This work develops a unified mathematical framework for understanding how recursive cognition—such as hierarchical syntax, nested visual concepts, and multi-level planning—can emerge from non-recursive perceptual and sensorimotor processes. Cognitive representations are modeled as metric–measure spaces equipped with coarse-graining operators, group-invariance projections, nonlinear cognitive transformations, and compositional operators. By formulating these components within Banach-space approximation theory and non-commutative operator dynamics, the paper identifies precise conditions under which recursive structure becomes a stable fixed point of a renormalization flow. The analysis shows that non-contractive coarse-graining, non-expansive cognitive transformations, and controlled compositional distortion are jointly necessary for the emergence and stability of recursive representations. The framework connects naturally to concepts in control theory, state-space stability, and Kalman filtering, offering a bridge between cognitive science, mathematical physics, machine learning, and computational neuroscience. This cross-disciplinary formulation provides a principled foundation for explaining why recursive cognition is rare, how it develops under resource constraints, and why many artificial systems struggle with recursive generalization.