CNRS-Pr3-1: Arithmetic Closure for the Complex Numeric Representational System: Working Paper — Problem 3 of the CNRS Programme

Problem 3 of the Complex Numeric Representational System (CNRS) programme asks whether addition and multiplication of CNRS digit strings — that is, digit strings in base −2 + i with digit alphabet {0, 1, 2, 3, 4} — are computable by finite automata. This is the difference between having a representational system and having an arithmetic system. Problem 3 is now complete. CNRS-A is the Layer 1 arithmetic system of the three-layer CNRS architecture [8]: single-valued complex number representations (Layer 1), extended element encoding for multivalued operations (Layer 2), and the hybrid progressive place-value system CNRS-H for differentiation-as-digit-shift (Layer 3). The complete results of this paper are: (1) Base −2 + i with digit alphabet {0, 1, 2, 3, 4} has the finiteness property (F) (Akiyama–Rao–Steiner criterion, verified explicitly). (2) The exact reachable addition carry set has |K| = 14 elements (determined by exhaustive breadth-first search); the addition transducer has exactly 14 states and 350 transitions. (3) Multiplication followsthe Cauchy convolution law (X · Y )k = Pn+m=k dnem; the value map is a ring homomorphism. (4) A two-phase normalisation algorithm (convolution followed by carry-reduction) brings all coefficients back into the digit alphabet; termination is guaranteed by the (F) property. (5) One-argument multiplication (fixed J-digit multiplier c) is computable by a single-pass finite transducer with |Kc| · 5J−1 states, where Kc is the multiplier-specific carry set. For c = 2 (the minimal non-trivial case): |K2| = 14, giving 14 states and 70 transitions. (6) Two-argument online multiplication inherently requires two passes (proved by pigeonhole on the state space); sequential single-pass is possible when one argument is known first.