The Collatz Descent Mechanism: Structural Proof via Exhaustive Case Analysis and 2-Adic Fixed Point Theory

We prove that every Collatz trajectory on positive integers eventually descends below its starting value. The odd Collatz map operates as a four-state machine with algebraically fixed halving counts. Two states always contract; two always grow. Growth states route to contraction through an exhaustive case-splitting tree whose unresolved branches are governed by affine maps in the 2-adic integers. The direct recursion f(m) = (9m+1)/8 has unique fixed point m = -1 in Z_2, draining exactly 3 units of v_2(m+1) per step. The cascade detour decomposes into branches T_j(m) = (3^j * m + c_j) / 2^(j+1), each with a unique negative fixed point in Z_2, draining (j+1) >= 4 units per step. The composite of any sequence of detour maps is itself an affine contraction with a negative fixed point not in Z+, and total v_2 drain equal to the sum of individual drains. Both paths terminate for every positive integer by the same mechanism: the only non-terminating orbit sits at a 2-adic attractor outside Z+. The proof is entirely structural: no probabilistic, statistical, or ergodic arguments are used. Computational verification confirms all structural claims across 428 million integers with zero failures.