How to verify new digits of $\pi$? Bob makes a claim that he made a new record and computed $\pi$ to 10 trillion digits (or your favourite number here). How would Alice verify that the newly computed constant is actually a correct approximation of $\pi$?

Given a finite string $x,$ of $n+1$ decimal digits: $(3\ 1\ 4\ 1\ 5\ 9\ 2\ldots),$ is there an efficient algorithm to decide whether $x$ is an approximation of $\pi$ up to the $0.\underbrace{00 \ldots 01}_{n-1}$ decimal places?


Edit: Clarification.


*

*Alice does not have access to Bob's method (so she can't prove that his method is correct).

*Alice only receives $x$ from Bob (in any number bases), and wants to verify that $x$ is indeed a correct approximation. No further communications between them.

*Alice could look at all digits of $x$ but should be able to verify in time $\ll$ than what it takes to compute $x$.

*Motivation: Assume Alexander J. Yee did not publish his code nor his method. He only publish $x$ in many number bases. He said it took him 3 months to compute $x.$ How could we verify his claims that $x$ is correct in a day or week or two without access to his code and formulas? Is there such a verification formula or algorithm?


Since you allow any base, (16 in particular) and randomized algorithm, you can use the Bailey-Borwein-Plouffe formula which allows you to compute the $n^{th}$ digit of $\pi$, without having to compute the earlier $n-1$ digits! (Alas, such a algorithm seems to have been discovered only for base-16.)
All Alice needs to do is pick "some" random digits and compare.
