{
  "abstract": "This paper extends the causal modular scattering proof of the Riemann Hypothesis under Topological Vacuum Electrodynamics to the Grand Riemann Hypothesis for all Dirichlet, Hecke, and cuspidal automorphic $L$-functions. While the ground-state unknot waveguide generates the full modular surface $Y=\\operatorname{SL}(2,\\mathbb{Z})\\backslash\\mathbb{H}$, higher braided prime-knot configurations of conductor level $N$ and their internal Aharonov–Bohm phase holonomies are identified, via Ghys’ modular knot theorem and the braid projection $B_3\\to\\operatorname{SL}(2,\\mathbb{Z})$, with the congruence covering surfaces $Y_0(N)=\\Gamma_0(N)\\backslash\\mathbb{H}$ twisted by unitary characters $\\chi$ and automorphic local systems $\\mathcal{L}_\\pi$. The minor-radius cutoff $y=a$ supplies the Lax–Phillips truncation of the cusps. Imposing the Sommerfeld outgoing radiation condition at $y=a$ and applying Green’s identity on the Lax–Phillips cavity subspace $K$ proves that the generator $B$ is maximally dissipative before any scattering matrix is evaluated. Direct unfolding of the twisted coset sum over $\\Gamma_\\infty\\backslash\\Gamma_0(N)$ and the Langlands–Shahidi constant-term integral on the truncated domain yields the diagonalized characteristic scattering matrix $S_{a,\\pi}(s)=a^{2s-1}\\epsilon(s,\\pi)\\Lambda(2s-1,\\widetilde{\\pi})/\\Lambda(2s,\\pi)$. By the Lumer–Phillips theorem, $\\mathbf{S}_{a,\\pi}(s)$ is analytic and contractive ($|S_{a,\\pi}(s)|\\le1$) in the half-plane $\\operatorname{Re}s>1/2$. Substitution into the contractive bound forbids zeros of the completed automorphic $L$-function $\\Lambda(2s,\\pi)$ for $\\operatorname{Re}s>1/2$. The functional equation and the critical strip bound then force every non-trivial zero of $\\Lambda(\\rho,\\pi)$ onto the critical line $\\operatorname{Re}\\rho=1/2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Truth Vector"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e28j2txezjf98.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23459024",
  "fulltext_url": null,
  "id": "e28j2txezjf98",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23459024",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://zenodo.org/api/records/23459024/files/TVE_%20Grand%20Riemann%20Hypothesis%2020261018.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e28j2txezjf98.ris",
  "sources": [
    {
      "first_seen": "2026-11-01T12:20:55Z",
      "native_id": "23459024",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23459024"
    }
  ],
  "subjects": [],
  "title": "Grand Riemann Hypothesis Proof via Causal Scattering on Congruence Modular Coverings under Topological Vacuum Electrodynamics",
  "title_text": "Grand Riemann Hypothesis Proof via Causal Scattering on Congruence Modular Coverings under Topological Vacuum Electrodynamics",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:55Z",
  "url": "https://research.draconitic.com/paper/e28j2txezjf98",
  "venue": "Zenodo",
  "versions": []
}