What was already known
Rivière asked whether antisymmetry and critical Lⁿ control alone imply continuity. Martino and Schikorra had proved continuity under stronger Lorentz and curl assumptions; they independently found a counterexample from the same family during preparation of this paper.
AI involvement
ChatGPT 5.6 Sol proposed the winding-map construction. Dominik Schlagenhauf recognized its relevance to Rivière’s broader system, simplified it, wrote the paper, checked the proof and explains the mechanism. The independent Martino–Schikorra work retains separate credit.
The model supplied the retained counterexample, while the human identified the precise theorem it answered and completed the manuscript.
Importance
Gives a counterexample to the broad antisymmetric-potential regularity question.
Limitations
It does not resolve regularity for genuine weakly n-harmonic maps or higher-dimensional H-systems. Martino and Schikorra independently obtained the same example family and a stronger gradient-potential example; their result retains separate credit.
Technical verification
Schlagenhauf explicitly states that the full proof of the stated counterexample was checked; this is a source-reported complete human proof check, separate from the independent paper.
Formal checking
human check of the complete construction proof
- Checker
- Dominik Schlagenhauf, author
- Scope
- Theorem 1.2 discontinuous W^{1,n} solution with antisymmetric Lⁿ potential
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The author explains how logarithmic winding causes discontinuity and how extra angular components and a wedge product produce the required antisymmetric potential.
Correction and publication history
First report of Schlagenhauf’s counterexample.
Read this published revision