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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
2 · Permanent link
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Overview

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. No unrestricted Chollet inequality is proved.

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

Scope of the work

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. Generic rank-six positivity, unrestricted Chollet, optimal radii, global algorithmic complexity and separation from the exact shifted cofactor surrogate remain unproved here.

Reproduction notes

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 do not prove a universal inequality by sampling. Keep extensions.tex beside main.tex when compiling the paper.

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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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A stable scholarly record

Cite this edition

Scott Schubr. (2026). Occupation bounds for coupled permanent cofactors (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/occupation-bounds-coupled-permanent-cofactors/v/2

BibTeX
@techreport{luminary_occupation_bounds_coupled_permanent_cofactors_v2,
  title = {Occupation bounds for coupled permanent cofactors},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 2},
  url = {https://research.luminaryoddities.com/occupation-bounds-coupled-permanent-cofactors/v/2}
}

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