Negative Feedback: The Collatz Conjecture as a Discrete Control System

The Collatz conjecture is reframed as a discrete feedback control problem. The plant is the set of odd positive integers; the controller is the Syracuse map T(n) = (3n + 1)/2^k; the reference signal is the constant 1; and the feedback element is the +1 operation. The sign of this feedback — positive or negative — is shown to determine convergence completely. Negative feedback (3n + 1) routes every expansion class (S3) unconditionally into contraction classes (S5, S1), constituting a one-step error correction at every Syracuse iteration. Positive feedback (3n − 1) routes contraction classes back into expansion, producing the known cycles of that system. The Lyapunov function V = n/2^(Σk), proved strictly…