An Odd-Arch Family of Perturbed Hofstadter Q-Recursions: Gaussian Profiles and Exact Frequency Laws
Abstract
For every integer p≥2, we construct a globally well-defined positive odd sequence Qp satisfying, after an explicit finite initial block, Qp(n)=Qp(n−Qp(n−1))+Qp(n−Qp(n−2))−(−1)n. Its binary block words admit the exact factorization Up,k=(AkA̅k+1)p−1Ak+2, producing exactly 2p−1 alternating signed arches per natural block. The elementary word heights converge uniformly to the Gaussian isoperimetric profile φ∘Φ−1. For each fixed p, this yields an explicit dyadic limiting profile with 2p−1 sign transitions and a sharp relative-error constant Λp=bp/2, where bp is determined by bp((3p−1)/2+Φ(−bp))=φ(bp). We determine all zeros and parity crossings in sufficiently large dyadic blocks. For p≥3, 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 z3/(2−z) is universal, although the highest finite multiplicities depend on p. Finally, the summatory function has main terms N2/4+(N/2)log2N, an explicit parameter-dependent log-periodic term of order N, and remainder Op(N/√(log N)).
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BibTeX
@misc{mantovanelli2026oddarch,
title = {{An Odd-Arch Family of Perturbed Hofstadter Q-Recursions: Gaussian Profiles and Exact Frequency Laws}},
author = {Marco Mantovanelli},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01453},
eprint = {2611.01453},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01453},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Odd-Arch Family of Perturbed Hofstadter Q-Recursions: Gaussian Profiles and Exact Frequency Laws
AU - Marco Mantovanelli
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01453
UR - https://arxiv.org/abs/2611.01453
AB - For every integer p≥2, 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_kA̅_k+1)^p−1A_k+2, producing exactly 2p−1 alternating signed arches per natural block. The elementary word heights converge uniformly to the Gaussian isoperimetric profile φ∘Φ^−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≥3, 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/√(log N)).
ER -