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
- 1 · 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. Arbitrary complex phases away from the certified regions, the other homogeneous derivative blocks and the full Chollet conjecture remain open.
Keywords permanent · positive semidefinite matrices · omission variance · compensation · complex phases · matrix inequalities
Scope of the work
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. This does not prove global generic complex compensation, the other derivative blocks, or the full Chollet conjecture. Constants are conservative.
Reproduction notes
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.
The manuscript
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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 1). Luminary Research Archive. https://research.luminaryoddities.com/real-section-permanent-derivative-compensation-quadratic-phase-control/v/1
BibTeX
@techreport{luminary_real_section_permanent_derivative_compensation_quadratic_phase_control_v1,
title = {Real-section permanent-derivative compensation and quadratic phase control},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/real-section-permanent-derivative-compensation-quadratic-phase-control/v/1}
}Loading discussion…