Research Article · Preprint Mathematics
Phase-boundary reduction and low-rank compensation for permanent derivatives
- 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 study a compensated homogeneous permanent derivative on centered Hermitian positive semidefinite matrices of order six. An exact Hermitian sum-of-squares certificate proves F₄(K) > 8000(tr K)⁴ for every nonzero centered complex positive semidefinite rank-one matrix, with no conjugate-disjointness assumption. We also prove a quantitative phase-boundary reduction: positivity of the actual derivative propagates from a saturated phase endpoint to its entire feasible ray, with a strictly positive reserve. Combining the two results gives F₄(K) > 5040(tr K)⁴ whenever rank(Re K) ≤ 2, including unequal real spectra and singular boundaries. A complementary common-state tensor dual certifies every feasible phase over an explicit noncirculant real base and a closed neighborhood outside the previously established spectral domain. The proofs use an opposite bound for the pure quartic phase term, exact rational certificates, and complete original-permanent coefficient reconstruction. Generic saturated compensation with higher real-part rank, the other homogeneous terms, and the full Chollet conjecture remain open.
Keywords permanent · positive semidefinite matrix · phase boundary · Hermitian sum of squares · matrix inequalities · exact certificate
Scope of the work
Computer-assisted exact theorems for the auxiliary compensated derivative F₄ on centered Hermitian positive semidefinite matrices of order six: unrestricted complex rank-one bound >8000(tr K)⁴; all real-part-rank-at-most-two phase fibers >5040(tr K)⁴; paid phase-boundary reduction; an explicit nonpinched all-phase neighborhood >16000(tr K)⁴. Generic saturated ranks two through four, other homogeneous blocks, and the full Chollet conjecture remain open. Constants and multiplier degree are not claimed optimal.
Reproduction notes
The complete publication package includes the editable TeX manuscript, identical PDF, exact coefficient and Gram data, portable Python verification code, pinned SymPy 1.14.0/python-flint 0.9.0 dependencies and a SHA-256 member manifest. Run python verify.py and python -O verify.py. Complete original coefficient identities and exact characteristic-polynomial/Bareiss certificates are checked; finite controls are not universal proofs.
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. Exact computations and internal checks are not external refereeing or formal proof-assistant verification.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Phase-boundary reduction and low-rank compensation for permanent derivatives (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/phase-boundary-reduction-low-rank-compensation-permanent-derivatives/v/1
BibTeX
@techreport{luminary_phase_boundary_reduction_low_rank_compensation_permanent_derivatives_v1,
title = {Phase-boundary reduction and low-rank compensation for permanent derivatives},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/phase-boundary-reduction-low-rank-compensation-permanent-derivatives/v/1}
}Loading discussion…