ChatGPT 5.6-Generated Work Finds Product-Positivity Counterexamples for Primitive Forms
Yuhang Liu proves product positivity for positive primitive forms when the first symplectic space has real dimension at most six and constructs explicit counterexamples in complex dimensions at least 30; he states that ChatGPT 5.6 generated the main proof content and that he verified it, while intermediate dimensions remain open.
In a manuscript dated August 15, 2026, Yuhang Liu studies Haiden's question about whether a positive primitive form that is not almost geometric can be paired with another positive primitive form so that their exterior product loses positivity. The paper proves that such failure cannot occur when the first symplectic space has real dimension at most six, but constructs explicit higher-dimensional counterexamples.
The construction gives positive primitive forms on whose product vanishes on the graph of an anti-symplectic involution, and extends this failure to every complex dimension at least 30. It also supplies an exact product criterion valid in every dimension. The cases of real dimensions , including the smallest dimension in which failure can occur, are not settled.
Liu states that ChatGPT 5.6 generated the main content of the proof and that he verified it. The preprint is marked AI-generated and human-verified, with comments invited.