CNRS-Pr1: Toward a Definition of Base−b for All b∈R+: Extending the Ito–Sadahiro (−β)-Expansion Framework

We work toward a rigorous definition of base −b for all b ∈ R+, extending the (−β)-expansion framework of Ito and Sadahiro (2009), which is restricted to β ∈ (1, 2]. This problem — designated Problem 1 of the Complex Numeric Representational System programme [8] — is the necessary first step toward a numeric system in which complex numbers are represented as single values, via the mechanism that fractional powers of negative bases naturally introduce complex-valued structures, suggesting a possible route toward unified positional representations. We identify the precise location of the β ≤ 2 restriction in the Ito–Sadahiro construction: it lives in the digit alphabet, not in the interval width. We propose the natural generalisation: for β ∈ (n, n + 1] with n = ⌊β⌋, the digit alphabet is Σ(β) = {0, 1, . . . , n} and the fundamental interval is Iβ = [−n/(β + 1), 1/(β + 1)), with width (n + 1)/(β + 1). This width equals 1 at integer β and dips below 1 between integers, creating a coverage problem absent from the Ito–Sadahiro domain. We formulate three open sub-questionsconstituting the full statement of Problem 1. We then carry out complete orbit computations for two Pisot numbers in (2, 3]: β = 1 + √2 (Theorem 7.2) and β = 1 + √3 (Theorem 8.1). In both cases the endpoint orbits are eventually periodic and an explicit, decidable admissibility condition is given. A systematic computation across six values of β reveals a sharp pattern: every Pisot β tested produces a finite-period orbit and a clean admissibility condition; the non-Pisot case β = 2.5 does not. We state a strengthened conjecture (Conjecture 9.2) that eventual periodicity is equivalent to the Pisot property, with period bounded by a computable function of the minimal polynomial.