CNRS-Pr2(1-3): A Single-Value Complex Numeral System via Canonical Number Systems and the Riemann Surface of the Logarithm

Abstract

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 three components. Layer 1 (Positional encoding). In base z0 =−2+iwith 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 as a two-sided positional string with a finite integer part and an infinite fractional tail — loosely “bi-infinite,” though only finitely many integer-side digits are nonzero (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, Laurent convolution of coefficients), and the value map is a ring homomorphism on the finite Laurent coefficient ring RA. Layer 2 (Branch state). Attaching a branch index k ∈Z to a complex number produces an extended element z = [r,θ,k] — an equivalence class of triples under (r,θ,k)∼(r,θ+ 2π,k−1), canonically represented with θ∈(−π,π] — 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. The Layer 2 operations defined here do not form a ring (the carrier has no additive identity, and distributivity fails at the branch-index level); we do not claim that no ring extension of any kind exists. Layer 3 (Divided-power differential interface). Replacing the geometric place values zn 0 by divided powers Πn = ρn/n! gives a second positional system, CNRS-H — formally the divided-power basis ρ[n] over RA, analytically ρn/n! after embedding — in which the outer digit shift (DH(Π0) = 0, Πn →Πn−1 for n ≥1) agrees with d/dρ on evaluation: differentiation becomes a structural operation on the numeral rather than an externally applied operator. The basis is lowering (divided-power, Appell), not an eigenbasis, and ρ is the independent variable. Multiplication is binomial convolution Πn⋆Πm = (n+m/n) Π(n+m). Binomial convolution gives the formal ring structure on H(RA); analytic evaluation is multiplicative for finite-support or suitably convergent sequences. The formal carrier is the Hurwitz-series ring H(RA) over the CNRS-A coefficient ring RA = Z[i][z^(−1)0 ], each coefficient carrying a canonical finite CNRS-A encoding; CNRS-H is a nested formal differential representation, not a bounded-digit covering of C. One new notational symbol is required for the Layer 1 string: 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: a fixed multiplier is finite-state, unrestricted two-argument multiplication is exact by convolution with carry normalisation, and no synchronous finite-state two-input multiplier exists. Operational completeness under addition and multiplication is proved in a companion paper. Differentiation is already realised as a primitive positional operation, by the divided-power (CNRS-H) system, in which the outer digit shift (DH(Π0) = 0, Πn →Πn−1 for n≥1) agrees with d/dρon evaluation; the associated Hurwitz-series ring H(RA) over RA = Z[i][z^(−1)0 ] is established, with canonical finite CNRS-A encoding of its coefficients. The principal remaining programme work is therefore efficient nested implementation of that structure — complexity and convergence control — rather than an existence question. Extending the Layer 1 construction to a transcendental e-base remains of independent interest for the canonical-number-system literature, but is subsidiary: its resolution would not alter the status of differentiation as a primitive.