We introduce the State Transition Graph Sa and the Fabric Fa – a bijective, recursive weave of odd numbers in which multiples of a act as terminal leaves with no children, while every non-terminal node generates an infinite balcony of descendants. We provide proof of the Collatz conjecture. The Terminal Leaf Proof identifies multiples of 3 as the unique entry points of all trajectories and shows that they lock the door to infinity. The Graph Proof, expressed in the language of S3, establishes the same result through weak connectivity and the absence of non-trivial cycles. The two proofs are fully equivalent, mirroring the bijection between the Fabric and the Graph. We classify canonical roots for all odd a, describe the modular genetic code (MGC) of the Fabric (the 2:1 rhythm for a=3), and prove the Localness of Divergence Theorem. We show that terminal leaves always exist in any ax+1 system, so terminal leaf is only unique universal invariant of ax+1 systems. Their proportion among all odd numbers is exactly D(a)=1-φ(a)/a, where φ is Euler's totient function. For the classical case a=3 this gives exactly 1/3. Consequently, as a grows, the number of branches shrinks and their average depth increases without bound. In the limit a → ∞ Fabric of a-systems with terminal leaf density D(a) → 0 degenerates into a single branch connecting 1 to infinity – the axis of all natural numbers – exposing the illusion of divergence that plagues computer experiments for systems such as 7x+1 and beyond. Whereas Fabrics of a-systems with terminal leaf density D(a) → 1 are overgrown with terminal leaves; in the limit of highly composite numbers they degenerate into the super-dandelion, where almost every number is a silver leaf directly attached to the root. We extend the concept of silver leaves (1 step to root) to systems with multiple attractors (e.g., 5x+1), where each cycle possesses its own infinite periodic family of silver leaves. We observe an isolated but striking cross-connection: the smallest silver leaf 5 of 5x+1 is simultaneously a child of the root in 3x+1, suggesting to a deeper resonance between Mersenne and Mersenne+2 Fabrics. Among the other objects we introduce are golden leaves, the fractal fan of fans, the Prime Weave, and the Green Leaf Comet – the scatter plot of stopping times of terminal leaves, exhibiting cascades of nested comets, interference patterns modulated by residue classes modulo 3, and a distinct two-peak depth distribution (a new measure of deterministic memory). Extensive computation reveals deep structure in the distribution of depths, interference patterns, and a structured sieve for prime numbers. A complete list of 34 original contributions is provided.
Andrei Fedotkin (Mon,) studied this question.