CNRS-Pr2: Complex Numbers as Single Values: A Three-Layer Architecture for the Complex Numeric Representational System

We present a three-layer architecture for Problem 2 of the Complex Numeric Representational System (CNRS) programme: the construction of a positional numeral system in which a single sequence of real integer digits encodes a single complex number, with no ordered-pair structure, no separator symbol between real and imaginary parts, and single-valued logarithm and exponentiation. Layer 1 (Positional encoding, from literature). In base z0 =−2 + i with digit alphabet {0,1,2,3,4}, every Gaussian integer has a unique finite representation as a single digit string, and every complex number has a bi-infinite representation via the Thurston tiling theorem. One digit string, one complex number, no separator (K´atai–Szab´o 1975, Thurston 1989). The Layer 1 instance is denoted CNRS-A(ρ, D) (arithmetic prototype): place values Πn = ρ^n; multiplication law Πn·Πm = Πn+m; value map Val a ring homomorphism. Layer 2 (Branch state, new construction). Attaching a branch index k ∈ Z to a complex number produces an extended element z = (r,θ,k) that lives on the Riemann surface of the logarithm. A multiplication law with carry tracks winding around the origin. The logarithm log(z) = ln r+ i(θ+ 2πk) and exponentiation exp(c) = (e^(ℜc),ℑc mod 2π, ⌊ℑc/2π⌋) are single-valued and mutually inverse. Layer 3 (Hybrid progressive system CNRS-H — proved; original e-base — open). The hybrid progressive system Πn = ρn/n!, denoted CNRS-H (ρ, D) (calculus prototype), resolves the Layer 3 transcendence obstruction: convergence is unconditional by factorial decay; the digit-shift operator DH is differentiation d/dρ on evaluation (under the EGF digit convention — seeRemark 5.17); the base ρ is the eigenvalue of DH on the canonical digit stream. Multiplication law: Πn ⋆ Πm = (n+m n)Πn+m(binomial convolution); value map Val a ring homomorphism. The CNRS-H system is fully proved. The original fixed e-base formulation — a CNS-type theorem with a finite real digit alphabet for z0 = e1+iβ — remains a separate open problem (see Section 7) The two-system architecture. CNRS-A serves Layer 1 (arithmetic: finite digit alphabet, Gaussian integer representation, addition and multiplication by finite transducer). CNRS-H serves Layer 3 (calculus: differentiation as primitive, unconditional convergence, eigenvalue connection). Together with Layer 2, they constitute the complete CNRS architecture. An analysis of addition in the extended system (Section 4.5) establishes that Layer 2 is a commutative group with compatible partial addition: multiplication is a full group operation; addition is defined when the sum is non-zero (a partial operation, since the Riemann surface has a puncture at the origin); the distributive law holds for underlying complex values but not for branch indices. This is the correct and expected algebraic type for a system encoding the Riemann surface of the logarithm. It is not a ring, and does not need to be.