Regular Totally Nonnegative Loops Without Entire Matrix Logarithms
Manuscript 3 October 2026 · Online 3 October 2026
Abstract
We construct an explicit family of regular, entire, two-periodic totally nonnegative loops with determinant one whose inverses are also entire. Each loop is the product of two positive exponential phases. We prove that this product admits an entire complex matrix logarithm if and only if the phase generators commute. Noncommuting phases give a counterexample to Lemma 8.4 of the inspected arXiv version 0812.0840v3. The obstruction compares the trace parity forced on a logarithm at nontrivial positive and negative unipotent values. Coefficient asymptotics prove regularity, and an explicit rational-parameter example has an exact Jordan certificate. The known general two-exponential factorization theorem is sharp within this regular totally nonnegative class. This does not settle the broader AIM block-Toeplitz factorization request.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Regular-loop logarithm counterexample
- Categories
- math.CV · math.CA · math.CO
- Manuscript
- 3 October 2026
- Online release
- 3 October 2026
- Version
- 1.0
- License
- Creative Commons Attribution 4.0 International
Files and verification
The PDF is the canonical reading copy. The source archive contains the LaTeX manuscript, bibliography, reproducibility material, and audit documents without build artefacts.
Citation
Alper Ferudun, “Regular Totally Nonnegative Loops Without Entire Matrix Logarithms,” EulerSolve Research Papers, AIM-LINEAR_ALGEBRA-0007, 2026. https://doi.org/10.5281/zenodo.23122702.
BibTeX
@misc{Ferudun2026RegularLoops,
author = {Ferudun, Alper},
title = {Regular Totally Nonnegative Loops Without Entire Matrix Logarithms},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/aim-linear-algebra-0007/},
doi = {10.5281/zenodo.23122702},
note = {AIM-LINEAR_ALGEBRA-0007; unrefereed preprint}
}