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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
3 · Permanent link
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Overview

Abstract

For a complex positive semidefinite Gram matrix B = VV*, we couple the permanent-cofactor contractions of B and |B|² and prove the sharp all-order bound 0 ⪯ U − aT/n! ⪯ (n − 1)(c − a²/n!)I, where a = per(B) and c = per(|B|²). For the actual bordered permanent defect, a nonnegative cofactor-weighted variance E has a geometric zero set. A trace-contraction argument gives an explicit nonlinear lower bound for arbitrary trace-one positive operators. For six unit Gram factors it implies E ≥ v²/1600, and E ≥ v²/600 at rank at most five, where v is the total occupation variance. The quadratic power is sharp. Arbitrary complex phases, singular limits and kernel mixtures are included.

A coherence-dependent base-defect reserve includes a uniform rank-five reserve. An all-border matrix certificate holds at coherence at most 1/15. A self-contained reconstruction of the attributed small-order coefficient induction supplies the scalar inequalities through order six used in these certificates. Relative Gram-factor stability, occupied-vertex domains, neighborhoods of repeated orthogonal blocks and exact adaptive occupation-box certificates are retained and augmented by the uniform variance bound. The original residual occupation polynomial, thinning law and nonreal-root examples remain. The generic rank-six compensation inequality and unrestricted Chollet conjecture remain open.

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

Scope of the work

Same-paper substantive strengthening. Uniform variance: E ≥ v²/1600 for six unit Gram factors, improved to v²/600 at rank at most five; mixed trace-one extension uses actual operator variance and support dimension. Quadratic power sharp, constants not claimed optimal. Includes complex phases, all kernel mixtures and rank limits. Coherence-dependent base reserve and coherence ≤ 1/15 matrix certificate are proved. Included reconstruction discharges the order-five premise in the order-seven one-high-row transfer; the general transfer retains its complementary self-Chollet hypothesis. Generic rank-six compensation, unrestricted Chollet, a complete uniform box cover, optimal radii, global complexity and separation from the exact shifted cofactor surrogate remain unproved.

Reproduction notes

The eleven-member package contains the 16-page PDF, main.tex, extensions.tex, uniform.tex, three standard-library Python checkers, rational adaptive certificate data, citation metadata, reader instructions and a checksum manifest. Run verify.py, verify_adaptive.py adaptive_certificates.json --self-test, and verify_uniform.py in normal and optimized modes with Python 3.10+. Exact finite checks corroborate identities, constants and examples; universal statements follow the displayed analytic proofs. Keep both included TeX files beside main.tex. The adaptive geometric checker conservatively retains its two-high-row rule.

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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 3). Luminary Research Archive. https://research.luminaryoddities.com/occupation-bounds-coupled-permanent-cofactors/v/3

BibTeX
@techreport{luminary_occupation_bounds_coupled_permanent_cofactors_v3,
  title = {Occupation bounds for coupled permanent cofactors},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 3},
  url = {https://research.luminaryoddities.com/occupation-bounds-coupled-permanent-cofactors/v/3}
}

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