The Circularity Is a Property of the Criterion

Been chasing a proof strategy for months and it feels like it should be working? Check whether your certificate is just your conclusion wearing a different vocabulary. Takes one minute. If you work on positivity, semidefinite programming, moment problems, or positive-definite functions in signal processing, you've probably been offered one of these: sum-of-squares, positive Cholesky, Bochner positive-definiteness, Carathéodory–Toeplitz, the Toeplitz symbol. All five are provably the same statement as the conclusion, so a search for any of them can never return "not found, keep looking" — the search is the test, and it terminates instantly. Ask one question: does the certificate exist exactly when the property holds? Yes means the route is dead before you start. And a warning if you test this numerically: we built a fake certificate that agreed with the truth on 100% of 20,000 random matrices and was wrong on thousands once we aime