Variance cancellation and two-direction bounds for six-row permanent derivatives
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Abstract
We prove a uniform positive bound for an actual compensated permanent derivative on the entire rank-one conjugate-disjoint locus of centered complex positive semidefinite matrices of order six. With trace r, the bound is [1248-(1672/3)√2]r^4>459r^4 for nonzero matrices, without equal-modulus or phase restrictions. The proof derives an exact paired-moment identity and cancels its difficult signed cubic moment against ten times the same ordinary omission variance. Centered projection and moment bounds then pay the remainder. We also prove the sharp bound (14688-208√2)r^4 on the full two-direction cube-root family, including unequal norms, arbitrary complex overlaps and singular endpoints, together with a sharp spectrum-resolved variance tradeoff. The earlier explicit Lipschitz estimate gives full-dimensional neighborhoods and quantitative exclusion of the rank-one boundary from the rank-two extension.
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Research details
- Subjects
- Mathematics · Linear Algebra and Matrix Inequalities
- Keywords
- permanent, variance cancellation, positive semidefinite matrix, conjugate-disjoint Gram locus, Fischer norm, rank-one compensation
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
Uniform rank-one conjugate-disjoint compensation through exact variance cancellation, together with a sharp bound and equality classification for the full two-direction cube-root family.
Disclosures
AI assistance was used in research, drafting and checking. The author is responsible for the mathematical statements and presentation. The exact computations and internal checks do not constitute external refereeing or formal proof-assistant verification.
Reproduction
12-page manuscript and 8 package members. Compile main.tex with Tectonic or compatible LaTeX. Run python verify.py or python -O verify.py using Python 3.10+ and its standard library; the program regenerates verification.json with exact rank-one, two-direction and independent permutation controls.