We present a full contradiction-based proof of the Collatz Conjecture using classicaltools from number theory and integer dynamics. The argument is built around a compressedtransformation operator that captures full growth–decay cycles of the standard3n + 1 map in a single step. We define a bit-length entropy function to measure thecomplexity of iterated values and show that entropy decreases in expectation under thecompressed operator for odd inputs. This expected descent contradicts the possibilityof infinite or divergent orbits. The analysis is entirely deterministic, formalizable inPeano Arithmetic, and does not rely on probabilistic heuristics. The result confirmsthat all positive integers eventually reach the known cycle {4, 2, 1} under the Collatzmap.