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 = ρⁿ; 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.