{
  "schema_version": 1,
  "problem_number": "AIM-COMBINATORICS-0093",
  "title": "B-Spline Laws and Nonreal Zeros for Weighted-Cycle Internal DLA",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give an exact finite-time distribution for single-source internal diffusion limited aggregation driven by an arbitrary positive nearest-neighbour chain on the integers. The probability of each occupied interval is a B-spline value on the harmonic-scale knots of the chain. Lifting a cycle until its penultimate occupation yields a last-site law and a divided-difference formula for arbitrary positive edge weights. For the four-cycle we characterize the full attainable probability region and every reversible realizing resistance vector up to scale. The shifted last-site polynomial has only negative real zeros for every positive nearest-neighbour chain on three or four vertices, but a five-cycle with resistances (10,10,1,1,20) has a nonreal conjugate pair. Thus five is the smallest possible cycle size for this obstruction.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "keywords": [
    "internal diffusion limited aggregation",
    "weighted cycle",
    "harmonic scale",
    "B-spline",
    "real-rootedness",
    "AIM-COMBINATORICS-0093",
    "math.PR",
    "math.CO"
  ],
  "manuscript_version_date": "2026-09-29",
  "publication_date": "2026-09-29",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-29",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-combinatorics-0093/",
  "pdf_url": "https://eulersolve.org/papers/aim-combinatorics-0093/paper.pdf?v=cd0ca02e8a0c",
  "doi": "10.5281/zenodo.23032028",
  "zenodo_record_url": "https://zenodo.org/records/23032028",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "One-dimensional nearest-neighbour branch only; homogeneous q-law is known; possible folklore, no absolute priority. Self-audited and unrefereed, with AI assistance; no independent peer review or absolute priority is claimed.",
  "files": {
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    "source.zip": {
      "sha256": "ae974d483a9865dec88adab7f3f989ae7a35ec90b3f93507e9455098d0b0c42b"
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    "verification_report.md": {
      "sha256": "968c63422ad2bf4c9931a21c3830ded9e3d15780f9f87b9ee28a8a9a07e55372"
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  },
  "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."
}