer. Math. Soc. 69 (1963), 494–496.
  16. D. Lu, Nonlinear eigenvector methods for convex minimization over the numerical range, SIAM J. Matrix Anal. Appl. 41 (2020), 1771–1796, doi:10.1137/18M1234473.
  17. A. Neumaier, Complete search in continuous global optimization and constraint satisfaction, Acta Numer. 13 (2004), 271–369, doi:10.1017/S0962492904000194.
  18. E. A. Carlen, E. H. Lieb and M. Loss, An inequality of Hadamard type for permanents, Methods Appl. Anal. 13 (2006), 1–17.
  19. R. Sinkhorn and P. Knopp, Concerning nonnegative matrices and doubly stochastic matrices, Pacific J. Math. 21 (1967), 343–348.
  20. S. Sa-nguansin and K. Rodtes, Permanents of correlation matrices and the Chollet permanental conjecture, Linear Multilinear Algebra 73 (2025), 4084–4096.

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Occupation bounds for coupled permanent cofactors (Version 9). Luminary Research Archive. https://research.luminaryoddities.com/occupation-bounds-coupled-permanent-cofactors/v/9

BibTeX
@techreport{luminary_occupation_bounds_coupled_permanent_cofactors_v9,
  title = {Occupation bounds for coupled permanent cofactors},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 9},
  url = {https://research.luminaryoddities.com/occupation-bounds-coupled-permanent-cofactors/v/9}
}

Loading discussion…