I wonder how many attempted proofs attempt to solve through this bottom up approach rather than top-down.
I started with powers of 2 that obviously reduce to the cycle and tried to apply these transitions inferring larger and larger sets of numbers whose binary representation satisfies specific regular expressions on 0 and 1s, but got lost pretty quickly.
My intuition is that, given that several generalizations of Collatz Conjecture are undecidable (equivalent to a halting problem), this process is in the territory of being not yet Turing-complete but already undecidable. But I am pretty sure many people way smarter than me tried this approach as well.