Irreversibility as a Structural Law of Mathematics

We establish that mathematical systems obeying finite describability and operational irreversibility necessarily exhibit entropy dissipation—a structural constraint independent of probability, physics, or number theory. We prove that no finitely describable irreversible system can sustain infinite information loops: unbounded forward trajectories that avoid finite attractors must accumulate unbounded backward entropy, contradicting finite composition. This positions mathematics as an operational subsystem of nature, constrained by the same informational laws governing physical processes. As an application, we derive a conditional theorem for the Collatz conjecture, reducing its truth to a single entropy compatibility condition (ECC). We prove ECC holds for all trajectories with net dissipative parity structure, narrowing the remaining obstruction to a constrained class of parity-balanced infinite sequences.