AI 中文总结
本文研究奇偶扰动的Hofstadter Q递归及其双变量狄利克雷级数，推导相关恒等式，得到t=-1切片的渐近结果与边界共振晶格，分析负偶拱通道的高斯极限与负拱质量的显式表达式。
AI 中文摘要
我们研究奇偶扰动的Hofstadter Q递归：$\tilde{Q}(1)=\tilde{Q}(2)=1$，$\tilde{Q}(n)=\tilde{Q}(n-\tilde{Q}(n-1))+\tilde{Q}(n-\tilde{Q}(n-2))+(-1)^n$，以及相关的双变量狄利克雷级数$Z_{\tilde{Q}}(s,t)=\textstyle\bigoplus_{n\boldsymbol{\textgreater}=1}n^{-s}\tilde{Q}(n)^{-t}$。估计$\tilde{Q}(n)=n/2+O(n/\boldsymbol{\textbackslash}sqrt{\boldsymbol{\textbackslash}log n})$给出了绝对收敛的精确域$\text{Re}(s+t)\boldsymbol{\textgreater}1$。令$w=s+t$，我们分离出通用项$2^t\boldsymbol{\textbackslash}zeta(w)$，并推导了精确的传输、频率-位置和二进重整化恒等式。主要结果涉及$t=-1$。对于$E(n)=2\tilde{Q}(n)-n$和$A(X)=\textstyle\bigoplus_{n\boldsymbol{\textless}=X}E(n)$，二进制拱时钟给出$A(X)=X\boldsymbol{\textbackslash}log_2X+X\boldsymbol{\textbackslash}Omega\boldsymbol{\textleft}(\boldsymbol{\textbackslash}log_2\frac{3X}{32}\boldsymbol{\textright})+O\boldsymbol{\textleft}(\frac{X}{\boldsymbol{\textbackslash}sqrt{\boldsymbol{\textbackslash}log X}}\boldsymbol{\textright})$，其中$\boldsymbol{\textbackslash}Omega$是显式连续周期函数。这延续了对$\text{Re}w\boldsymbol{\textgreater}0$的归一化修正，并产生了边界共振晶格：在$w=0$处的双重共振和在$2\boldsymbol{\textbackslash}pi i m/\boldsymbol{\textbackslash}log2$处的简单共振。在减去全切片的$X$阶骨架后，我们分析了负偶拱通道。其伴生森林层具有弱高斯极限，且一个规范子序列以显式符号常数实现了最优的$n/\boldsymbol{\textbackslash}sqrt{\boldsymbol{\textbackslash}log n}$逐点尺度。负拱质量满足$A_r=\frac{512}{9\boldsymbol{\textbackslash}sqrt{2\boldsymbol{\textbackslash}pi}}\frac{16^r}{\boldsymbol{\textbackslash}sqrt{r}}\boldsymbol{\textleft}(1-\frac{13}{16r}+O(r^{-2})\boldsymbol{\textright})$。我们不主张在$\text{Re}w=0$上的全切片延拓。
英文摘要
We study the parity-perturbed Hofstadter $Q$-recursion $$ \widetilde Q(1)=\widetilde Q(2)=1,\qquad \widetilde Q(n)=\widetilde Q(n-\widetilde Q(n-1)) +\widetilde Q(n-\widetilde Q(n-2))+(-1)^n, $$ and the associated two-variable Dirichlet series $$ Z_{\widetilde Q}(s,t)=\sum_{n\ge1}n^{-s}\widetilde Q(n)^{-t}. $$ The estimate $\widetilde Q(n)=n/2+O(n/\sqrt{\log n})$ gives the exact domain of absolute convergence $\operatorname{Re}(s+t)>1$. 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 $E(n)=2\widetilde Q(n)-n$ and $A(X)=\sum_{n\le X}E(n)$, the binary-arch clock yields $$ A(X)=X\log_2X+XΩ\!\left(\log_2\frac{3X}{32}\right) +O\!\left(\frac{X}{\sqrt{\log X}}\right), $$ where $Ω$ is an explicit continuous periodic function. This continues the normalized correction to $\operatorname{Re}w>0$ 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 $n/\sqrt{\log n}$ pointwise scale with an explicit signed constant. The negative-arch mass satisfies $$ A_r=\frac{512}{9\sqrt{2π}}\frac{16^r}{\sqrt r} \left(1-\frac{13}{16r}+O(r^{-2})\right). $$ We do not claim a full-slice continuation across $\operatorname{Re}w=0$.
Comments40 pages, 3 figures. Includes an appendix with the second integrated Edgeworth coefficient. Reproducibility materials are archived at Zenodo: doi:10.5281/zenodo.22250518
