Variance cancellation proves rank-one compensation on the six-row conjugate-disjoint locus
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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]r⁴ > 459r⁴ 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)r⁴ 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 remaining rank-two problem.
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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 lower bound for the auxiliary derivative F₄ on centered PSD6 rank-one K with K K̄ = 0, including arbitrary phases, unequal moduli and zero rows. Sharp unequal-complex-trine bound and equality; explicit neighborhood corollaries. The rank-one constant is conservative.
Disclosures
AI assistance was used in research, drafting and checking. The author is responsible for the mathematical statements and presentation.
Reproduction
The eight-member supplementary package contains the PDF, complete editable LaTeX source, a standalone standard-library Python verifier, exact output, reader instructions, citation metadata and a SHA-256 manifest. The verifier checks seven coefficient-identification fixtures, 29 feasible trine controls, ten rank-one controls and three separate 720-permutation checks. Universal partition, symmetry and cancellation proofs are printed in the manuscript; finite controls corroborate arithmetic.