The Collatzogin Tree: A Structural Framework with Conditional Proofs and Open Problems for the Collatz Conjecture

We present a comprehensive structural framework for the Collatz conjecture based on the Collatzogin Tree, a directed graph that partitions all positive integers by residue classes modulo powers of two.

Our proven contributions include:
(1) Fibonacci branching $N_k = F_{k+2}$ with Golden Ratio convergence;
(2) the Universal Transition Lemma: every node reaches a Single-Child Node (SCN);
(3) depth function analysis $D = E - O\log_2 3$ and the 2-adic Accumulation Lemma;
(4) no non-trivial cycles;
(5) every SCN contains at least one element reaching the Golden Path $\G = \{(2^{2r}-1)/3 : r \ge 1\}$.

We prove conditional results: the Collatz conjecture follows from either the Golden Path Conjecture or the Global Depth Conjecture.

Scope: This paper provides a rigorous structural framework and conditional proofs. The Collatz conjecture remains open. We identify the precise open problem: proving that every element in every SCN reaches the Golden Path.