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

Abstract

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 two-sided positional representation with a finite integer part and an infinite fractional tail 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(z0, D) (arithmetic prototype): place values Πn = zn 0 ; multiplication law Πn·Πm = Πn+m; value map ValA a ring homomorphism on RA. Layer 2 (Branch state, CNRS branch-state realisation). 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) = [eℜc ,ℑc,0] (a representative of the quotient class, whose canonical form reduces the angle into (−π,π]) are single-valued and mutually inverse. Layer 3 (Hybrid progressive system CNRS-H — proved; original e-base — open). The hybrid progressive system — formally the divided-power basis ρ[n] over RA, with analytic realisation Πn = ρn/n! after embedding — denoted CNRS-H (ρ; RA) (calculus prototype), lies outside the representation class the transcendence obstruction constrains — it uses a divided-power basis over the coefficient ring RA, not one coefficient per exponential place at a fixed transcendental radix with a finite alphabet. Convergence of the analytic realisation is unconditional by factorial decay for bounded scalar digits; thedigit-shift operator DH is differentiation d/dρ on evaluation (under the EGF digit convention — see Remark 5.17); differentiation is realised structurally as the outer digit shift on this divided-power (lowering) basis. Multiplication law:Πn⋆Πm = (n+m/n) Π(n+m) (binomial convolution); binomial convolution gives the formal ring structure on H(RA), with analytic evaluation multiplicative for finite-support or suitably convergent sequences. The CNRS-H system has an established formal differential structure — the Hurwitz carrier H(RA) with canonical coefficient encoding — while efficient nested implementation and analytic control remain open. The original fixed e-base formulation — the bounded-digit transcendental-radix expansion problem, with a finite real digit alphabet for the transcendental radix qe = e1+iβ (written qe throughout, reserving z0 for the fixed CNRS-A radix−2 + i) — 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 fixed-multiplier operations by finite transducer; unrestricted multiplication exact by convolution and normalisation, with no synchronous finite-state two-input multiplier). CNRS-H serves Layer 3 (calculus: differentiation as primitive, and the divided-power lowering structure). 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.