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 (Q2, metric)
requires p-adic specialist input and remains open. The third (Q3, operational) is proved here in the partial sense defined precisely below.

With Problem 3 now complete [9], we can state and prove partial operational completeness of the CNRS system across its three layers:

    • Q3a (Layer 1, CNRS-A): CNRS-A is closed as a ring under addition, subtraction, and multiplication. All three operations are computable by explicit finite automata. The addition transducer has exactly 14 states and 350 transitions. Restricted division (by base powers and Gaussian units) is also closed. General division remains open.

    • Q3b (Layer 2): Logarithm and exponentiation are single-valued and operationally closed in the extended element representation.

    • Q3c (Layer 3, CNRS-H): Differentiation and integration are operationally closed as exact fixed digit-shift operators.

What remains open in Problem 4: metric completeness (Q2); general division in CNRS-A; differentiation directly on CNRS-A digit strings (blocked on the e-base CNS theorem, the central open problem of the programme).