CNRS-Pr4: Partial Operational Completeness of the Complex Numeric Representational System Working Paper — Problem 4 (Sub-question Q3) of the CNRS Programme

Problem 4 of the Complex Numeric Representational System (CNRS) programme asks whether the system is complete in three distinct senses: coverage of C, metric completeness in the (−2 + i)-adic topology, and operational closure under all intended arithmetic operations. The first sense (coverage) is essentially resolved by the Thurston tiling theorem. The second (metric) requires p-adic specialist input and remains open. The third — operational completeness — was partially blocked on the completion of Problem 3. With Problem 3 now substantially complete [7], we can state and prove the operational completeness of CNRS-A (the arithmetic layer, base−2 + i, digit alphabet {0, 1, 2, 3, 4}) under addition and multiplication. The addition transducer is a finite automaton with exactly 14 states and 350 transitions, computing any sum within the system. Multiplication is characterised in three tiers: one-argument multiplication is a single-pass transducer; two-argument sequential multiplication is a single-pass transducer when one operand is known first; and two-argument online multiplication requires two passes, provably and necessarily. Division by base powers and units reduces to these operations. The CNRS-H calculus layer is separately operationally closed under differentiation and integration, both realised as single digit-shift operations. Logarithm and exponentiation are single-valued in Layer 2. We state precisely what remains open in Problem 4: metric completeness(requiring p-adic analysis) and the closure of CNRS under general division and under differentiation as an arithmetic operation on CNRS-A strings (blocked on the e-base CNS theorem, the central open problem of the programme).