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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
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8 · Permanent link

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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. A uniform six-row theorem proves compensation whenever one selected occupation is at most 1/31, giving an order-seven weak-pair consequence and a bounded residual chart. Weighted coherent projection and its occupation resolution give actual phase-dependent lower certificates for the ordinary derivative. The occupation-projection defect is bounded by (364373/5) eta^4 for eta at most one, without conditioning or residual-scale cutoffs. On their stated nonnegative ordinary-loss domains, unequal opposite-pair and Fourier families admit strict occupation-capacity comparisons. On nonnegative ordinary loss, for the full family of two third-root orbits, arbitrary unequal norms and complex overlap phases are allowed at every amplitude: the actual occupation projection satisfies Psi^2 > 2 Uopt, while a separate explicit floor satisfies B0^2 > (3/2) Uopt. Bounded adaptive duals and fixed feasible capacity weights extend this control to an entrywise 2^-100 neighborhood of bounded orbit references, with Psi^2-Uopt > 2^21, including complex noncirculant rank-six and rank-five strata. Complete rational covers, analytic unbounded tails and portable exact readers accompany the proofs. 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

The paper proves the stated all-order cofactor bounds, occupation and curvature reserves, 1/31 sufficient domain, uniform Newton projection-defect bound, specified structured-family comparisons and explicit higher-rank neighborhoods. The full two-third-root-orbit factor-two and B0 lower-floor theorems require nonnegative ordinary loss; their amplitudes and overlap phases are unrestricted. The adaptive tube requires PSD actual/reference cores with diagonal at most 30, nonnegative reference loss and entry distance at most 2^-100. No global localization or generic rank-six occupation-capacity/compensation theorem is claimed. The 2025 Sa-nguansin–Rodtes full proof was inaccessible for comparison; no exhaustive containment or first-ever priority is claimed. Classical projection, balancing, CLL, Jensen and complete-search tools are attributed. Internal AI-assisted verification is not external human refereeing.

Reproduction notes

The complete 58-page manuscript and 38-member package contain twelve editable LaTeX sources, twenty-one Python standard-library checkers, necessary exact fixture data, reader/build instructions, citation metadata and a complete public checksum manifest. Forty-two extracted normal/optimized reader executions passed. An isolated source rebuild reproduced all mathematical text and page geometry. The orbit reader verifies complete rational 48/30 and 586/455 covers with analytic unbounded tails; the adaptive-tube reader checks low/high overlap constants and exact complex active-rank-six/rank-five examples. Shared local arithmetic dependencies and the distinction between analytic proofs, complete finite covers and finite nonemptiness checks are explicit. Generic coverage is not inferred from examples.

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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

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  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.
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Cite this edition

Scott Schubr. (2026). Occupation bounds for coupled permanent cofactors (Version 8). Luminary Research Archive. https://research.luminaryoddities.com/occupation-bounds-coupled-permanent-cofactors/v/8

BibTeX
@techreport{luminary_occupation_bounds_coupled_permanent_cofactors_v8,
  title = {Occupation bounds for coupled permanent cofactors},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 8},
  url = {https://research.luminaryoddities.com/occupation-bounds-coupled-permanent-cofactors/v/8}
}

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