{
  "abstract": "Wirsching (2003) reduces uniform positive predecessor density for the $3n+1$ map to a chain of five conditions, organized into three conjectures. We prove two of them. Conjecture 1 concerns his path-counting generators: in the convolution carrying them to his Elka functions the partition weights grow subexponentially while the binomial ratio decays geometrically, so a window of width $O(\\sqrt{\\ell})$ dominates and its radius fits inside the hypothesis. Conjecture 3 concerns the asymptotics near 0 of an invariant density $φ$, a base-3 analogue of the Fabius density, against an explicit $φ_0$ due to Berg and Krüppel. The exact log-Laplace transform of $φ$ splits into a smooth part, a log 3-periodic correction $H$, and a doubly exponentially small remainder. Berg and Krüppel represented that correction as an infinite product in 1998; their analysis did not determine whether it is constant. We give $H$ as a Fourier series with coefficients in closed form in $Γ$ and $ζ$; a classical zero-free theorem for $ζ$ shows it is not constant, and we enclose its oscillation rigorously. Wirsching's comparison class fixes one phase of $H$, and there Conjecture 3 holds with limit $e^{H(0)}$, certified to lie in $(0.53412203666478,0.53412203666479)$. Off that class the phase sweeps a full period, so the unrestricted asymptotic $φ(t)\\simκφ_0(t)$ fails. Condition $(\\star4)$ follows, at every window radius, with $μ=1/3$: what Wirsching's argument needs is weaker than Conjecture 3 itself, and the same saddlepoint chain settles it directly. With Conjecture 1 the chain reduces to the single condition $(\\star3)$. Conjecture 2 is his route to it and remains open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27617",
  "author_total": 1,
  "authors": [
    {
      "name": "Renato Augusto Tavares"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b4gw422gzmpcc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27617",
  "fulltext_url": null,
  "id": "b4gw422gzmpcc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27617",
  "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",
      "math.GM"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7204824573",
  "pdf_url": "https://arxiv.org/pdf/2608.27617",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b4gw422gzmpcc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.27617",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27617"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2610|Mathematical Dynamics and Fractals",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Wirsching's Positive-Predecessor-Density Program: Proofs of Conjectures 1 and 3",
  "title_text": "Wirsching's Positive-Predecessor-Density Program: Proofs of Conjectures 1 and 3",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/b4gw422gzmpcc",
  "venue": "arXiv",
  "versions": []
}