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 F₄(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 F₄(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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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
Computer-assisted exact theorems for the auxiliary compensated derivative F₄ on centered Hermitian positive semidefinite matrices of order six: unrestricted complex rank-one bound >8000(tr K)⁴; all real-part-rank-at-most-two phase fibers >5040(tr K)⁴; paid phase-boundary reduction; an explicit nonpinched all-phase neighborhood >16000(tr K)⁴.
Disclosures
AI assistance was used in research, drafting and checking. The author is responsible for the mathematical statements and presentation.
Reproduction
The complete publication package includes the editable TeX manuscript, identical PDF, exact coefficient and Gram data, portable Python verification code, pinned SymPy 1.14.0/python-flint 0.9.0 dependencies and a SHA-256 member manifest. Run python verify.py and python -O verify.py. Complete original coefficient identities and exact characteristic-polynomial/Bareiss certificates are checked.