I think in the base2 reformulation I propose we do not know for certain if the list of numbers terminates, as all Fermat primes are in the set and we don't know

I think in the base2 reformulation I propose we do not know for certain if the list of numbers terminates, as all Fermat primes are in the set and we don't know if there are infinitely many Fermat primes. For base-10 and the original rules the set is provably closed. "Drilling through zeros" makes the branching unbounded.