The Navier–Stokes existence and smoothness problem is one of the seven Clay Millennium Prize Problems, and as of 2026, the only Millennium Prize problem to have been solved is the Poincaré conjecture. This market asks whether Anthropic will publicly claim that one of its AI systems has resolved it.

Resolves YES if: before 11:59 PM ET December 31, 2026, Anthropic publicly announces — or a preprint with Anthropic-affiliated authors or Anthropic’s official endorsement presents — a claimed complete resolution of the Clay Mathematics Institute’s Navier–Stokes existence and smoothness problem, with Claude or another Anthropic AI system credited as the primary source of the proof.

Terms:

• Either direction counts. A claimed proof of global existence and smoothness, OR a claimed counterexample (finite-time blow-up from smooth, finite-energy initial data), in either the R³ or periodic setting as specified in Clay’s official problem statement. Both resolve the Millennium problem and both count.

• Must be a claim of full resolution. Partial or conditional progress does NOT count: blow-up for modified, hyperdissipative, or averaged Navier–Stokes; Euler blow-up; results under additional assumptions; “substantial progress toward”; or numerical evidence without a claimed proof.

• AI as primary prover. Claude must be credited as having produced the proof or its central argument. A human-authored proof that used Claude as an assistant does not count. Resolver judgment on this element, stated in advance: if Anthropic’s own announcement describes the result as Claude’s, that suffices.

• Announcement, not verification. Resolves YES on announcement regardless of whether the proof is later found flawed. Peer review, expert acceptance, and Clay recognition are not required.

• Anthropic only. A Navier–Stokes claim from Google DeepMind, OpenAI, or any other lab does not count. Anthropic claiming a different Millennium problem does not count.

• Post-creation only. No pre-creation statement satisfies these criteria; as of creation, no such announcement exists.

• Resolution by the primary announcement or preprint, linked in the comments.