A Fibonacci-Automatic Set Whose Enumerator Image Is Not Context-Free

Abstract

Let S be the Golden Exclusion set, let T be its complement, let B(n)=∣T∩[1,n]∣, and let q(k) be the increasing enumeration of T. This paper studies R2=q(T), the first nontrivial self-enumerator image in the associated Kimberling dispersion. On the regular landmark family Z(pa)=(100)a, an exact rank formula leads to Fibonacci reset points Aj=(F3j+4+3)/2 and the canonical representation Z(B(pa))=(100)a−j−3101P3j+4(h). The representation collapses each reset block to one state of the parent Zeckendorf automaton and yields an exact block law: the j-th block has length F3j+5, with exactly F3j+3−1 landmarks in R2. Along the exact block endpoints, the induced unary characteristic sequence has limiting proportion 1/φ2. By standard closure properties of context-free languages and Parikh’s theorem, the canonical Zeckendorf languages of R2 and L1=T∖R2 are therefore not context-free. Secondary consequences show that the parent rank B is Fibonacci-regular but not Fibonacci-synchronized, the enumerator q is not Fibonacci-synchronized, and B∘B is not Fibonacci-regular. The paper explicitly places the rank iteration in Kimberling’s dispersion framework and distinguishes the result from nearby golden-ratio sieve constructions.