A Dyadic Frequency Law for a Perturbed Hofstadter $Q$-Recursion

Abstract

We study the perturbed Hofstadter $Q$-recursion $$ Q(1)=Q(2)=1,\qquad Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2))+(-1)^n \quad (n\ge 3). $$ Clo\^itre proved that the recursion is globally well-defined and encoded its odd- and even-indexed subsequences by exact binary arches and canonical plane forests. Since every value is odd, let $$ F(s)=\#\{n\ge 1:Q(n)=2s-1\}. $$ We prove that, for every $k\ge 0$, $$ \{F(s):2^k\le s<2^{k+1}\} = \{3+\nu_2(j):1\le j\le 2^k\} $$ as multisets. Thus the theorem determines the multiplicities of the frequencies in each dyadic block, but not their order. The proof converts frequencies into plateau local times, folds paired gap degrees under reversal-complementation, and identifies the resulting multiset with the degree multiset of a canonical tree. An ordered central-pair lemma is the boundary step that makes the dyadic cut exact.