Uniform permanent-derivative compensation on the conjugate-disjoint Gram locus
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Abstract
We prove a uniform lower bound for a compensated homogeneous permanent derivative on the complete conjugate-disjoint locus of centered complex positive semidefinite matrices of order six. The locus has rank at most two, and the theorem includes arbitrary phases, nonuniform diagonals, unequal eigenvalues and singular limits. If r is the trace, p the product of the two possible nonzero eigenvalues and δ the ordinary omission variance, the lower bound is [1248 − (1672/3)√2]r⁴ + [29568 − (33920/3)√2]pr² + (20160 − 80√2)δ. All three coefficients are positive. The proof extends the earlier rank-one cancellation identity before imposing spectral relations, then uses an exterior-square Gram identity to pay the coupled remainder with the same variance reserve. Combined with an earlier conditioning-free projection and explicit Lipschitz estimate, this gives a uniform small-overlap region in the full centered positive semidefinite section.
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Research details
- Subjects
- Mathematics · Linear Algebra and Matrix Inequalities
- Keywords
- permanent, positive semidefinite matrix, conjugate-disjoint Gram locus, variance cancellation, exterior-square identity, compensated derivative
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
Uniform positive lower bound for F₄ on the entire centered complex PSD6 locus K K̄ = 0, with positive spectral-product and omission-variance reserves; arbitrary phases, zero rows and singular limits. Explicit uniform small-overlap corollary. Constants are 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 includes the complete ten-page PDF, editable LaTeX source, standalone standard-library exact verifier and reader data, instructions, citation metadata and SHA-256 manifest. Seven exact realizable controls, three separate 720-permutation checks, zero and scaling endpoints, the eleven-row coefficient-collection certificate and a negative mixed-coefficient obstruction corroborate the printed universal proof. Normal and optimized Python produce identical exact output.