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

Abstract

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 resolved: exact arithmetic algorithms and the finite-state boundary are both characterised precisely, addition inside it and unrestricted multiplication outside it. CNRS-A is the Layer 1 arithmetic carrier of the CNRS architecture [6, 7]: a positional single-valued representation of every Gaussian integer by a finite string, extending to every complex number via an infinite fractional tail (Layer 1 itself, not Layer 2). Layer 2 separately supplies a branch-index lift making multivalued analytic operations (logarithm, exponentiation, powers) single-valued. Together with the Hurwitz differential extension H(RA) that carries CNRS-H’s divided-power calculus structure [8], this is the full CNRS architecture. This paper concerns Layer 1 arithmetic only; it has no dependence on the Layer 3/ CNRS-H material. The results of this paper are: (1) Base−2+i with digit alphabet {0, 1, 2, 3, 4} has the finiteness property (F), directly by the K´atai–Szab´o canonical-number-system theorem for Gaussian integers. (2) The exact reachable addition carry set has |K|= 14 elements (determined by exhaustive breadth-first search, independently re-verified); the addition transducer has exactly 14 states and 350 transitions. (3) Multiplication of finite-support CNRS-A strings representing Gaussian integers follows the Cauchy convolution law (X·Y )k = n+m=kdnem; the value map is a ring homomorphism on Z[i], extending to a homomorphism on all of RA = Z[i][z^(−1)0 ] via the shift argument given in the Problem 2 capstone [6]. (4) Convolution and carry-normalisation are two logically distinct computational stages; together they give an exact algorithm for multiplication, with termination proved directly from the (F) property applied to the outstanding carry (a Gaussian integer). (5) One-argument multiplication (fixed J -digit multiplier c) is computable by a single-pass finite transducer with exactly |Qc|reachable states in the specified construction, where Qc ⊆Kc ×{0, . . . , 4}^(J−1) and Kc is the multiplier-specific carry set, so |Qc|≤|Kc|·5^(J−1). For c = 2 (the minimal non-trivial case): |K2|= 14, exactly, giving 14 states and 70 transitions. (6) No single-pass finite-state transducer computes two-argument online multiplication (proved by pigeonhole on the state space); the convolution-plus- normalisation algorithm is a sufficient effective procedure, not itself finite-state, and sequential single-pass is possible when one argument is known first.