Recursive Stability in the Collatz 3n + 1 Mapping

This paper applies the Law of Admissibility to the Collatz 3n + 1 mapping, framing the conjecture as a discrete dynamical system operating within a constrained recursive container. By modeling the mapping as a coupled system of amplification and damping, the work demonstrates how trajectories function as self-correcting energy flows driven toward a zero-debt ground state. This analysis identifies the 4 → 2 → 1 loop as the unique admissible sink where net energy flux vanishes and structural integrity is restored. The framework predicts a specific autocorrelation signature for the Collatz mapping: sustained signal coherence during 3n + 1 amplification phases (energy debt accumulation), followed by a rapid correlation decay as the trajectory resolves into the 4 → 2 → 1 ground state.