{
  "abstract": "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 $\\operatorname{Re}w=0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02412",
  "author_total": 1,
  "authors": [
    {
      "name": "Marco Mantovanelli"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2x60nv5p21xe2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02412",
  "fulltext_url": null,
  "id": "2x60nv5p21xe2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02412",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7207814689",
  "pdf_url": "https://arxiv.org/pdf/2609.02412",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2x60nv5p21xe2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02412",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02412"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Two-Variable Zeta Function for a Parity-Perturbed Hofstadter Q-Recursion: The Exceptional t = -1 Slice and Gaussian Boundary Layers",
  "title_text": "A Two-Variable Zeta Function for a Parity-Perturbed Hofstadter Q-Recursion: The Exceptional t = -1 Slice and Gaussian Boundary Layers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/2x60nv5p21xe2",
  "venue": "arXiv",
  "versions": []
}