Sharp trace-degree certificates for rank-two permanent-derivative compensation
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Abstract
This paper proves a uniform lower bound for a compensated homogeneous permanent derivative on every centered complex positive semidefinite matrix of order six and rank at most two. The bound is F_4(K)>1000(tr K)^4 for nonzero K, with arbitrary complex phases, unequal spectra and singular limits. In a two-column Gram factorization, the fifth trace multiplier gives a positive-definite Hermitian sum-of-squares certificate. An explicit degree-eight coefficient-test polynomial has strictly negative coefficient-form value -110160-576√2, proving that five is the least nonnegative integer trace exponent in this multiplier hierarchy. Column-unitary symmetry and row permutation symmetry reduce the positive certificate to 28 primitive forms of dimensions at most 72, with exact coefficient reconstruction and rational positive-definiteness checks. The previously established phase-boundary inequality then gives the same uniform bound on every feasible phase fiber whose real part has rank at most three.
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- Rank_Two_Compensation.pdf339.0 KiB · pdf
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6d545a98e508bf39b299b4a6eac27093d1bead3f448c504e9ffed3c419bf2bc4 - Rank_Two_Compensation_Publication_Package.zip680.8 KiB · package
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Research details
- Subjects
- Mathematics · Linear Algebra and Matrix Inequalities
- Keywords
- permanent derivatives, positive semidefinite matrices, Hermitian sums of squares, trace multipliers, exact certificates, representation theory
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
For nonzero centered complex PSD matrices of order six and rank at most two, F₄(K)>1000(tr K)⁴; exact Hermitian SOS certificates make five the least nonnegative integer trace exponent in the stated hierarchy.
Disclosures
AI systems assisted with algebraic development, source comparison, exact-code checking and manuscript preparation. The author is responsible for the statements and attribution.
Reproduction
The nine-page manuscript and complete editable source are supplied with the portable exact polynomial constructors, 28 primitive coefficient forms and trace-four witness. README.md gives distinct build and certificate-reproduction commands; the manifest authenticates every member.