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

Abstract

We study the perturbed Hofstadter $Q$-recursion $$Q(1)=Q(2)=1,Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2))+(-1{)}^n(n\ge 3).$$ Cloître 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 $k\ge 0$, $$\{F(s):2^k\le s<2^{k+1}\}=\{3+\hat{I}{½}_2(j):1\le j\le 2^k\}$$ 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.

20 pages, 3 figures; Substantially revised and shortened version. The previous rank-lifting and mass-closure proof has been replaced by a direct folded-degree proof based on Cloitres canonical forest structure. Two incorrect ancillary statements concerning bounded first differences and non-dyadic cumulative frequencies have been removed. The main dyadic frequency multiset theorem is unchanged