Full-phase permanent-derivative compensation under real-part spectral pinching
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Abstract
We prove positivity of a compensated homogeneous permanent derivative for centered complex positive semidefinite matrices of order six whose real parts satisfy a fixed spectral pinch. With trace r>0 and P=I-J/6, the closed inequalities (7r/40)P≼Re K≼(9r/40)P imply F_4(K)>487r^4. Every positive-semidefinite-allowed imaginary part is included, without a small-phase assumption. The matrices may be singular or arbitrarily ill-conditioned. The proof combines a universal quartic phase bound, certified by an exact rational sum of squares, with the complete quadratic phase coefficient, its operator-norm variations and monotonicity of actual omission coefficients. On the isotropic real base the full phase fiber has the stronger bound 24455r^4. An inherited negative mixed coefficient also excludes a class of global independent-factor tensor certificates, without refuting the repeated-state inequality.
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Research details
- Subjects
- Mathematics · Linear Algebra and Matrix Inequalities
- Keywords
- permanent, Chollet conjecture, Gram matrix, spectral pinching, sum of squares, complex phases
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
Uniform full-phase compensated permanent-derivative positivity on the stated real-part spectral-pinching domain of centered six-row complex PSD matrices, including singular limits.
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 do not constitute external refereeing or formal proof-assistant verification.
Reproduction
10-page manuscript and 13 source-package members. Compile main.tex with its included sections. Python 3.10+ and its standard library run python verify.py or python -O verify.py to check the quartic sum-of-squares, quadratic phase and original permanent-coefficient identities.