This is, I think, the moral equivalent of checking to see if the program has started executing a “while(1) ;” loop (e.g. by finding that the current instruction

This is, I think, the moral equivalent of checking to see if the program has started executing a “while(1) ;” loop (e.g. by finding that the current instruction is a “jump $pc”, ignoring multithreading). Although it definitely proves that a program doesn’t terminate, the converse is not true - most “interesting” non-terminating programs aren’t just busy-looping infinitely, they’re just performing computations that happen to have no way of ending. Similarly, a Life machine might just sit there and push gliders around in a circle forever, without sending out an infinite stream.