{
  "abstract": "For coprime $a+b=c$, the parity class in which $a$ and $b$ are odd and $c$ is even admits an exact treatment. Triples are measured by the radical excess $E_{\\varepsilon}=\\log c-(1+\\varepsilon)\\log\\operatorname{rad}(abc)$, and called transgressive at fixed $\\varepsilon>0$ when it is non-negative. The $abc$ conjecture asserts that such triples are finite in number. The excess is written exactly in terms of the defect, the mass of repeated primes the radical discards, and rearranges into a linear threshold in which the smaller summand $s=\\min\\{a,b\\}$ appears explicitly. Either $s$ stays bounded along a subsequence, or the defect must overshoot the threshold and rejoin it at an amplified exponent. For the Mersenne family $\\mathcal{V}_m = (1, 2^m-1, 2^m)$ transgression at $\\varepsilon=0$ holds precisely when $2^m-1$ fails to be squarefree, on a set of exponents of density $47/210$. The defect obeys the exact law $Δ_m = Ω_m + \\log G_m$, where $Ω_m$ collects the Wieferich primes dividing $2^m-1$ and $G_m$ is the largest divisor of $m$ whose prime factors divide $2^m-1$. This divisor is trivial on prime-power exponents and largest on $m_k = \\operatorname{lcm}(1,\\dots,k)$, where the margin $q(\\mathcal{V}_{m_k}) - 1 \\geq (1-o(1))\\log m_k / (m_k\\log 2)$ is the most any Wieferich-free mechanism can give. An $abc$ counterexample family along the Mersenne line would require infinitely many Wieferich primes with exponential order--defect growth. A final section separates two elliptic curves attached to the class: the Frey curve, whose minimal discriminant expresses the total defect with a bounded correction at $2$, and a congruent-number Jacobian, whose Szpiro quotient stays below $3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00039",
  "author_total": 1,
  "authors": [
    {
      "name": "R. Laniewski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/11xqqwgx1v06m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00039",
  "fulltext_url": null,
  "id": "11xqqwgx1v06m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00039",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7206150971",
  "pdf_url": "https://arxiv.org/pdf/2609.00039",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/11xqqwgx1v06m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.00039",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00039"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Analytic Number Theory Research",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Radical defects, Wieferich primes, and the $abc$ conjecture",
  "title_text": "Radical defects, Wieferich primes, and the abc conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/11xqqwgx1v06m",
  "venue": "arXiv",
  "versions": []
}