There is no proof of that for our crypto systems. There is a proof you can’t prove ZF theory, but you also probably can’t prove it false. If you hide that question in the thing you iterate over, you can take it on faith you won’t find a counterexample and say the program won’t terminate; but you haven’t, and in fact cannot, prove it.

There are also lesser problems that have been around a while which we haven’t been able to prove, but might be provable… you can iterate over numbers looking for an counter example to the Collatz conjecture… I think that’s the best example of a simple program that is outside our current knowledge of math (better than the crypto one), but maybe someday we will have an answer.