{
  "schema_version": 1,
  "problem_number": "AMR-022-5057",
  "title": "Finite-Area Stolz Images at Almost Every Plessner Point",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We extend Oleg Ivrii’s Gaussian lacunary construction by slowing its variance clock. If deterministic positive variances satisfy v_n <= C/n, sum v_n = infinity, and sum v_n/log(n+1) < infinity, then almost surely the holomorphic series sum sqrt(v_n) xi_n z^(2^n) has almost every boundary point Plessner, while every Stolz image there has finite planar Lebesgue measure and the areas of truncated images tend to zero. Area is taken without multiplicity. The hypotheses hold for v_n = 1/[n(log(n+1))^beta] for every fixed 0 < beta <= 1. The proof adapts Ivrii’s dyadic shadowing and Brownian embedding, adding a summable covering by unit-time Wiener sausages. Ivrii’s September 2026 preprint already answered the Collingwood and Baernstein questions (Hayman-Lingham 5.20 and 5.57) negatively; this note is a finite-area strengthening, not a first solution of those questions. The five-page English manuscript contains the complete proof. AI-assisted and self-audited; no independent peer review, formal verification, or exhaustive priority certification is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CV",
    "math.PR"
  ],
  "keywords": [
    "AMR-022-5057",
    "Plessner point",
    "Stolz angle",
    "Gaussian lacunary series",
    "Brownian motion",
    "Wiener sausage",
    "finite area"
  ],
  "manuscript_version_date": "2026-10-26",
  "publication_date": "2026-10-26",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-26",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-022-5057/",
  "pdf_url": "https://eulersolve.org/papers/amr-022-5057/paper.pdf?v=28d23a95c526",
  "doi": "10.5281/zenodo.23659387",
  "zenodo_record_url": "https://zenodo.org/records/23659387",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For deterministic positive variances v_n<=C/n with divergent sum and sum v_n/log(n+1)<infinity, the Gaussian lacunary series sum sqrt(v_n) xi_n z^(2^n) almost surely has almost every boundary point Plessner, every Stolz image there has finite set area, and truncated image areas tend to zero. One full-measure set works for every aperture. The explicit logarithmic family has 0<beta<=1. The original negative resolution of Hayman-Lingham 5.57 is due to Oleg Ivrii (arXiv:2609.18785v1, September 2026). This manuscript extends his method to finite total image area and vanishing truncated area. Area is without multiplicity; this is not a new first source closure. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "28d23a95c526fd0bdd92b8f5972a8105ca4c3c4b9aa3a4cbd1576f53c374f9ae"
    },
    "source.zip": {
      "sha256": "2c8a0a15c9770d9ad521bc98f4526e184b967e5ee0cd536dbb2efa20701caf37"
    },
    "verification_report.md": {
      "sha256": "a621d66ed1acaa5e0bbee20c74c8d7b2351c13369d29f704652422b64240f37f"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}