Tsimerman: AI can outlast mathematicians in treacherous waters

The unit-distance disproof drew the richest commentary, and Gowers framed proof difficulty as the shortest sequence of hints that lets experts reconstruct it, a measure he calls Kolmogorov complexity modulo experts. His own sequence had three parts: look for a counterexample, generalize the best known construction, and try number fields of increasing degree with prime ideals of bounded norm. The order matters. Start at the third hint and you do not even know the exercise is disproof, which is why listing a thousand techniques and trying every combination does not work.

What stopped Jacob Tsimerman was not skill but the feeling of being several weeks into a swamp with no evidence of another side. Psychology found the same split decades ago: on insight problems, solvers report no sense of progress until they find the answer. For an LLM, failures are cheap. OpenAI's harness uses a drafter that turns failed paths into negative hints, a solver that runs parallel agents, and a verifier that prunes completed trajectories. Its Navier-Stokes proof took 10,000 agents and 300B tokens in 88 hours.

The divergence may be complementary. Humans shrink branches, models shrink the time it takes to prune one branch, and experts can ask a model to explore an idea before deciding whether it is worth three months. Problems needing many insights stay hard for both. Terence Tao describes solving problems as a proxy for developing understanding, and a proof is a community event, so announcing only the final solution mistakes the destination for the journey.