{
  "abstract": "A population of $N$ identical pulse-coupled oscillators ultimately settles into one of two outcomes: full synchrony or a state of co-existing synchronized clusters. We show that which outcome occurs is controlled by the prime factorization of $N$. At the critical charging curve --- linear, the boundary between the synchronizing and clustering regimes --- the synchronization basin acquires exact arithmetic structure. The synchronization probability is $\\Psync=A_{N,1}/N^N$ for all $N$, where $A_{N,1}$ satisfies an exact recurrence relation. For prime $N$, $A_{N,1}=N^N-1$ giving the closed form $\\Psync=1-1/N^N$; for composite $N$, the observed asymptotic scaling is $1-\\Psync\\sim C_m N^{-(m-1)}$, where $m$ is the smallest prime divisor. The result adds a new member to the atlas of exotic basin geometries: alongside fractal, riddled, and tentacled basins, we now have a basin that is arithmetic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01668",
  "author_total": 1,
  "authors": [
    {
      "name": "K. P. O'Keeffe"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/97zt0qgx4ckp8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01668",
  "fulltext_url": null,
  "id": "97zt0qgx4ckp8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01668",
  "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",
      "nlin.AO"
    ],
    [
      "arxiv",
      "nlin.CD"
    ],
    [
      "openalex",
      "1705|Nonlinear Dynamics and Pattern Formation"
    ]
  ],
  "openalex": "W7207873262",
  "pdf_url": "https://arxiv.org/pdf/2609.01668",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/97zt0qgx4ckp8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:48Z",
      "native_id": "2609.01668",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01668"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:nlin.AO",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Arithmetic of the sync basin for pulse-coupled oscillators",
  "title_text": "Arithmetic of the sync basin for pulse-coupled oscillators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:48Z",
  "url": "https://research.draconitic.com/paper/97zt0qgx4ckp8",
  "venue": "arXiv",
  "versions": []
}