Cody T. Dianopoulos here. My proof was extremely flawed as natural, but my research has shown up some results. Considering the even numbers is just a bump in the road, completely eliminating them from the problem by modifying the algorithm simplifies the problem vastly. Secondly, my progress has shown that CC is not a problem in number theory, but instead a string transformation on binary strings. I'm working on a form of strong induction (in turn finding an upper bound) in order to prove CC. A sketch: assume CC holds for strings of length 0<=L<=k. Then show that given iteration of the algorithm in my most recent paper will eventually end up reducing the length to some 0<=L<=k.