Scaling Fixed Points in Hierarchical Recursive Estimation Systems

We establish the existence and uniqueness of a scaling fixed point in hierarchical recursive estimation systems, where parameters are governed by concentration-of-measure inequalities. Under a self-similarity assumption, the deep-layer dynamics converge to a regime characterized by the canonical scaling ₖ x^*/Nₖ for accuracy and a constant failure probability ₖ 2e^- (x^*) ²/C. While the 1/N law is fundamental in single-shot estimation, its emergence as the unique attractor of a multi-level, interactive recursive process is a non-trivial dynamical phenomenon. This fixed point is interpreted as a discrete renormalization-group fixed point, revealing an emergent scaling structure. Our results provide a foundational asymptotic framework for analyzing error propagation and resource-performance trade-offs in hierarchical coding, distributed computation, and recursive statistical inference.