Fourier Shadows and Tracial Free Infinite Divisibility
Manuscript 3 October 2026 · Online 3 October 2026
Abstract
We compare tensor and free addition of bounded tracial tuples whose joint laws are freely infinitely divisible, after projecting both laws to scalar Fourier series. The Fourier shadows obtainable from free sums form a proper subset of products of such shadows in every dimension at least two. A witness is the product of two variance-one semicircle Fourier transforms on separate coordinate axes. Every tracial joint lift of this product fails to have a free fifth root: a fixed sum of Hermitian squares has expectation −2/25 at that formal root. At free exponent t its expectation is 2t(4t−1), so positivity requires t ≥ 1/4; no endpoint sufficiency is asserted. We also prove commuting free-root rank-one rigidity and construct distinct bounded tracial free compound Poisson laws with identical Fourier shadows.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Bounded-tracial theorem
- Categories
- math.OA · math.PR
- 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, “Fourier Shadows and Tracial Free Infinite Divisibility,” EulerSolve Research Papers, AIM-PROBABILITY-0131, 2026. https://doi.org/10.5281/zenodo.23114092.
BibTeX
@misc{Ferudun2026FourierTracialFID,
author = {Ferudun, Alper},
title = {Fourier Shadows and Tracial Free Infinite Divisibility},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/aim-probability-0131/},
doi = {10.5281/zenodo.23114092},
note = {AIM-PROBABILITY-0131; unrefereed preprint}
}