{
  "abstract": "Automating AI research creates a positive feedback loop between software capability and research effort. We study a stochastic realisation of this loop in which independent, positive, multiplicative improvements arrive at a rate proportional to a power of current capability. Within this model, randomness leaves the deterministic explosion threshold unchanged but replaces the blow-up date by an explicit distribution. For equal-sized improvements the explosion time is a geometrically weighted sum of independent exponentials; its $q$-series representation gives its transforms, moments and tail behaviour, including a two-term small-time asymptotic. For Lomax-distributed improvement sizes the explosion time is exactly Gamma distributed. Self-similarity implies that, with parameters known, the relative uncertainty about the remaining time does not vanish as explosion approaches. We formulate a fixed-sample test between constant and geometrically increasing idea-arrival rates. If the baseline rate and the alternative log-rate increment $\\ell$ are known, the minimum sample size at error level $α$ is asymptotic to $(12z_α^2)^{1/3}\\ell^{-2/3}$, where $z_α$ is the upper-$α$ standard normal quantile. Uncertainty about the growth exponent near its critical value produces a $1/t$ tail with an explicit constant, and hence an infinite conditional mean waiting time, although explosion is certain given a positive exponent. Separate extensions show that a fixed deployment delay prevents explosion in the specified delayed-feedback models, that power-law diversion of effort shifts the growth threshold, and that in a deterministic oversight race a fixed allocation share does not change the asymptotic regime, except at a critical equality of exponents. Numerical illustrations, which are not empirical forecasts, show how timing and detectability depend on the granularity of improvements.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18445",
  "author_total": 4,
  "authors": [
    {
      "name": "Xiaoyu Li"
    },
    {
      "name": "Zheng Gao"
    },
    {
      "name": "Chiwun Yang"
    },
    {
      "name": "Zhizhou Sha"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4wbp7fm05p83z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18445",
  "fulltext_url": null,
  "id": "4wbp7fm05p83z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18445",
  "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",
      "cs.AI"
    ],
    [
      "arxiv",
      "cs.CY"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "1803|Innovation Diffusion and Forecasting"
    ]
  ],
  "openalex": "W7221827468",
  "pdf_url": "https://arxiv.org/pdf/2610.18445",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4wbp7fm05p83z.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:20Z",
      "native_id": "2610.18445",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18445"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The Singularity Has a Distribution: Exact Timing Laws and Detection Limits for a Stochastic Intelligence Explosion",
  "title_text": "The Singularity Has a Distribution: Exact Timing Laws and Detection Limits for a Stochastic Intelligence Explosion",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:20Z",
  "url": "https://research.draconitic.com/paper/4wbp7fm05p83z",
  "venue": "arXiv",
  "versions": []
}