This synopsis will hardly even introduce the subject, but it will mention the highlights and suggest a way in which a proof of the conjecture might be teased out of this viewpoint. The nomenclature used here derives not from the predecessor tree , rooted in 1, in which every integer which is a predecessor in the Collatz sequence of a parental node appears in ascending order as its children, but from its transform into a binary tree .
Extensions head every left descent and every node in the tree has an infinite set of extensions.
The binary predecessor tree is collapsed into an abstract predecessor tree , a tree whose nodes are infinite sets of integers. Its root node consists of all the extensions. The growth continues indefinitely, defining all possible combinations of b and s paths for the left descents. These formulas are derived by traversing in each left descent down to leaf nodes and up again to the parental extension. These formulas represent mutually disjoint subsets.