The Complex Numeric Representational System (CNRS) programme represents a complex number c ∈ C as a pair consisting of a digit string (Layer 1) and a branch index k ∈ Z (Layer 2). The digit string encodes c positionally in base −2 + i; the branch index k specifies the sheet of the Riemann surface of the logarithm on which c lives, making the logarithm single-valued. The current system is therefore not a pure digit string: it is a digit string plus an integer. This paper investigates whether the branch index k can be incorporated into the digit string itself, yielding a single unified notational object with no auxiliary data. Three approaches are explored: (1) a dedicated marker position at a fixed location in the digit string; (2) an extended digit alphabet incorporating branch information; (3) a structural identification of the branchindex with specific digit patterns in the Layer 1 expansion. The primary finding is a unified two-segment CNRS representation: the branch index k can be incorporated at a designated position in a bi-infinite digit string as a marked branch segment, using a special branch marker symbol | analogous to the complex point. The k-segment uses the same digit alphabet {0, 1, 2, 3, 4} and base −2+i as the c-segment; its finite length is guaranteed by the (F) property of base −2 + i (every integer has a finite (−2 + i)-expansion). This gives a representational system requiring exactly two new symbols: the complex point . (already proposed) and the branch marker |. Uniqueness is obtained by suppressing leading zeros in the k-segment. The result is more precisely described as a tagged two-segment representation than as a full absorption of k into the ordinary digit expansion of c: the branch marker creates a compound but unified notation. A stronger form — incorporating k into the ordinary digit positions with no new symbols — is shown to face an obstruction from the discrete nature of k ∈ Z versus thefinite digit alphabet. The physical interpretation (branch index as quantum phase sheet, programme-level connection to sI in the (x, y, z, s) framework) is discussed as a programme-level interpretation, not a proved result of this paper.

The mathematical development in this paper was produced in dialogue with Claude.ai (Anthropic) in Spring 2026, directed by the author. Use of AI assistance is acknowledged in accordance with standard scholarly practice.