A Two-Variable Zeta Function for a Parity-Perturbed Hofstadter Q-Recursion: The Exceptional t = -1 Slice and Gaussian Boundary Layers
Abstract
We study the parity-perturbed Hofstadter Q-recursion Q̃(1)=Q̃(2)=1, Q̃(n)=Q̃(n−Q̃(n−1)) +Q̃(n−Q̃(n−2))+(−1)n, and the associated two-variable Dirichlet series ZQ̃(s,t)=∑n≥1n−sQ̃(n)−t. The estimate Q̃(n)=n/2+O(n/√(log n)) gives the exact domain of absolute convergence Re(s+t)>1. With w=s+t, we separate the universal term 2tζ(w) and derive exact transport, frequency-position, and dyadic renormalization identities. The main result concerns t=−1. For E(n)=2Q̃(n)−n and A(X)=∑n≤ XE(n), the binary-arch clock yields A(X)=Xlog2X+XΩ (log23X/32) +O (X/(√(log X))), where Ω is an explicit continuous periodic function. This continues the normalized correction to Rew>0 and yields a boundary resonance lattice: a double resonance at w=0 and simple resonances at 2πi m/log2. After subtracting the full-slice order-X skeleton, we analyze the negative-even arch channel. Its companion-forest layers have a weak Gaussian limit, and a canonical subsequence realizes the optimal n/√(log n) pointwise scale with an explicit signed constant. The negative-arch mass satisfies Ar=512/9√(2π)16r/√r (1−13/16r+O(r−2)). We do not claim a full-slice continuation across Rew=0.
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BibTeX
@misc{mantovanelli2026twovariable,
title = {{A Two-Variable Zeta Function for a Parity-Perturbed Hofstadter Q-Recursion: The Exceptional t = -1 Slice and Gaussian Boundary Layers}},
author = {Marco Mantovanelli},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02412},
eprint = {2609.02412},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02412},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Two-Variable Zeta Function for a Parity-Perturbed Hofstadter Q-Recursion: The Exceptional t = -1 Slice and Gaussian Boundary Layers
AU - Marco Mantovanelli
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02412
UR - https://arxiv.org/abs/2609.02412
AB - We study the parity-perturbed Hofstadter Q-recursion Q̃(1)=Q̃(2)=1, Q̃(n)=Q̃(n−Q̃(n−1)) +Q̃(n−Q̃(n−2))+(−1)^n, and the associated two-variable Dirichlet series Z_Q̃(s,t)=∑_n≥1n^−sQ̃(n)^−t. The estimate Q̃(n)=n/2+O(n/√(log n)) gives the exact domain of absolute convergence Re(s+t)>1. With w=s+t, we separate the universal term 2^tζ(w) and derive exact transport, frequency-position, and dyadic renormalization identities. The main result concerns t=−1. For E(n)=2Q̃(n)−n and A(X)=∑_n≤ XE(n), the binary-arch clock yields A(X)=Xlog_2X+XΩ (log_23X/32) +O (X/(√(log X))), where Ω is an explicit continuous periodic function. This continues the normalized correction to Rew>0 and yields a boundary resonance lattice: a double resonance at w=0 and simple resonances at 2πi m/log2. After subtracting the full-slice order-X skeleton, we analyze the negative-even arch channel. Its companion-forest layers have a weak Gaussian limit, and a canonical subsequence realizes the optimal n/√(log n) pointwise scale with an explicit signed constant. The negative-arch mass satisfies A_r=512/9√(2π)16^r/√r (1−13/16r+O(r^−2)). We do not claim a full-slice continuation across Rew=0.
ER -