{
  "schema_version": 1,
  "problem_number": "AIM-PROBABILITY-0093",
  "title": "Uniform Height Bounds for Brownian Excursions with an Exact Area Constraint",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a standard Brownian excursion of duration T and exact area rho T, with fixed rho>0, we prove an exponential maximum tail with a T^(3/2) prefactor and logarithmic bounds for every positive maximum moment. A positivity-preserving Gaussian bridge shift compares increasing functionals under exact-area conditioning with an exponential area tilt; scalar Airy-area asymptotics and kernel and reflection estimates complete the proof. For the literal duration-2N, area-N normalization in AIM item 2.18, the entire curve divided by N^(1/3) converges uniformly to zero. Thus an N^(2/3) time window cannot yield a nondegenerate random limit at that height scale. No changed-area, moving-wall, multiline or constant-scale process-limit theorem is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.PR"
  ],
  "keywords": [
    "Brownian excursion",
    "exact area",
    "maximum height",
    "Airy function",
    "Gaussian bridge",
    "AIM-PROBABILITY-0093"
  ],
  "manuscript_version_date": "2026-10-05",
  "publication_date": "2026-10-05",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-05",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-probability-0093/",
  "pdf_url": "https://eulersolve.org/papers/aim-probability-0093/paper.pdf?v=ee7a542b7f59",
  "doi": "10.5281/zenodo.23148017",
  "zenodo_record_url": "https://zenodo.org/records/23148017",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete negative theorem only for the literal single-excursion normalization in AIM KPZ item 2.18 (dataset AIM-PROBABILITY-0093). This does not resolve corrected-area, moving-wall or multiline questions, a sharp maximum scale, or a constant-scale Ferrari-Spohn process limit. Known one-point asymptotics are credited. Novelty and absolute priority remain undetermined. AI-assisted, self-audited and unrefereed; no independent peer review or formal verification is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "ee7a542b7f5985159c6dd554b68475a98b6573317c4d0286dc568f511aa67900"
    },
    "source.zip": {
      "sha256": "c52e108c0832101860bd86fd208faef5b6f9830e5129feedd08232b4ed2d800f"
    },
    "verification_report.md": {
      "sha256": "27c31629267b0b190eca299c79bd189ecdab6c0853ca3b8f8bbedd0c2ead09b2"
    }
  },
  "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."
}