The Collatz like behaviour may be specific to the choice of Turing Machines as computational model. In functional busy beavers we see the emergence of repeated exponentiation [1] at similar sizes (the TM in the article takes 32 bits to describe given the statecount).<p>[1] <a href="https://oeis.org/A333479" rel="nofollow">https://oeis.org/A333479</a>
The surprising thing about the program discussed in the article is that it isn't complicated at all. It's well-structured and elegant, with no "spaghetti" to speak of. It almost looks like something you could have written yourself, provided you knew the that this specific Collatz-like function takes such a long orbit on the input 2.<p>It could be that larger Busy Beaver programs engage in programming tricks to run up high scores, but it could also be that the champion programs are always elegant, instead winning by exploiting increasingly wild mathematical facts. That's what we might expect based on the results of the Bigger Number Game [1] and the C Bignum Bakeoff [2].<p>[1] <a href="https://www.scottaaronson.com/writings/bignumbers.html" rel="nofollow">https://www.scottaaronson.com/writings/bignumbers.html</a><p>[2] <a href="http://djm.cc/bignum-results.txt" rel="nofollow">http://djm.cc/bignum-results.txt</a>
Conway showed that Collatz-like functions are as hard as Halting, by creating a language called FRACTRAN whose programs are Collatz-like functions: <a href="https://esolangs.org/wiki/Fractran" rel="nofollow">https://esolangs.org/wiki/Fractran</a>