Comment by charlieyu1

This answer from o3 looks better. There are still some holes, Lemma 1 works on the four right-most columns, then the model tries to apply it to any four columns. "Hence at least four columns lack a vertical line; take the last four columns above." is a slip in logic, and Lemma 1 doesn't work for any two columns. For example, if we choose columns 1, 3, 5, 7 as columns lacking a vertical line, then we can take a sunny line with slope -1/2 and this will meets at all these four columns. Still, this looks more promising than what I got.

Interesting result from Gemini, I don't know its thought process but it seemed like Gemini tried to improve from its own previous answer and then got there.