ChatGPT-5.5-Pro provided a formal proof for a rare math error that human mathematicians then verified as a major discovery.
This research documents a specific case where a cutting-edge language model generated a mathematically useful lead that was actually true. The model identified a differential equation where the standard rules of uniqueness fail, a result that is highly non-trivial. While many assume AI only summarizes existing knowledge, this system acted as a true research partner by suggesting a valid conjecture. Humans then took that lead and turned it into a formal, verified proof for the field. It shows that AI is starting to contribute original, high-level mathematical insights that move the needle for science.
Non-uniqueness for a differential equation and a proof by ChatGPT
arXiv · 2605.04810
Let $f(t,x),M(t,x)\in C([0,1]^2)$ with $M(t,x)>0$. We consider differential equations of the form \[\frac{\partial f}{\partial t}(t,x)=\frac{M(t,x)f(t,x)-M(t,0)f(t,0)}{x},\quad x>0. \] For a fixed positive weight $M$, we ask whether the condition $f(0,x)=0$ forces $f\equiv 0$. We show the answer is negative for smooth functions: there exist $f(t,x),M(t,x)\in C^{\infty}([0,1]^2)$ with $f(0,x)=0$, $f(t,0)\not\equiv 0$, and $M(t,x)>0$ satisfying the above equation. However, we show that for a large