CNRS Pr2-2: Incorporating the Branch Index into the CNRS Digit String

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.