Zero geometry and sharp compensated permanent derivatives on a complex Gram stratum
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Abstract
We study a homogeneous compensated derivative arising from permanent inequalities for centered complex positive semidefinite matrices of order six. On the conjugate-disjoint locus K K̄ = 0, the ordinary omission variance has exactly forty trace-normalized zeros, all sparse rank-one cube-root configurations. On the entire flat, equal-spectrum rank-two stratum, including all compatible complex phases, we determine the compensated derivative explicitly and prove its sharp lower bound 18912 − 140√2. The proof retains the common Gram structure: a real balanced-frame complement gives a complete complex-structure chart, and threefold permutation symmetry eliminates the remaining phase from the paired permanent derivative. An explicit Frobenius tube of radius 10⁻⁶ has a positive uniform margin and contains full-dimensional families of centered rank-five matrices, whose positive rank-one anchors have rank six. A conditioning-free positive-part construction also reduces small conjugate overlap to the exact locus.
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Research details
- Subjects
- Mathematics · Linear Algebra and Matrix Inequalities
- Keywords
- permanent, positive semidefinite matrix, Gram matrix, compensated derivative, Fischer norm, balanced frame
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
Exactly forty omission-variance zeros on the centered conjugate-disjoint trace-one locus; sharp phase-complete bound F₄ ≥ 18912 − 140√2 on the flat equal-spectrum rank-two stratum; explicit full-rank neighborhood and positive-part reduction.
Disclosures
AI assistance was used in research, drafting and checking. The author is responsible for the mathematical statements and presentation.
Reproduction
The supplementary ZIP includes the manuscript PDF, complete editable LaTeX source, a standard-library Python exact verifier, exact output, reader instructions, citation metadata and SHA-256 member manifest. The verifier computes a symbolic coefficient and checks an independent 720-permutation implementation; the universal geometry and phase argument are proved in the paper.