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 - Olog₂(3), where E is the number of halving steps and O is the number of odd steps, and prove that: (a) The 3 4 regime is the only source of negative drift in the depth function; (b) D > 0 descent (Terras' Criterion). 2-adic Accumulation Lemma: We prove that the length of consecutive 3 4 blocks is bounded: s ≤ log₂(a+1). Residue Dynamics: We prove the complete transition table modulo 4: (a) 0, 2 4 → smaller odd number; (b) 1 4 → 0 4; (c) 3 4 → 2 4; (d) Every 3 4 eventually reaches 1 4. No Non-Trivial Cycles: We prove there are no non-trivial cycles using the irrationality of log₂ 3. Equivalence to Collatz: We prove that the Collatz conjecture is equivalent to the Global Depth Conjecture: forall n ∈ Z>0,\; limt → ∞ D(t) = +∞. Scope: This paper provides the dynamical tools needed for the complete proof in Paper 3. The final step---proving that every 1 4 number reaches the Golden Path---is addressed in Paper 3.