Collapsed Boundary Data Do Not Force Sobolev Labeling
Manuscript 24 October 2026 · Online 24 October 2026
Abstract
We construct a smooth polynomial, uniformly gradient-convex energy on two-valued maps of the unit disk whose minimizers have completely collapsed zero boundary data but admit no global labeling by two Sobolev functions. The potential is lambda times the squared modulus of y^2-(1-|z|^2)^2z. An unordered square-root competitor has energy 16*pi/5, independently of lambda, whereas scalar minima grow on the order of sqrt(lambda). At lambda=2^20 an explicit energy gap excludes every globally labeled minimizer. The density is quasiconvex in the multiple-valued sense. This answers OWR-13948-006 negatively within the original general value-dependent framework. Autonomous gradient-only energies are not addressed. The argument uses established Sobolev, semicontinuity and orientability methods, with prior work credited. This five-page English preprint is by Alper Ferudun, Mercury Software GmbH. It is AI-assisted, self-audited and unrefereed. No independent review, formal verification or absolute priority is claimed. The source package includes exact arithmetic diagnostics; finite checks do not replace the analytical proof. Version 1.0, 24 October 2026. CC BY 4.0. Source dataset record: OWR-13948-006 (ulamai/UnsolvedMath v1.6.0). The manuscript discloses OpenAI Codex assistance.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Complete counterexample
- Categories
- math.AP · math.CA
- Manuscript
- 24 October 2026
- Online release
- 24 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, “Collapsed Boundary Data Do Not Force Sobolev Labeling,” EulerSolve Research Papers, OWR-13948-006, 2026. https://doi.org/10.5281/zenodo.23621671.
BibTeX
@misc{ferudun2026sobolevlabeling,
author = {Ferudun, Alper},
title = {Collapsed Boundary Data Do Not Force Sobolev Labeling},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/owr-13948-006/},
doi = {10.5281/zenodo.23621671},
note = {OWR-13948-006; unrefereed preprint}
}