
The Clay Mathematics Institute must officially announce that a Millennium Prize has been awarded for the complete solution of the specified problem. The solution must be a positive proof as originally stated, not a counterexample or disproof (unless the market specifically includes counterexamples). Announcements of submitted solutions, peer-reviewed publications without CMI verification, or partial solutions do not qualify. If a mathematician solves a problem but declines the monetary prize, the market still resolves to Yes if CMI officially announces the prize award. The Poincaré Conjecture has already been solved by Grigori Perelman.