A Recursive Descent Proof of the Collatz Conjecture via Parity Collapse and Bounded Drift

We present a formal proof strategy for the Collatz Conjecture using recursive descent,parity-based reduction, and bounded drift analysis. We define a potential functionthat captures logarithmic entropy and demonstrate that all orbits under the Collatzmap must eventually decrease, eliminating the possibility of divergent trajectories.Unlike previous heuristic or computational approaches, this method builds a rigorousfoundation for inductive descent and proves that all positive integers eventually reachthe 4–2–1 loop. The proof arises from a structure that, while formally mathematical,is motivated by a deeper logical principle: that any escape path under the Collatzfunction ultimately contradicts its own generative rules.