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Occupation bounds for coupled permanent cofactors

Scott Schubr

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Abstract

For a complex positive semidefinite Gram matrix B = VV*, with a = per(B) and c = per(|B|²), 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. A central-transposition statistic gives an order-six refinement. For the actual bordered permanent defect, a nonnegative cofactor variance has a zero set characterized by parallel and orthogonal Gram lines, while every kernel mixture is retained. Two-sided relative stability in the unit Gram factors gives explicit occupied-vertex domains without spectral-gap assumptions. Transfer is conditional on the smaller complementary self-Chollet inequality and is unconditional when that complement has order at most four. A general conditional occupation radius of order n⁻³ gives ratio at most 2/3; larger explicit radii at order seven give ratio at most 7/8. An unconditional entrywise neighborhood of repeated orthogonal blocks, of radius of order n⁻⁴, certifies every admissible border, with an explicit larger rank-five neighborhood at order seven. Exact occupation-box matrix certificates give finite completeness at strictly positive fixed bases and a conditional all-low extension through kernel zeros. Exact adaptive examples illustrate subdivision. The original residual occupation polynomial, thinning law and nonreal-root examples are retained.

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  1. Edition 11 · 2026-10-27 · public
  2. Edition 10 · 2026-10-25 · public
  3. Edition 9 · 2026-10-24 · public
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  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
permanent inequalities, positive semidefinite matrices, coupled cofactors, occupation bounds, Gram factors, adaptive certificates
Manuscript date
2026-10-23
Reader reuse
No additional permission granted

Scope

Same-paper substantive strengthening of the sharp residual cofactor bound. Occupied-vertex transfer retains the complementary self-Chollet hypothesis; it is automatic for complementary order at most four. The order-seven one-high-row case retains its order-five hypothesis. All-low and kernel-mixture conclusions retain their stated defect-sign assumptions.

Disclosures

AI tools assisted mathematical exploration, drafting and checking. The author is responsible for the statements and presentation.

Reproduction

The nine-member package includes the complete PDF, main.tex, extensions.tex, two standard-library checkers, rational adaptive certificate data and a checksum manifest. With Python 3.10 or later, run verify.py and verify_adaptive.py adaptive_certificates.json --self-test in normal and optimized modes. The checkers reconstruct exact examples and complete occupation-box covers, enforce the stated hypotheses and reject invalid trees. Finite certificates support the stated analytic arguments. Keep extensions.tex beside main.tex when compiling the paper.

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.
  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.