{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRY-0322",
  "title": "One Compact Amoeba Intersection, Several Mixed Cells",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give explicit obstructions to assigning one binomial system to a connected compact intersection of amoebas. For f=w-1 and g=(z-10)(z-12)(z+11)(z+13)-525w, the entire amoeba intersection is a single compact interval with four distinct real torus solutions. The canonical Ronkin spines have four transverse intersections, each dual to a fully mixed cell of volume one, while the normalized mixed Monge-Ampere mass is four. Every binomial system from any single fully mixed cell for these Newton polytopes has at most two real solutions, even after coarsening. Thus the single-binomial real-preserving correspondence in AIM A.2 (UnsolvedMath v1.6.0, AIM-GEOMETRY-0322) is false. For every d>=2, a related construction gives one compact component of mass d with d unit canonical-spine cells. The volume comparison fixes the canonical Ronkin-spine convention: arbitrary coarsenings may match complex mass. We do not claim a whole-record closure of the source's unspecified volume formulation. The first Newton polytope is a segment; unions of start systems, full-dimensional-support modifications and valid patchworking/lifting theorems are excluded. Classical local mass, spine facts and known real-homotopy limitations are credited. This six-page English preprint includes 1,104 exact standard-library checks supporting the written analytic proofs. Developed with AI assistance and self-audited, it is unrefereed, without independent review, proof-assistant verification or absolute-priority certification.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.AG",
    "math.CV"
  ],
  "keywords": [
    "AIM-GEOMETRY-0322",
    "amoeba",
    "Ronkin spine",
    "mixed cell",
    "real root",
    "binomial system"
  ],
  "manuscript_version_date": "2026-10-27",
  "publication_date": "2026-10-27",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-27",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-geometry-0322/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometry-0322/paper.pdf?v=e49d1027c4fa",
  "doi": "10.5281/zenodo.23699246",
  "zenodo_record_url": "https://zenodo.org/records/23699246",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "An explicit compact amoeba intersection has four real torus solutions and canonical Ronkin-spine mixed cells of volume one. Every single binomial system from these Newton polytopes has at most two real solutions under any lattice subdivision. For each d>=2, a single compact component can have normalized mass d and d unit canonical-spine cells. The single-binomial real-preserving question is refuted for every lattice subdivision of these Newton polytopes. The volume comparison fixes canonical Ronkin spines; coarsenings can match complex mass. The first support is a segment. This is not a whole-record closure; unions of start systems, full-dimensional-support variants and valid lifting or patchworking results are excluded. 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": "e49d1027c4faac155b75667b5c7c086bdc6deb87b40199456d301c502127c32d"
    },
    "source.zip": {
      "sha256": "b464faf72e7c4201655a8295454ef965c0c21976dbd41a7a81a1725474739244"
    },
    "verification_report.md": {
      "sha256": "b056a98293fe5708747ee8d52f26cb7627f5f785f2d4ab21d4620dd8a27e05b8"
    }
  },
  "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."
}