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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
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10 · Permanent link
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Overview

Abstract

For a complex positive semidefinite Gram matrix B=VV*, we couple its permanent-cofactor contractions with those of |B|² and prove the sharp all-order operator 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, we identify a nonnegative cofactor-weighted variance E and characterize its zero set. Ordinary occupations of the same Gram factors give the sharp all-order bound E≥(αₙ/n!)(na−t), with an occupation-resolved refinement. Joint trace and frame-variance bounds retain singular factors, arbitrary complex phases and ambient kernel mixtures. At order six the variance, inherited base defect and forward curvature yield explicit sufficient compensation domains, with the sharp curvature floor R_(√2−1)≥(19/240)α₆². A common-factor argument proves compensation when one selected occupation is at most 1/31, giving the corresponding self-conjugate order-seven domain.

For the actual six-factor core F(t)=per(K+tJ), set P=F(1)+F′(1) and 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 a diagonal cutoff. A separate 1/32 entrywise tube around the equality family has a strict optimized-capacity margin. The intrinsic criterion L≥0, P+3L−7200≤2⁻¹⁶⁰ derives bounded diagonals and two small residual rows, rather than assuming them, and supplies another explicit capacity margin. The tube and intrinsic regions also satisfy the strict self-bound per|K+√2J|²<(3/4)F(√2)².

Weighted coherent projections, occupation geometry and classical permanent bounds provide further phase-dependent certificates, including full-amplitude two-mode and unequal two-third-root-orbit families with explicit higher-rank neighborhoods. Exact obstructions show why several scalar, fixed-dual and coarse-envelope extensions fail. The attributed small-order induction is reconstructed self-containedly, and the original residual occupation and thinning identities are retained. The generic rank-six compensation inequality, the generic three-quarter self-bound and unrestricted Chollet remain open. A uniform six-row creation-operator reserve further gives strict diagonal monotonicity and convexity of the transformed deficit. Two upper coefficients of a two-coordinate residual pencil are controlled, while an exact complex positive-semidefinite example rules out unrestricted directional monotonicity. The first and middle two-coordinate payments remain open.

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

Scope of the work

All previous stated domains and hypotheses are retained. Section 30 proves the uniform six-row row reserve P>(Kii+√2)per(K(i|i)+√2J5), strict diagonal monotonicity and convexity of D=P²−C, and the two-coordinate upper payments c3≤p1p2 and c4≤p2²/2. An exact positive-loss complex PSD pair A≼B with D(A)>D(B)>0 rules out unrestricted PSD-direction monotonicity. The first and middle two-coordinate payments, generic rank-six compensation, generic three-quarter self-bound and unrestricted Chollet remain open. This does not newly remove the singular rank-five obligation or justify repeated sparse descent. Classical inputs are attributed. External novelty remains qualified, including the inaccessible complete 2025 Sa-nguansin–Rodtes proof. Internal AI checking is not external human refereeing.

Reproduction notes

Complete 64-page manuscript and 42-member package: fourteen editable TeX sources, twenty-three Python standard-library checkers, exact adaptive data, reader instructions, citation metadata and a complete 41-entry checksum manifest. All 46 extracted normal/optimized checker runs passed. The row checker reconstructs all 21 Jacobi leading minors and 76 positive coefficients exactly, and the named complex PSD obstruction. The row proof includes the whole degree-five space and singular boundaries. An isolated packaged-source rebuild matches every page’s extracted text and geometry. Finite examples and structured certificates do not establish generic coverage. Shared arithmetic and internal-review limits are disclosed.

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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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Cite this edition

Scott Schubr. (2026). Occupation bounds for coupled permanent cofactors (Version 10). Luminary Research Archive. https://research.luminaryoddities.com/occupation-bounds-coupled-permanent-cofactors/v/10

BibTeX
@techreport{luminary_occupation_bounds_coupled_permanent_cofactors_v10,
  title = {Occupation bounds for coupled permanent cofactors},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 10},
  url = {https://research.luminaryoddities.com/occupation-bounds-coupled-permanent-cofactors/v/10}
}

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