CNRS-Pr1-2: A Single-Value Complex Numeral System via Canonical Number Systems and the Riemann Surface of the Logarithm

We construct a positional numeral system in which a single sequence of real integer digits encodes a single complex number — with no ordered-pair structure and no separator symbol between real and imaginary parts — and on which the logarithm and exponentiation are single-valued. The system has two components. Layer 1 (Positional encoding). 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 representation via a bi-infinite digit string (one string, one complex number, no separator). These results follow from K´atai–Szab´o (1975) and Thurston (1989), which we connect to the single-value problem. The multiplication law is Πn·Πm = Πn+m (index addition, Cauchy convolution of coefficients), and the value map is a ring homomorphism. Layer 2 (Branch state). Attaching a branch index k ∈ Z to a complex number produces an extended element z = (r,θ,k) living on the Riemann surface of the logarithm. The logarithm log(z) = ln r+ i(θ+ 2πk) and exponentiation are single-valued and mutually inverse. All powers of an extended element are single-valued without branch cuts. The algebraic structure is a commutative group under multiplication with a compatible partial addition — a commutative group with partial addition — the correct type for the Riemann surface of the logarithm, which does not admit a ring structure. One new notational symbol is required: the complex point . (the positional analogue of the decimal point, separating positive-index from negative-index digit positions). We discuss three open problems. The arithmetic closure of Layer 1 (Problem 3 of the CNRS programme) is now substantially complete: addition is computable by a 14-state, 350-transition transducer; multiplication is characterised in three tiers with two passes inherently necessary for online two-argument multiplication. Operational completeness under addition and multiplication is proved in a companion paper. The most significant remaining open problem is extending the Layer 1 construction from base z0 =−2 + i to a transcendental e-base that would realise differentiation as a primitive positional operation.