Offers a model of problem difficulty as the shortest chain of insights plus work, to predict which open problems LLMs are likely to crack and where to aim them.
Key takeaways · AI-distilled
The essay adopts W.T. Gowers' view of difficulty as the shortest hint sequence experts need plus the work between hints, roughly 2^k + t. For the unit-distance disproof the hints were: seek a counterexample, generalize the best construction, use number fields of rising degree.
The author argues humans search breadth first because developing an idea can take months, so weak hints get dropped early. Models go depth first, and an automated verifier that prunes finished arguments makes it rational to test implausible ideas in minutes.
Failed attempts become negative hints. The essay quotes OpenAI's Cycle Double Cover prompt, which lists insufficient substitutes and known failure modes as standing checks and tells the model not to stop merely because current approaches fail.
Adding a branching factor b, the author predicts models do well where b^k times compute cost is small. That would explain why cybersecurity fell first, with intermediate branching of about 1-2, and why seeking a counterexample collapsed the unit-distance search.
The author expects models to excel at finishing work whose conceptual leaps exist, citing OpenAI's Navier-Stokes proof completing Córdoba and Martínez-Zoroa's program with 10,000 agents and 300B tokenThe chunk of text a model reads and writes in — roughly three-quarters of a word — and the unit AI usage is billed in.Full definition → in 88 hours. Gowers expects any human edge to be on deep, branching searches.
Terms in this piece · Glossary
token — The chunk of text a model reads and writes in — roughly three-quarters of a word — and the unit AI usage is billed in.
