Nick Trefthen's Coffee Table PDE book provides the following proof of the Weierstrass Approximation Theorem:

Similar questions can be found e.g. here.

This market resolves to yes if an LLM is able to provide a coherent and correct proof in response to a prompt of the form:

Prove the Weierstrass Approximation Theorem using the Simple Heat Equation.

You can use simple prompt engineering tricks, e.g.
- Asking the chatbot to use LaTeX notation.
- Asking the chatbot to pretend to be a well-known mathematician.
- Asking the chatbot to think step by step.

---

A solution that requires specific fine tuning, leading the chatbot towards the correct answer, etc. is insufficient to resolve yes.