http://www-personal.ksu.edu/~kconrow/" [Broken] I mean really read it, to where they understand it, not just to the point where their eyes glaze over? I see his site cited quite frequently, but I don't know if I've ever seen any critique. I admit, I was intimiitaded by his site at first. But over the years, I'm become increasingly skeptical, especially now (Jul 2009) that he thinks he's finally worked out the proof. He's still asking for help to formalize his proof into a publishable format and I can't help him there. We have corresponded by e-mail frequently in the past and I've helped him with little things over the years like fixing his faulty state machines. But I can't see the validity of this interger density thing. And he seems reluctant to talk about my objections. I would like to think my point devastates his theory so much he's speechless, but maybe he thinks it's so trivially wrong he won't waste his time (although with the help I've given in the past, you would think he would have the courtesy to tell me what's wrong). So I'm soliciting other opinions since Ken seems reluctant to offer his. I'll try to keep this as concise as possible, I hope I get it right. Let's start with the basis of Ken's proof, the Left Descent Assemblies (LDA). Ken breaks up his Collatz graph into branches that start with odd numbers 0 (mod 3) and ends them on the first place an odd number is followed by 3 consecutive evens. This odd number is called the LDA Header. After an LDA Header, the sequence merges into other LDAs until they all eventually merge to 1. Ken points out that all LDA Headers are 5 (mod 8) and if all LDA Headers are gathered into a set and used as the root of something he calls the Abstract Predecessor Tree (APT), he can prove that this tree has an integer density of 1 which he claims proves that every positive integer is on the Trivia Collatz Graph, thus, proving it. Now, I don't have any problem with this integre density thing. If Collatz is true, of course the Collatz graph contains all the integers. But he's not using the Collatz graph, he's using the APT. And this is the nub of my gist. I say the truth of Collatz does not follow from integer density 1 of the APT even if it's true. Suppose I use 3n+7 instead of 3n+1. Now, we know 3n+7 fails the conjecture, it's Collatz graph has multiple disjoint pieces and any number not on the Trivial Graph is a counterexample. We know its LDA Headers are 3 (mod 8). We know that all the disjoint graph pieces have LDA Headers. So, collecting LDA Headers into a set must necessarily include the counterexamples on the APT. All APTs for all 3n+C systems must have an integer density of 1, there's no where else for an integer to be. It simply does not follow that Collatz is true just because its APT has an integer density of 1, since 3n+7 has and it isn't true. I say Ken has abstracted away the very thing he's trying to prove, that integer density of the APT does not prove the graph components of the underlying Collatz graph are not disjoint. Does any of this sound right?