It's already being done. Has been done for decades now. Definitely wouldn't be a good use of an LLM-type model if that's what you're proposing

If you look at the placement of Journal of Cellular Automata in SciMago's Shape of Science visualization[0] you'll see that it's completely surrounded by machine learning/AI journals

[0] https://www.scimagojr.com/shapeofscience/

“Easily verifiable”… Not really, you have to simulate 133_076_755_768 steps. Sure it’s doable. But if the AI suggests a thousands patterns, then it will be useless.

You can change "might not" to "have not".

The Game of Life is Turing complete. And therefore a complete analysis of how to write programs in it would imply a solution to the Halting problem. Which is impossible.

Turing completeness relies on infinite state.

With finite state, one could theoretically brute force search every possible 1D sequence to find a glider shorter than the one discovered here.

Obviously that's impractical, but turns the whole thing into a search problem - find the best/a good solution in a huge search space.

This is specifically a task that LLMs would be utterly useless at. Because they can't execute loops.