Dyadic Frequency Laws, Clock Dynamics, and Defect Scaling in a Perturbed Hofstadter $Q$-Recursion

Abstract

We study the perturbed Hofstadter $Q$-recursion \[ Q(1)=Q(2)=1,\qquad Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2))+(-1)^n \quad (n\ge3). \] We investigate its value frequencies and dyadic fluctuation structure. Our first main result is an explicit dyadic frequency law: if $F(s)$ denotes the number of occurrences of the value $2s-1$, then for every $k\ge0$, \[ \{F(s):2^k\le s<2^{k+1}\} = \{3+ν_2(j):1\le j\le2^k\} \] as multisets. The proof uses Cloître&#39;s binary interleaving structure, dyadic hitting-time identities, and an induced rank-lifting mechanism for plateau zero-runs. We also study deviations from exact dyadic scaling through the renormalized defect $R(n)=Q(2n)-2Q(n)$. Introducing the auxiliary clock process $t_1(n)=n-Q(n-1)$, we prove the exact identity \[ R(n)=2t_1(n+1)-t_1(2n+1)-1, \] which expresses the dyadic defects entirely in terms of a single delayed clock dynamics. Numerical computations further indicate coherent fluctuation profiles across dyadic scales and approximate logarithmic self-similarity on the $\log_2 n$-scale. Together with Cloître&#39;s asymptotic estimate $Q(n)=n/2+O(n/\sqrt{\log n})$, these results suggest a nontrivial recursive dyadic scaling structure in the perturbed recursion.