Busy Beaver & the Halting Problem

A supposedly "omnipotent" AI would still need to answer a very small question correctly: given an arbitrary program, will it ever stop running? This simulator runs real Turing machines — including two provably-optimal "busy beaver" champions — step by step on a 3D tape, so you can watch the mechanism directly: a state, a read/write head, and a lookup table. Switch to the trivial non-halting machine and you can watch it run forever without ever being able to prove, from observation alone, that it won't stop on the very next step. That gap is not an engineering limitation — Alan Turing proved in 1936 that no algorithm can decide halting for every machine, and the busy-beaver numbers above show just how explosively the true answer can hide even inside machines with only four states.