Research Article · Preprint Mathematics
Sharp trace-degree certificates for rank-two permanent-derivative compensation
- 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 every centered complex positive semidefinite matrix of order six and rank at most two. The bound is F_4(K)>1000(tr K)^4 for nonzero K, with arbitrary complex phases, unequal spectra and singular limits. In a two-column Gram factorization, the fifth trace multiplier gives a positive-definite Hermitian sum-of-squares certificate. An explicit degree-eight coefficient-test polynomial has strictly negative coefficient-form value -110160-576√2, proving that five is the least nonnegative integer trace exponent in this multiplier hierarchy. Column-unitary symmetry and row permutation symmetry reduce the positive certificate to 28 primitive forms of dimensions at most 72, with exact coefficient reconstruction and rational positive-definiteness checks. The previously established phase-boundary inequality then gives the same uniform bound on every feasible phase fiber whose real part has rank at most three.
Keywords permanent derivatives · positive semidefinite matrices · Hermitian sums of squares · trace multipliers · exact certificates · representation theory
Scope of the work
For nonzero centered complex PSD matrices of order six and rank at most two, F₄(K)>1000(tr K)⁴; exact Hermitian SOS certificates make five the least nonnegative integer trace exponent in the stated hierarchy.
Reproduction notes
The 9-page manuscript and complete editable LaTeX sources are supplied in this 20-member package. README.md gives the build and reproduction commands. Reproducibility code: contractions.py, operator_blocks.py, original.py, positive.py, pure_type.py, receiving.py, representations.py, trace_four.py, verify.py. The accompanying manuscript and reader instructions specify the mathematical scope and required dependencies.
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Author disclosures
AI systems assisted with algebraic development, source comparison, exact-code checking and manuscript preparation. The author is responsible for the statements and attribution.
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Cite this edition
Scott Schubr. (2026). Sharp trace-degree certificates for rank-two permanent-derivative compensation (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/sharp-trace-degree-certificates-rank-two-permanent-derivative-compensation/v/2
BibTeX
@techreport{luminary_sharp_trace_degree_certificates_rank_two_permanent_derivative_compensation_v2,
title = {Sharp trace-degree certificates for rank-two permanent-derivative compensation},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/sharp-trace-degree-certificates-rank-two-permanent-derivative-compensation/v/2}
}Loading discussion…