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 branch index 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. Version 3 layout: the k-segment is sandwiched between the branch marker and the complex point, . . . d2d1d0 | km. . . k1k0. d−1d−2. . ., giving k a bounded slot between two fixed anchors regardless of how large the integer part of c or k itself grow. (Version 2 placed k and the integer part of c on the same, unbounded side of the marker, creating a real parsing ambiguity with no fixed anchor for locating the boundary between them; this is corrected here.) 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 the finite digit alphabet.

Two marker symbols. The single bar | is used when the k-segment is itself expanded in base−2 + i (the long-term target, and the form analysed throughout this paper). A second, explicitly transitional marker, the broken bar ∥, is reserved for the case where k is given in ordinary decimal notation
rather than expanded in base−2 + i — useful during any period in which base-(−2 + i) expansion of k is not yet implemented in practice (see the discussion of cnrs layer2.py in Section 7). The broken-bar form is not a third representational approach: it is the same unified string with the k-segment left unexpanded, and is expected to be retired once base-(−2 + i) expansion of k is implemented throughout.

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.