Phase-boundary reduction and low-rank compensation for permanent derivatives
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Abstract
We study a compensated homogeneous permanent derivative on centered Hermitian positive semidefinite matrices of order six. An exact Hermitian sum-of-squares certificate proves F_4(K)>8000(tr K)^4 for every nonzero centered complex positive semidefinite rank-one matrix, with no conjugate-disjointness assumption. We also prove a quantitative phase-boundary reduction: positivity of the actual derivative propagates from a saturated phase endpoint to its entire feasible ray, with a strictly positive reserve. Combining the two results gives F_4(K)>5040(tr K)^4 whenever rank(Re K)≤2, including unequal real spectra and singular boundaries. A complementary common-state tensor dual certifies every feasible phase over an explicit noncirculant real base and a closed neighborhood outside the previously established spectral domain. The proofs use an opposite bound for the pure quartic phase term, exact rational certificates, and complete original-permanent coefficient reconstruction.
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59b6df197fbb3815bab7271368436893f972098174a79a6bfb1766efb441af00
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Research details
- Subjects
- Mathematics · Linear Algebra and Matrix Inequalities
- Keywords
- permanent, positive semidefinite matrix, phase boundary, Hermitian sum of squares, matrix inequalities, exact certificate
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
Uniform compensated permanent-derivative positivity on centered complex rank-one and real-part-rank-at-most-two six-row PSD matrices, with a phase-boundary reduction and additional certified phase domains.
Disclosures
AI assistance was used in research, drafting and checking. The author is responsible for the mathematical statements and presentation. Exact computations and internal checks are not external refereeing or formal proof-assistant verification.
Reproduction
13-page manuscript and 20 source-package members. Compile main.tex with its section files. Install requirements.txt, including SymPy 1.14.0 and python-flint 0.9.0, and run python verify.py and python -O verify.py to reconstruct the exact permanent, phase and certificate controls.