CNRS: A Combined Architecture for Complex Numeric Representation Systems: Arithmetic, Analytic Continuation, Operator Calculus, and Physical Structure

We present a unified architecture for Complex Numeric Representation Systems (CNRS), integrating three layers: CNRS-A (arithmetic, base−2 + i, place values ρⁿ), Layer 2 (analytic continuation via branch index k), and CNRS-H (calculus, place values ρⁿ/n!, digit shift as differentiation). The combined state X = (a, k, h) separates arithmetic, branch, and differential structure into interoperable components governed by six consistency axioms. A type system with four types and explicit conversion functors (A scalar, H series, A poly, H poly) defines the practical interface. The operator-theoretic structure of CNRS-H includes a Hilbert-space inner product with weights wn = 1/ (n!) ², the Weyl algebra commutation relation D, I = id on H/Cδ0, and a spectral decomposition connecting eigen-streams to coherent states. The digit shift admits a renormalisation-group interpretation in which eigen-streams are RG fixed points. Four open problems and a prioritised seven-item roadmap are stated.