Sharp permanent-derivative compensation on the flat equal-spectrum conjugate-disjoint 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
Complete forty-point omission-variance zero classification, sharp compensation on the flat equal-spectrum rank-two stratum, and quantitative full-rank neighborhoods and positive-part reduction.
Disclosures
AI assistance was used in research, drafting and checking. The author is responsible for the mathematical statements and presentation. This author-published preprint does not claim external refereeing or formal proof-assistant verification.
Reproduction
10-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 to regenerate exact paired-polynomial, permutation, sparse-fixture and continuity-constant controls in verification.json.