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 positivity of the auxiliary compensated homogeneous derivative F4 on the entire real centered PSD section of order six, with retained omission variances; a closed complex phase tube and a general algebraic correction identity. All centered ranks and singular boundaries are included. Constants are conservative.
Disclosures
Author-published preprint. AI assistance was used in research, drafting and checking; the author is responsible for the mathematical statements and presentation.
Reproduction
The complete package contains the manuscript PDF, editable LaTeX source, standalone Python standard-library exact verifier, reproducible JSON, reader instructions, citation metadata and SHA-256 member manifest. Run python verify.py or python -O verify.py; compile with tectonic main.tex. The universal proofs are analytic; finite controls corroborate arithmetic and normalization.