Okay, I'll just tripfag.
The idea is to split the positive integers into a collection of infinitely many sets, where each set consists of all solutions to (2n-1)(2^x) for positive integer x and a single n from the positive integers. So the first set contains (1,2,4,8...), the second set contains (3,6,12,24...) and so on for all of the positive integers.
Under the 3n+1 transformation, the first set points to itself, the second set points to the third set (an increase of 1), the third set points to the first set (a decrease of 2), the fourth set to the third set and so on. OEIS A256425 counts how much each set jumps up or down the collection of sets under the transformation.
If you count how many jumps between sets it takes from each starting set until you hit the first set you get a different series: 0,2,1,5,6,5,2,5,3,7,1,4,8... which I ran on excel out to 3000 entries to find the upper bound grows at between x and x^2, with the distance between "peaks" growing at a constant rate in the long run.
More of a disproof than a proof, this implies that the number of 3n+1 jumps from any starting number to reach the 1>2>4>1 loop grows faster than any constant multiple of the number of terms.
