CNRS-Pr3: Arithmetic Closure for the Complex Numeric Representational System: A Finite Transducer for Addition and Complete Characterisation of Multiplication in Base−2 + i

The Complex Numeric Representational System (CNRS) encodes every complex number as a single digit string in base z0 =−2 + i with digit alphabet 0, 1, 2, 3, 4. We establish that this representational system is also an arithmetic system: addition and multiplication of digit strings are computable by finite automata. Addition. The finiteness property (F) of base−2 + i implies the carry set is bounded. Exhaustive BFS determines the exact reachable carry set: |K|= 14 elements, lying in the rectangle |Re|≤3, |Im|≤2. The addition transducer has exactly 14 states and 350 transitions; the complete transition table is given and verified on six test cases. Multiplication. A two-phase algorithm computes the product of any two digit strings: Phase 1 (Cauchy convolution) produces naive coefficients; Phase 2 (carry normalisation) reduces them to the digit alphabet via the 14-state transducer. The one-pass versus two-pass question is resolved definitively: two passes are inherently necessary for online two-argument multiplication (proved by pigeonhole on the state space) ; sequential single-pass is possible when one argument is known first. For one-argument multiplication (fixed multiplier c with J -digit encoding), a single-pass transducer exists with |Kc|·5^ (J−1) states, where Kc is the multiplier-specific carry set. The ×2 transducer (14 states, 70 transitions) is constructed explicitly and verified.