The [3x+1] problem and its generalizations: A computation of spatial means through the iterates affine representation

We extend Terras’ affine representation of Collatz-type iterates to generalized $[\mathbf{a}x+\mathbf{b}]$ dynamical systems over Modulo-d Residue Classwise Affine (RCWA) mappings. By exploiting the underlying spatial periodicities, we derive explicit closed-form formulas for the spatial arithmetic and geometric means, higher-order tensor moments, variance matrices, and joint Multiplier-Remainder cross-coupling structures. We show that the classical [3x + 1] problem occupies a critical algebraic singularity characterized by a non-diagonalizable Jordan normal form of multiplicity 2. Furthermore, we introduce a parameterizable probabilistic framework based on independent Bernoulli vectors that reproduces the deterministic spatial statistics. We uncover a saddle-point behavior. We then map these trajectories into the dual space of Moment Generating Functions (MGFs) to study the system's global dynamics.