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. Publication is currently paused by the author.
