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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*, 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.

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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
  4. Edition 8 · 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. 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. The quadratic power is sharp. 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.

Disclosures

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

Reproduction

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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