Collatzogin Tree Level Descent: Depth Function, 2-adic Accumulation, and Residue Dynamics

This paper builds on the structural framework of the Collatzogin Tree (Paper 1) and develops the Tree Level Descent analysis. We organize Collatz dynamics into a hierarchy of residue levels modulo powers of two. Our main contributions are: Depth Function: We introduce $D = E - O\log_2(3)$, where $E$ is the number of halving steps and $O$ is the number of odd steps, and prove that: (a) The $3 \pmod 4$ regime is the only source of negative drift in the depth function; (b) $D > 0 \iff$ descent (Terras' Criterion). 2-adic Accumulation Lemma: We prove that the length of consecutive $3 \pmod 4$ blocks is bounded: $s \le \lfloor \log_2(a+1) \rfloor$. Residue Dynamics: We prove the complete transition table modulo 4: (a) $0, 2 \pmod 4 \to$ smaller odd number; (b) $1 \pmod 4 \to 0 \pmod 4$; (c) $3 \pmod 4 \to 2 \pmod 4$; (d) Every $3 \pmod 4$ eventually reaches $1 \pmod 4$. No Non-Trivial Cycles: We prove there are no non-trivial cycles using the irrationality of $\log_2 3$. Equivalence to Collatz: We prove that the Collatz conjecture is equivalent to the Global Depth Conjecture: forall $n \in \mathbb{Z}_{>0},\; \lim_{t \to \infty} D(t) = +\infty$. Scope: This paper provides the dynamical tools needed for the complete proof in Paper 3. The final step---proving that every $1 \pmod 4$ number reaches the Golden Path---is addressed in Paper 3.