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 $$\tilde{Q}(1)=\tilde{Q}(2)=1,\tilde{Q}(n)=\tilde{Q}(n-\tilde{Q}(n-1))+\tilde{Q}(n-\tilde{Q}(n-2))+(-1{)}^n,$$ and the associated two-variable Dirichlet series $$Z_{\tilde{Q}}(s,t)=\sum _{n\ge 1}{n}^{-s}\tilde{Q}(n{)}^{-t}.$$ The estimate $\tilde{Q}(n)=n/2+O(n/\sqrt{\log n})$ gives the exact domain of absolute convergence $\mathrm{Re}(s+t)>1$. With $w=s+t$, we separate the universal term $2^t\zeta (w)$ and derive exact transport, frequency-position, and dyadic renormalization identities. The main result concerns $t=-1$. For $E(n)=2\tilde{Q}(n)-n$ and $A(X)={\sum }_{n\le X}E(n)$, the binary-arch clock yields $$A(X)=X{\log }_{2}X+X\Omega \left({\log }_2\frac{3X}{32}\right)+O\left(\frac{X}{\sqrt{\log X}}\right),$$ where $\Omega$ is an explicit continuous periodic function. This continues the normalized correction to $\mathrm{Re}w>0$ and yields a boundary resonance lattice: a double resonance at $w=0$ and simple resonances at $2\pi im/\log 2$. 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 $n/\sqrt{\log n}$ pointwise scale with an explicit signed constant. The negative-arch mass satisfies $$A_r=\frac{512}{9\sqrt{2\pi }}\frac{16^r}{\sqrt{r}}\left(1-\frac{13}{16r}+O(r^{-2})\right).$$ We do not claim a full-slice continuation across $\mathrm{Re}w=0$.

40 pages, 3 figures. Includes an appendix with the second integrated Edgeworth coefficient. Reproducibility materials are archived at Zenodo: doi:10.5281/zenodo.22250518