There are some good ideas in there, but I can see a few problems with this proof: 1. It doesn't show the result for all natural numbers, only some "high probab

There are some good ideas in there, but I can see a few problems with this proof: 1. It doesn't show the result for all natural numbers, only some "high probability" fraction of them (similar to what Tao did). 2. The expected value of the ratio being 1 doesn't imply that the actual number goes to 1, since it could be stuck in an endless loop with itself as the minimum. 3. 3n+1 doesn't necessarily have a uniform distribution over the even numbers. Tao gets around this by picking some "stable" subset of numbers that don't move too much under the transformation.