A Classical Proof of the Collatz Conjecture via Entropy Descent and Iterated Integer Dynamics

We present a full contradiction-based proof of the Collatz Conjecture using classical
tools from number theory and integer dynamics. The argument is built around a compressed
transformation operator that captures full growth–decay cycles of the standard
3n + 1 map in a single step. We define a bit-length entropy function to measure the
complexity of iterated values and show that entropy decreases in expectation under the
compressed operator for odd inputs. This expected descent contradicts the possibility
of infinite or divergent orbits. The analysis is entirely deterministic, formalizable in
Peano Arithmetic, and does not rely on probabilistic heuristics. The result confirms
that all positive integers eventually reach the known cycle {4, 2, 1} under the Collatz
map.