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 F4(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.
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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
For centered Hermitian positive semidefinite order-six K of positive trace r, the closed real-part spectral band (7r/40)P ≤ Re K ≤ (9r/40)P gives F4(K) > (124803/256)r^4 > 487r^4, for every feasible complex phase. This is an auxiliary compensated derivative theorem.
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 contains the editable TeX manuscript, standard-library Python verifier and exact rational SOS certificate, deterministic verification data, reader instructions, citations and SHA-256 manifest. Run python verify.py, optionally python -O verify.py, and compile main.tex with a TeX distribution. The verifier reconstructs the full quadratic coefficient identities, verifies all quartic polynomial coefficients and exact positivity, and checks normalization and singular-boundary controls.