Scale-Weighted 2-adic Haar Sobolev Spaces and Upper-Triangular Koopman Dynamics of the Syracuse Operator

This preprint develops a functional-analytic framework for studying the compressed Syracuse (Collatz) map over the ring of 2-adic integers. By introducing scale-weighted 2-adic Haar Sobolev spaces, the paper proves that the associated Koopman pull-back operator admits an explicit infinite-dimensional block upper-triangular representation with respect to the Haar basis. Exact spectral bounds are established: the operator is bounded on the mean-zero subspace for all s > (1/2) log2(3), and becomes a strict Sobolev contraction only when s > 1. These results provide new operator-theoretic insights into non-Archimedean dynamical systems and open problems related to Collatz dynamics.