We present a structural framework for the Collatz conjecture called the Collatzogin Tree---a directed graph constructed from the forward Collatz function. The tree partitions positive integers by their residue modulo 2^k-1, guaranteeing coverage of all integers by construction. Our main contributions are: Fibonacci Branching: The number of nodes at each level follows Nₖ = F₊+₂, where Fₖ is the Fibonacci sequence. Branch Distribution: The distribution of nodes between the 1 4 and 3 4 branches follows a Fibonacci pattern, with N₁ (k) = F₊+₂ and N₃ (k) = F₊+₁. The ratio N₁/N₃ converges to the Golden Ratio. Odd-Only Predecessor Chains: } From the tree, we extract odd-only chains that obey the recurrence aₓ+₁ = 4aₜ + 1, with closed-form formula aₙ = (3a₁ + 1) 4^n-1 - 13. Nest Induction: } We prove that for all n 0, 2 4 and all n 5 8, the trajectory descends to a smaller value. The only remaining case is n 1 8, which may enter the 3 4 regime and is addressed in companion papers. Total Stopping Time Pattern: } For all odd-only predecessor chains, the total stopping times form an arithmetic progression with common difference 2: (aₓ+₁) = (aₜ) + 2. Scope: This paper is structural and descriptive. It establishes the foundation for the dynamical analysis in Paper 2 and the complete proof in Paper 3.
Ogin Sugianto (Thu,) studied this question.