What was already known
The three-dimensional ellipticity-only W¹,¹ question and the corresponding approximation-limit question remained open; stronger two-dimensional and near-isotropic bounds did not settle them.
AI involvement
Le, Sun and Tran say the main results arose in chats with ChatGPT 5.6 Sol, which supplied key strategies. They rewrote, simplified and checked all arguments in the final paper.
The authors expressly credit the model with the key strategies for both main results, while retaining substantial human mathematical rewriting and checking.
Importance
The examples give negative answers to both stated three-dimensional regularity questions.
Limitations
The authors checked and rewrote the final arguments, but the disclosure credits ChatGPT with key strategies collectively rather than assigning particular proof steps.
Technical verification
The three authors expressly report checking all arguments of the final paper, which includes both main theorems. This is a source-reported complete human proof check, not an independent reproduction or machine formalization.
Independent checking
Complete author proof review and simplification of all arguments
- Checker
- Nam Q. Le, Qi Sun and Hung V. Tran, paper authors
- Scope
- Theorem 1.1's smooth-coefficient W1,1 counterexamples and Theorem 1.2's non-BV limit
- Basis
- Reported by the linked checker
Human understanding
The authors rewrote and simplified the proofs and give a concrete account of Hessian splitting, scale amplification and the saddle condition that keeps the coefficients uniformly elliptic.
Correction and publication history
Initial publication of this result.
Read this published revision