CNRS-MN: The Complex Number Representation System (CNRS): Foundations, Arithmetic, and Calculus

This paper constructs a positional numeric system in which complex numbers are assigned a single self-contained state value. Standard complex notation represents numbers either as coordinate pairs a+ bi, re^(iθ) or as expressions that are multi-valued without external branch specication, such as log z. In neither case does a single self-contained state determine a unique complex value without external context. The Complex Number Representation System (CNRS), built on the base z0 =−2 + i with digit alphabet {0,1,2,3,4}, provides such a state. The system is organised into three modules. The arithmetic module (CNRS-A) provides exact positional representation of the Gaussian integers Z[i] and their ring operations over nite canonical digit strings. Addition is realised by a 14-state nite-state transducer; multiplication by a two-pass Cauchy-convolution-plus-normalisation algorithm; negation natively via the canonical representation 144 of −1. Division is classied into three cases exact Gaussian-integer quotients, terminating shifted expansions, and eventually-periodic expansions determined by the factorisation of the reduced denominator in Z[i]. A bounded-input 14-state normalisation theorem is established for addition-type raw digit streams; arbitrary nite Gaussian-integer strings are canonically normalised by the general KátaiSzabó greedy procedure. A Multiplication Closure Theorem (ring isomorphism CNRS-A∼= Z[i]) and a Division Theorem are established. The branch-state module proves that a digit string alone cannot determine a single-valued logarithm on C^×, and that an explicit integer branch index is sucient for local branch-labelled logarithm evaluation (Branch-State Necessity Theorem). Full branch semantics under composition remains an open problem. The CNRS-H module realises a coecient calculus in which dierentiation is an exact EGF coecient drop requiring no arithmetic. EGF composition is native to CNRS-A via the Faà di Bruno formula and integrality of partial Bell polynomials; the chain rule holds at the digit-string level. A Lagrange Inversion Theorem establishes that compositional inverses are computable by a native CNRS-A recurrence, with all coecients in Z[i], provided f′(0) is a Gaussian unit. A Degree-Growth Theorem shows that no bounded-degree transducer can realise EGF composition for non-polynomial outer functions. The three modules are integrated into a single CNRS* state (a,k,h) a canonical digit string, an integer branch index, and an EGF coecient stream which constitutes a self-contained complex numerical state object: exact arithmetic, internalised branch state, and local analytic structure carried in a single structured triple. The paper concludes with six open problems and a CNRS* State Preservation Theorem.