This is a mathematical analysis of everything Collatz. I've come up with a revolutionary way of representing the natural counting numbers as an infinite set of equations. From these I am able to make some provable connections that not only show that all counting numbers are used once in the Collatz Tree structure; but where additional loops originate; the importance of 4x+1 and 2x+1; duality of even numbers; among others. I also show that there can only be one unbroken chain of continuous "3n+1 / 2" growing toward infinite number sizes approaching infinity but never actually getting there. This would be the 'only' counter-example that is possible and as odds would have it, it does not pan out. That only possible counter example is not to be.Using the induction method where we show that x = 1 is true (elementary, since it is part of the initial loop); from there we assume that x from 1 to k are also true building on the x=1 being true; then k+1 is also true. That is a complicated way of saying that if we know and assume all numbers from 1 to k are true, then the very next number k+1 is also true in as much as we apply the two rules correctly so the number reduces to one that is already in the proven set!The first three equations of my infinite set of equations are easy to apply this induction to and cover 87.5% of the counting number set. I change things up a bit for the upper level equations. I am able to prove through the same induction method that any number that is not a multiple of 3 ( falling in these levels/equations ) is also provable. Stepping outside the usual method of this proof I investigate the multiples of three separately to prove they are all following a similar induction proof. And they do. All said and done I am able to prove that 100% are provable. (4x+1) is important in this proof as well the application of (3x+1)/2. Read on to find out what I mean.I've covered off on the loop issue part of the proof by showing how additional loops come about in the Collatz Tree structure. There is only one loop in Collatz ( positive counting numbers ) and that is the trivial { 1 — 4 — 2 } loop. No others are possible no matter how close to infinity one gets and all numbers will reduce to this trivial loop.The detailed discussion of how I arrived at these different conclusions is outlined below. I apologize if some sections are difficult to follow. I am not a mathematician by nature or profession. I do love mathematics though. I hope you enjoy my approach of showing the 'self' enlightening process as I continued to explore. As my expertise improved, other intuitive aspects became readily useful in the proof. The reader will appreciate knowing why I went down each path I chose to pursue.This is an updated version of my original document with a new section near the end that gets into the details of the proof. The remainder of the original report remains intact for the most part but does have additional details and concepts introduced and dispersed therein.This is the third version where I have solved the outstanding subset of multiples of 3. I believe you will find that method eye-opening since it involves some under the sheets number manipulation by multiple applications of (3x+1)/2. I also introduce the 'duality' nature of even numbers that remain hidden in the Collatz tree structure... and that is that those even numbers can behave as if they were odd numbers; (Odd*3)+1=Even; (Even*3)+1=Odd; (4*Even)+1=Odd.After arriving at this proof I go back to my original set of equations to see if they behave the same way and may provide an easier more condensed method to the final proof. Low and behold they do! It boils down to one chart. Now that I see it on paper I am impressed with my progress. It has taken just over 3 years to finalize the process.In this, the forth version, I have looped back to my original infinite set of equations to formulate a single chart from whence a complete proof can be understood. I have also added a 'final' section that simplifies the induction method of all odd numbers. I think you'll be enlightened with that approach since it really simplifies the inductive proof. If I am right this simplified approach can be a much simplified proof in itself.This detailed analysis has led me to two alternate methods for complete proofs. Enjoy.