CNRS-Pr4: Partial Operational Completeness of a Positional Number System for Complex Numbers

Abstract

We study operational completeness of the canonical number system (CNS) for the base z0 =−2 + i in Z[i], extended to a positional system for the complex numbers, together with a differential extension. The system comprises: Layer 1 (CNRS-A), the CNS itself with digit alphabet {0,1,2,3,4}; Layer 2, a single-valued encoding of logarithm and exponentiation via a branch-index extended element; and the Hurwitz differential extension (CNRS-H), a formal place-value system with divided-power positions ρ^[n] (in a formal variable ρ, with analytic realisation ρ^[n] →ρ^n/n! after embedding the coefficient ring in C) that realises differentiation as an exact outer-coefficient-shift operation. We prove partial operational completeness across the two-layer carrier and the Hurwitz differential extension, and resolve the metric-completion questions (Q2a–Q2b): the vβ-adic completion of RA = Z[i][z^(−1) 0 ] and Q(i) is the local field Q5, and every element of its valuation ring has a unique expansion n≥0 dn β^n with dn ∈{0,1,2,3,4}; every nonzero element of Q5 has a unique such expansion with only finitely many negative-index digits. Layer 1: the finite CNRS-A strings are closed under addition, subtraction, and multiplication; addition and fixed-multiplier multiplication are computable by explicit finite transducers, and general two-argument multiplication by an effective two-phase algorithm that is provably not realisable as a single-pass finite-state transducer in the synchronous least-significant-digit-first model. Restricted division by base powers and Gaussian units is also closed; general division by a fixed divisor has a deterministic exact digit recurrence, with eventual periodicity and a reduced-denominator termination criterion, though realizing this as an online finite-state transducer remains open. Layer 2: logarithm and exponentiation on C×are operationally closed in the extended element representation, together with principal-lift partial addition. CNRS-H: differentiation and integration are operationally closed as exact fixed outer-coefficient-shift operators. The remaining open items are: an online finite-state transducer for general fixed-divisor division; arithmetic on arbitrary infinite fractional streams; and efficient nested implementation of the combined CNRS-A/CNRS-H differential structure H(RA), whose existence, canonical coefficient encoding, and finite-truncation arithmetic are already established — a fixed transcendental-radix representation is not required for this differential closure.