Real-section permanent-derivative compensation and quadratic phase control
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Abstract
We prove a uniform lower bound for a compensated homogeneous permanent derivative on the entire real centered positive semidefinite section of order six. If r is the trace, the bound is strictly greater than 5000r^4 for nonzero matrices, with explicit positive reserves from two actual omission variances. The proof cancels two signed contractions by subtracting fifty-eight times the fourth-degree omission variance and pays the ordinary side with the third-degree omission variance. No spectral gap, rank or entry-sign restriction is needed. Conjugation symmetry makes the loss along an imaginary perturbation quadratic; explicit coefficient and multilinear norm budgets then prove the lower bound 219r^4 on the closed complex tube ‖Im K‖_F≤ r/1000. We also record the unrestricted complex cancellation identity and its quadratic-state correction. Its remaining coupled remainder is not paid globally.
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Research details
- Subjects
- Mathematics · Linear Algebra and Matrix Inequalities
- Keywords
- permanent, positive semidefinite matrices, omission variance, compensation, complex phases, matrix inequalities
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
Uniform compensated permanent-derivative positivity on the full centered six-row real PSD section and an explicit quadratic-loss complex neighborhood.
Disclosures
Author-published preprint. AI assistance was used in research, drafting and checking; the author is responsible for the mathematical statements and presentation. Internal checks and exact finite computations are not external human refereeing or formal proof-assistant verification.
Reproduction
10-page manuscript and 8 source-package members. Compile main.tex. Standard-library python verify.py, also under -O, reconstructs the finite coefficient identities and variance controls and writes verification.json.