Observers on N: A Lyapunov–Entropy Taxonomy of Integer Programs

We introduce a Lyapunov–entropy framework for classifying accelerated integer programs as minimizers, oscillators, or exploders. The method constructs a state-only functional L(n) that decreases strictly under acceleration. This provides a taxonomy of integer dynamical systems, clarifies the role of cancellation and injection, and situates Collatz as a canonical minimizer. Extensions include blockwise surplus criteria, explicit examples, and an operator embedding into prime-based Hilbert spaces.