the mathematical proof of the halting problem, as I understand it, would in this context be based on the notion that user input could somehow direct the program

the mathematical proof of the halting problem, as I understand it, would in this context be based on the notion that user input could somehow direct the program to write a program in ram and get it to execute. This has in fact been achieved in games like super mario world or pokemon! It seems to me sometimes that educated programmers call halting problem too early on situations like this where you don't actually need a rock solid "provably correct" compiler, just a compiler that does something reasonable in this bounded set of cases, and we don't mind that much if it crashes sometimes, since it's not like an NES game is a rocket or a therac.