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## Divergents vs Convergents

The statement that "all non-trivial branches hold their own" seems to be valid in all Collatz worlds which are entirely convergent. But things start to get a little more outrageous when we examine worlds which contain divergence. Consider the two particular Collatz worlds. x4.4+1.4 lies on one of the

little blue vertical slashes lost in the sea of pink and is entirely convergent. Meanwhile its cousin x4.4+1.0 lies just below it but falls between two of the blue slashes and is pink. Pink dots, if you recall, contain both convergence and divergence.

Let's graph the percent membership of (somewhat arbitrarily) the odd numbers 1 through 15 in each of the two worlds:

x4.4+1.4

(Entirely convergent)

x4.4+1.0

(Contains Divergence)

This fundamental dichotomy persists across all Collatz Worlds. In entirely convergent Collatz worlds all non-trivial branches hold their own (aka % membership doesn't go to 0). But in any world where there is divergence, all branches' % memberships drop to 0...

Why is that?

We have found a major structural difference between worlds that are all convergent and worlds that contain divergence. If we could truly understand and prove this dichotomy and also show that the x3+1 world's branches wiggle and hold their own, then we would have proven that the x3+1 world has no divergence! This would be proving half of the legendary Collatz Conjecture!

For the first time ever, I had glimpsed the tail ends of a possible proof. I cannot even tell you with what excitement and maniacal effort I chased after these pieces of proof. I searched long and hard and wrote many more Python programs to explore this phenomenon than I'll detail here. Half a year passed. It was a fascinating journey with many sharp ups and downs. Sometimes I'd find tremendous evidence for something and I'd be filled with hope; other times I'd suddenly find a counterexample and all that hope would be dashed against the rocks. I kept searching and I kept digging...

Until I found something...

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