There are some mathematical truths that a computer can identify statistically but can never actually reach or prove.
This research defines a new limit on what is knowable by a machine. It shows that certain sets of real numbers permit a procedure that gets closer and closer to the truth, but it can never stop and say I am finished. It establishes a fundamental wall between statistical identification and actual computation. This proves that more data or better algorithms cannot solve certain types of problems. For some truths, we are doomed to approach them forever without ever arriving.
Computability Limits of Sequential Hypothesis Testing
arXiv · 2605.02501
Sequential hypothesis testing asks for decision rules that update as data arrive. A natural goal is \emph{eventual correctness}: the rule may change its mind early on, but it should make only finitely many wrong decisions almost surely. Starting from Cover's theorem, which guarantees such behavior for membership in a countable set of candidate means, we ask a sharper question: \emph{which sets actually admit computable sequential decision procedures with finitely many errors?} We answer this opt