The elements at the ‘horizon’ of the diagram are the irrational elements >1 in the extension field Q[sqrt5] encoded in binary in the Stern-Brocot tree. In that tree between every two integer values to the right of 1 there is a subtree isomorphic to the whole left side (the subtree between 0 and 1). As encoded in the S-B tree every irrational element >1 in the extension field Q[sqrt5] has a finite prefix and an infinite alternating tail. The set of encoded irrational elements >1 in Q[sqrt5] that corresponds to ‘points’ at the ‘horizon’ exhausts the set of integer prefixes. The tree is rooted at 1. There are no cycles. The structure is hyperbolic.
The integer prefix equivalence classes consist of finite strings that share the identical prefix and a finite alternating tail. Each such class corresponds 1-to-1 to the encoded irrational element >1 in Q[sqrt5] that shares the identical prefix but has an infinite alternating tail. Every string of connected values that rises to the right (elements in the same class) corresponds to such an encoded element in Q[sqrt5].
Since every integer belongs to some class and exactly one class, and every integer is the prefix of some class, every class is present as a right, rising sequence of finite strings leading to an infinite string at the horizon with that class prefix. Therefore, every prefix class is present and connected, and there are no integers not connected to 1 in the cascade of classes represented in the diagram above.
In the odds-only Collatz graph every downward path from an OPI will eventually reach 1 under iteration of Cp .
21st Century Paradox 