Applications of Sequence Operator Methods: Arithmetic, Geometric, and Recurrence Dynamics Beyond the Collatz Problem

The previous manuscripts in this program developed a general theory of sequence operatorsystems, multivariate coupling trees, a parity-progression formalism for the 3x + 1 problem,and a conditional level-ergodic bridge for density-one Collatz behavior. The present paper isdesigned as the next step: it demonstrates that the operator methodology is not a privatelanguage for the Collatz problem, but a portable framework for organizing and comparingseveral families of arithmetic dynamics. We apply the same conceptual package—generators,sequence-building laws, classification schemes, splitting operators, inherited weights, localstate transitions, and potential functions—to three model classes: generalized qx + 1 maps,geometric and residue-dynamical sequence families, and recurrence-generated systems.The main contribution is methodological but mathematically precise. We define canonicalsequence operator systems for these families, prove their basic well-definedness and closureproperties, identify natural invariant and potential candidates, and isolate the exact obstruc-tions that distinguish tame finite-state systems from wild cross-level systems. The paperthereby supplies evidence that the sequence-operator viewpoint is a general mathematicallanguage rather than a Collatz-specific encoding. It also explains why the Collatz case isexceptional: it combines a rigid local arithmetic template with nonlinear cross-level feedback,a combination not present in many simpler arithmetic, geometric, or recurrence models.