Scaling Universality: A Framework for Function Representations

A fundamental question in function approximation theory is whether physical operator scale bounds permanently restrict representational capacity or merely reparameterize it. Standard universal approximation theorems evaluate function classes under fixed input and output conventions, leaving a theoretical gap for scale-bounded computational substrates even when target functions remain fully recoverable via output rescaling. We resolve this gap by introducing scaling universality, an algebraic framework that incorporates input dynamic ranges and output scale bounds directly into the representation criterion. The central theoretical insight is the formulation of an output scale threshold ($\tau$): rather than requiring representation at a fixed output scale, scaling universality requires representations to hold across all output scales exceeding $\tau$. This threshold requirement elevates output scaling from an ad-hoc heuristic to a closed algebraic framework, guaranteeing structural closure and representation existence for both exact and approximate function classes under linear precomposition, output scaling, addition, and composition across operational depth. To demonstrate the framework in the norm-bounded regime of quantum computing, we instantiate scaling universality via Generalized Quantum Signal Processing (GQSP), proving exact scaling universality for real polynomials on $[-1,1]$ via real-valued GQSP primitives and deriving the analytical normalization threshold governing this space. Extending to multivariate domains, we combine GQSP primitives with Linear Combinations of Unitaries (LCU) and a scale-compatible recursive superposition architecture to prove approximate scaling universality for continuous functions $C({\mathbb{R}}^n)$ on $[-1,1{]}^n$. This work establishes the representational completeness and algebraic closure of scale-bounded primitives, providing a rigorous theoretical foundation for functional representation decoupled from downstream resource trade-offs such as sampling complexity, compilation depth, and parameter optimization.