AI suddenly solves 80-year-old math problems: humans search breadth-first, LLMs go depth-first

Latent Space · rss · 2026-10-23

Latent Space examines why AI models suddenly got good at math: in May an internal OpenAI model found a counterexample to the 1946 Erdős unit distance conjecture; a month later OpenAI published a proof of the Cycle Double Cover problem, then solutions to ten more open problems by July and a Navier-Stokes proof in September. Anthropic published a formalization of Fermat's Last Theorem and advanced bounds on the Riemann hypothesis.

The core question: can we predict which problems LLMs will solve? The article builds a complexity mental model:

Problem complexity: solving requires a sequence of insights with work between them. Gowers frames difficulty as Kolmogorov complexity modulo experts, 2^k + t (k = shortest hint sequence, t = development work). Complexity grows exponentially with the number of insights needed.

Humans are breadth-first: time is the human constraint — developing an idea can take months, so researchers demand early promise before committing. Research software exists to shrink the mechanical work between insights.

LLMs are depth-first: one prompt, one solution; CoT is greedy depth-first. Parallel agents push ten paths to conclusions. Without a verifier this yields beautiful false proofs; with an automated verifier, spending huge compute on uncertain ideas becomes rational — you know in minutes whether a hint works, so the human quality bar for hints nearly disappears.

Failed attempts are negative hints: a typical harness has a Drafter (writes prompts from priors and past failures), a Solver (parallel test-time compute), and a Verifier. For CDC, "assume a complete affirmative proof exists" stops the model from quitting because it knows the problem is open; adversarial agents rule out substitute pseudo-proofs.

Conclusion: LLMs let us trade compute for time, pursuing far more problems in parallel.

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