Resolution criteria

This market resolves to YES if the first AI-driven computation (inference run) that successfully solves one of the six remaining Millennium Prize Problems is publicly documented or estimated to have cost over $1,000,000 USD in compute resource for inference. It resolves to NO if the first successful AI solver's run costs $1,000,000 USD or less, or if the first verified solution is found by a human without the direct output of an AI inference system.

Definitions and Details:

• Millennium Prize Problems: The six remaining unsolved problems designated by the Clay Mathematics Institute (CMI):

  1. Birch and Swinnerton-Dyer Conjecture

  2. Hodge Conjecture

  3. Navier–Stokes Existence and Smoothness

  4. P versus NP Problem

  5. Riemann Hypothesis

  6. Yang–Mills Existence and Mass Gap

• Successfully Solves: A solution is considered successful if it is officially awarded the Millennium Prize by the CMI, or if it receives widespread consensus from the global mathematics community (e.g., peer-reviewed publication in a top-tier mathematics journal such as the Annals of Mathematics, with general acceptance over a 2-year period following its initial public release).

• Inference Compute Cost:

  • This is the cost of running the AI model to generate, search for, or verify the successful proof (including test-time compute, reasoning loops, tree searches, parallel generation pipelines, and multi-agent systems).

  • It excludes the cost of pre-training, fine-tuning, or training the model or its base architectures.

  • If the organization behind the run (e.g., Google DeepMind, OpenAI, Anthropic) publishes the exact or estimated dollar cost of the run, that figure will be used.

  • If no dollar figure is published but the hardware used and the duration of the run are known, the cost will be estimated using the on-demand rental rates of major cloud providers (such as AWS, GCP, Azure) for that equivalent hardware at the time the run occurred.

• Timing of Resolution: This market will remain open until a validated solution to one of the remaining six problems is announced and verified. If the verified proof is initially posted as a preprint (e.g., on arXiv), the market will resolve once that specific proof is validated (even if validation takes several years), evaluating the run that produced the initial successful version.

Background

The Clay Mathematics Institute established the seven Millennium Prize Problems in 2000, offering a $1 million prize for each. Only the Poincaré Conjecture has been solved (by Grigori Perelman in 2002–2003, awarded in 2010).

With the rapid development of frontier AI models utilizing search, reasoning-time compute, and reinforcement learning, organizations like Google DeepMind are actively leveraging AI to target mathematical discoveries (such as efforts on the Navier-Stokes equations). While training these models costs tens or hundreds of millions of dollars, this market specifically focuses on the inference-time (or test-time) compute cost of the final run that successfully generates the winning mathematical proof.