An Odd-Arch Family of Perturbed Hofstadter Q-Recursions: Gaussian Profiles and Exact Frequency Laws

Abstract

For every integer $p\ge2$, we construct a globally well-defined positive odd sequence $Q_p$ satisfying, after an explicit finite initial block, $Q_p(n)=Q_p(n-Q_p(n-1))+Q_p(n-Q_p(n-2))-(-1)^n$. Its binary block words admit the exact factorization $U_{p,k}=(A_k\overline{A}_{k+1})^{p-1}A_{k+2}$, producing exactly $2p-1$ alternating signed arches per natural block. The elementary word heights converge uniformly to the Gaussian isoperimetric profile $φ\circΦ^{-1}$. For each fixed $p$, this yields an explicit dyadic limiting profile with $2p-1$ sign transitions and a sharp relative-error constant $Λ_p=b_p/2$, where $b_p$ is determined by $b_p((3p-1)/2+Φ(-b_p))=φ(b_p)$. We determine all zeros and parity crossings in sufficiently large dyadic blocks. For $p\ge3$, there are exactly $2p-1$ transverse even-index zeros, whereas the three-arch case $p=2$ exhibits an alternating four/six-zero pattern. An ordered frequency factorization gives exact dyadic histograms, moments, total-variation distances, and extremal multiplicities. The limiting frequency generating function $z^3/(2-z)$ is universal, although the highest finite multiplicities depend on $p$. Finally, the summatory function has main terms $N^2/4+(N/2)\log_2N$, an explicit parameter-dependent log-periodic term of order $N$, and remainder $O_p(N/\sqrt{\log N})$.