On behalf of all the authors, I am sharing this manuscript with everyone interested in the problem, in the spirit of sharing mathematical knowledge.
I do not claim any traditional academic credit for this paper.
This paper is not intended for publication in a traditional mathematics journal.
Story behind the paper
This paper grew out of a project involving Shitan Xu, Ni Tang, Bochao Kong, Yu Shen, and Minghao Zhao, which explored whether GPT-5.6 Sol could prove the main theorem in dimension five. After obtaining an initial proof generated entirely by AI, they invited me to join the project and help verify it. We subsequently checked and simplified the proof.
We later found an alternative proof of some of the results in the five-dimensional case and recognized that this approach might extend to higher dimensions. In pursuing this extension, we encountered a gap in the argument that the image of the limiting linear system must be a variety of minimal degree when its dimension lies strictly between one and the dimension of the original variety. We presented the incomplete argument to GPT, which filled the gap and completed the proof of this assertion. An essential input introduced by GPT was the use of slope semistability of the tangent sheaf arising from K-semistability. We subsequently verified the argument. This enabled us to rule out the intermediate-dimensional case, which proved to be the most difficult part of the proof in arbitrary dimension.
We verified the arguments and wrote the paper from scratch to highlight its central ideas. ChatGPT was also used for language editing. We take full responsibility for all claims and arguments presented in the paper.
An independent development
After we obtained the proof in dimension five, I posted it on this webpage. Long Pan subsequently contacted me, suggesting that the argument could be extended to dimension six and informing me that he had completed such a proof.
After we obtained a proof in arbitrary dimension, I informed Long Pan of our result. He told me that, in the meantime, he and Haoyu Wu had also obtained a proof in arbitrary dimension using GPT. A brief exchange indicated that their proof likewise used slope semistability arising from K-semistability as an essential new input.
This experience illustrates how different groups presenting the same mathematical problem to GPT can arrive at similar arguments with the same essential ingredients.