Research Article · Preprint Mathematics
Real-section permanent-derivative compensation and quadratic phase control
- Primary subject
- Mathematics
- Subject subfields
- Linear Algebra and Matrix Inequalities
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
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.
Keywords permanent · positive semidefinite matrices · omission variance · compensation · complex phases · matrix inequalities
Scope of the work
Uniform compensated permanent-derivative positivity on the full centered six-row real PSD section and an explicit quadratic-loss complex neighborhood.
Reproduction notes
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.
The manuscript
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The complete manuscript, as published in this edition.
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Author 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.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Real-section permanent-derivative compensation and quadratic phase control (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/real-section-permanent-derivative-compensation-quadratic-phase-control/v/2
BibTeX
@techreport{luminary_real_section_permanent_derivative_compensation_quadratic_phase_control_v2,
title = {Real-section permanent-derivative compensation and quadratic phase control},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/real-section-permanent-derivative-compensation-quadratic-phase-control/v/2}
}Loading discussion…