The following things come to mind when you look into research in this area, especially the search for <BB(5)> which was hard but doable:
* it is largely <recreational mathematics>, i.e. done by non-professionals, a bit like the <aperiodic tiling>. Humbly, they tend to call their results <lemma (mathematics)>[lemmas]
* complex structure emerges from simple rules, leading to a complex <classification (mathematics)> with a few edge cases, much like the <classification of finite simple groups>

Bibliography:
* https://cs.stackexchange.com/questions/59344/what-are-very-short-programs-with-unknown-halting-status
  * https://cs.stackexchange.com/questions/44869/what-are-the-simplest-examples-of-programs-that-we-do-not-know-whether-they-term imprecise duplicate
* https://cstheory.stackexchange.com/questions/20978/what-is-the-smallest-turing-machine-where-it-is-unknown-if-it-halts-or-not


 Specific values of the Busy beaver function