CNRS: The Complex Numeric Representational System Programme; Comprehensive Summary

This document provides a comprehensive account of the Complex Numeric Representational System (CNRS) programme: its physical and historical motivation, its four-problem mathematical structure, every result that has been established, every result that remains open, and the complete set of documents produced. It is intended as the single reference that gives a reader who is unfamiliar with the programme a complete picture of where it stands and why, as of March 2026. The programme proposes the next step in the historical sequence of numeric system extensions: a positional system in which complex numbers appear as single values, with integration and differentiation as the primitive operations, analogous to the way the decimal system makes real numbers single values with exponentiation and logarithm as its natural primitives. The physical motivation is the scale coordinate zs ∈C of the (x, y, z, s) framework, whose imaginary part encodes the quantum phase currently discarded by the Born rule. The programme has established its founding claim structurally: every complex number is representable as a single digit string (Layer 1, proved), the logarithm is single-valued in the extended system (Layer 2, proved), and differentiation is a structural primitive operation via the hybrid progressive system Πn = ρⁿ/n! (Layer 3, proved in v6). Addition of digit strings is computable by a 14-state, 350-transition finite transducer (Problem 3, complete: exact carry set |K|= 14 computed by BFS). Multiplication is fully characterised: two-pass algorithm with the 14-state normalisation transducer; two passes inherently necessary for online mode; single-pass proved for sequential mode. The branch index kcan be unified into the digit string with one additional symbol (new). A leading-order quantitative bridge to the physics framework has been identified (triangulation conjecture, new). What remains open is: the invariance gap in the base definition (Problem 1), the original e-base formulation of Layer 3, and metric completeness (Problem 4).