{
  "abstract": "Erdős problem 364 asks whether three consecutive powerful numbers exist, where $n$ is powerful if $p \\mid n$ implies $p^2 \\mid n$. Erdős (1976) and, independently, Mollin and Walsh (1986) conjectured that none do; the $abc$ conjecture implies at most finitely many. The conjecture remains open. We present the first verification of the conjecture at any finite bound that is checked end to end by a proof kernel: machine-checked theorems in Lean 4 establishing that no triple of consecutive powerful numbers exists below $10^{12}$ and below $10^{14}$, with the axiom footprint of both theorems being exactly {propext, Classical.choice, Quot.sound} -- no sorry, no native_decide, no trusted external computation. The statements are phrased in the byte-identical vocabulary of the google-deepmind/formal-conjectures formalization of the problem, and we prove abstractly that the open conjecture implies each bounded form, pinning the statement correspondence. The proof reduces the search to odd numbers via a mod-4 argument, represents every odd powerful number as $a^2 b^3$ with $a, b$ odd and $b$ squarefree, enumerates all odd powerful numbers by a fueled, kernel-reducible generator whose completeness is proved once and instantiated across 3,524 per-interval Boolean certificates, and eliminates the seven surviving distance-2 pairs (the members of OEIS A076445 below $10^{14}$) by explicit non-powerfulness witnesses. Every certificate's expected values are computed independently by a Python engine, so each kernel-checked equality doubles as a cross-implementation agreement. Total certified kernel time is roughly 46 CPU-hours. Larger uncertified computations exist (exhaustive to $10^{22}$; conditionally to about $7.38 \\times 10^{28}$); our contribution is not a computational record but the elimination of trusted enumeration code from the evidence chain.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25011",
  "author_total": 2,
  "authors": [
    {
      "name": "Ibrahim Mian"
    },
    {
      "name": "Shayaan Siddique"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/faqnxncngp30a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25011",
  "fulltext_url": null,
  "id": "faqnxncngp30a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25011",
  "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.LO"
    ],
    [
      "arxiv",
      "math.GM"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7214085098",
  "pdf_url": "https://arxiv.org/pdf/2609.25011",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/faqnxncngp30a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25011",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25011"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Analytic Number Theory Research",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A Kernel-Certified Verification of the Erdős-Mollin-Walsh Conjecture below $10^{14}$",
  "title_text": "A Kernel-Certified Verification of the Erdős-Mollin-Walsh Conjecture below 10^14",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/faqnxncngp30a",
  "venue": "arXiv",
  "versions": []
}