Research Article · Preprint Mathematics
Sharp permanent-derivative compensation on the flat equal-spectrum conjugate-disjoint Gram stratum
- 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 study a homogeneous compensated derivative arising from permanent inequalities for centered complex positive semidefinite matrices of order six. On the conjugate-disjoint locus K K̄=0, the ordinary omission variance has exactly forty trace-normalized zeros, all sparse rank-one cube-root configurations. On the entire flat, equal-spectrum rank-two stratum, including all compatible complex phases, we determine the compensated derivative explicitly and prove its sharp lower bound 18912-140√2. The proof retains the common Gram structure: a real balanced-frame complement gives a complete complex-structure chart, and threefold permutation symmetry eliminates the remaining phase from the paired permanent derivative. An explicit Frobenius tube of radius 10^(-6) has a positive uniform margin and contains full-dimensional families of centered rank-five matrices, whose positive rank-one anchors have rank six. A conditioning-free positive-part construction also reduces small conjugate overlap to the exact locus.
Keywords permanent · positive semidefinite matrix · Gram matrix · compensated derivative · Fischer norm · balanced frame
Scope of the work
Complete forty-point omission-variance zero classification, sharp compensation on the flat equal-spectrum rank-two stratum, and quantitative full-rank neighborhoods and positive-part reduction.
Reproduction notes
10-page manuscript and 8 package members. Compile main.tex with Tectonic or compatible LaTeX. Run python verify.py or python -O verify.py using Python 3.10+ and its standard library to regenerate exact paired-polynomial, permutation, sparse-fixture and continuity-constant controls in verification.json.
The manuscript
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Author disclosures
AI assistance was used in research, drafting and checking. The author is responsible for the mathematical statements and presentation. This author-published preprint does not claim external refereeing or formal proof-assistant verification.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Sharp permanent-derivative compensation on the flat equal-spectrum conjugate-disjoint Gram stratum (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/zero-geometry-sharp-compensated-permanent-derivatives-complex-gram-stratum/v/2
BibTeX
@techreport{luminary_zero_geometry_sharp_compensated_permanent_derivatives_complex_gram_stratum_v2,
title = {Sharp permanent-derivative compensation on the flat equal-spectrum conjugate-disjoint Gram stratum},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/zero-geometry-sharp-compensated-permanent-derivatives-complex-gram-stratum/v/2}
}Loading discussion…