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
- 1 · 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 receiving 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 paid phase-boundary reduction then gives the same uniform bound on every feasible phase fiber whose real part has rank at most three. Higher real-support cases and the full Chollet conjecture remain open.
Keywords permanent derivatives · positive semidefinite matrices · Hermitian sums of squares · trace multipliers · exact certificates · representation theory
Scope of the work
Uniform compensated permanent-derivative positivity on centered complex PSD matrices of order six and rank at most two, and real-part rank at most three phase fibers. Least trace exponent five is proved only in the specified two-column integer trace-multiplier HSOS hierarchy. Higher real-support cases and the full Chollet conjecture remain open.
Reproduction notes
The package contains editable LaTeX, portable exact reconstruction code, primitive certificates and the explicit trace-four witness. Install the pinned requirements, then run python verify.py and python -O verify.py. The verifier regenerates the original identities, all 28 primitive forms and the exact negative receiving value using integer and rational arithmetic. The supplied README describes the proof scope, dependencies and PDF build.
The manuscript
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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.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Sharp trace-degree certificates for rank-two permanent-derivative compensation (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/sharp-trace-degree-certificates-rank-two-permanent-derivative-compensation/v/1
BibTeX
@techreport{luminary_sharp_trace_degree_certificates_rank_two_permanent_derivative_compensation_v1,
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 1},
url = {https://research.luminaryoddities.com/sharp-trace-degree-certificates-rank-two-permanent-derivative-compensation/v/1}
}Loading discussion…