idea
Random changes
Once the coin has been flipped and the rule added, that snapshot is just another fixed system. A coin also cannot repair the gap, because nothing steers it toward true statements. It has no access to truth.
There is one subtle exception: an infinite stream of truly random choices is, with probability 1, not computable, so the system as a whole would not be a program. But that is escape by accident. You would get a system no one can check or use, with no guarantee it contains the truths you care about.