AI Is Getting Good at Math. Here’s Why.

What it claims

The essay argues that LLMs succeed on open math problems when b^k × compute cost stays small. Here k is the number of insights a proof needs and b is the number of plausible next hints at each step. It cites OpenAI results it dates to this year without giving the year: a counterexample to Erdős's unit distance conjecture in May, a Cycle Double Cover (CDC) proof a month later, ten more open problems by July, and a Navier-Stokes proof in September. It also credits Anthropic with a Fermat's Last Theorem formalization and Riemann bounds. The forecast is the essay's own. Models should be strongest at finishing programs whose conceptual leaps humans already made. Gowers is quoted expecting humans to keep an edge where "the search tree is deep and has a large amount of branching."

How it works

The harness the essay reconstructs has three roles. A drafter writes solver prompts from model priors and from earlier failures. A solver runs parallel subagents. A verifier checks finished arguments only.

flowchart LR
  D[Drafter] --> S[Solver: parallel agents]
  S --> V{Verifier}
  V -->|valid| P[Proof]
  V -->|failed| N[Negative hints]
  N --> D

The essay reads the OpenAI CDC prompt as accumulated negative hints. It has four parts:

• An instruction to "assume ... a complete affirmative proof exists," so the model does not stop because pretraining says the problem is open.
• A list of substitutes that do not count: special graph classes, reductions to another unproved conjecture, and computational checks up to a fixed graph size.
• Adversarial checks for specific failure modes, such as repeated-edge closed trails passing as cycles, parallel-edge 2-cycles, and circular use of an equivalent CDC statement.
• A line against quitting: "Do not return merely because current approaches fail."

The essay names three model changes behind the results. Models chain primitives over long horizons, which lowers the work t between insights. OpenAI trained them on math verification, so they catch wrong steps within a trajectory. They also orchestrate subagents with memory, which lets negative hints build up.

Evidence

The essay says the Navier-Stokes proof completes Córdoba and Martínez-Zoroa's program. It runs 166 pages and used 10,000 agents and 300B tokens over 88 hours. For the unit-distance result it uses Gowers' three hints: look for a counterexample, generalize the best known construction, and use number fields of increasing degree with prime ideals of bounded norm.

Jacob Tsimerman, now at OpenAI, had tried the counterexample route with increasing degrees and stopped. The essay explains this with Metcalfe and Wiebe's finding that on insight problems, solvers' felt progress stays flat until they reach the solution. Cybersecurity is the essay's example of a low-b domain, with about 1 to 2 options per intermediate step because exploit chains already exist in the literature.

Limits

This is an opinion essay, not a measurement. The author calls Gowers' 2^k + t formula hard to put numbers into and only usable after the fact. The Navier-Stokes assessment rests on "a cursory reading." The verifier cannot judge half-developed ideas. Self-verification catches wrong steps but not valid steps that lead nowhere, so the middle of a search has no pruning signal. Labs rarely publish their failed attempts, so the insight in those failures is mostly lost. The author also expects labs to pick problems with narrow solution spaces when they want to show results.