{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRY-0036",
  "title": "Balanced Laurent Potentials for Finite Cluster Actions",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give counterexamples to the uniqueness assertion for maximum potentials in Theorem 4.7 of Ishibashi's Fixed point theorem for cluster modular groups, arXiv:2603.14338v2. Both a rank-one example and a full-rank type A2 example satisfy its filling conditions and have nontrivial finite modular groups, but their maximum potentials have continua of minimizers. More generally, adjoining a sufficiently large positive integral constant can make any prescribed compact set consist of minimizers. The finite-subgroup fixed-point conclusion is nevertheless preserved: the group-averaged sum of positive Laurent functions with balanced support is coercive and strictly convex in a global logarithmic seed chart. Its unique minimizer is fixed by the group. The logarithm of this sum has the same properties. This gives a replacement proof of the credited fixed-point conclusion using balanced support alone, without the additional slope-span condition. No properness result for infinite cluster modular groups or closure of the original AIM braid-variety question is claimed. This English preprint is AI-assisted, self-audited and unrefereed. Novelty remains undetermined; no independent human review, formal verification or absolute-priority claim is made.",
  "result_type": "COMPLETE_AUXILIARY_COUNTEREXAMPLE_AND_REPAIR",
  "categories": [
    "math.CO",
    "math.GR",
    "math.GT"
  ],
  "keywords": [
    "cluster modular groups",
    "positive Laurent polynomials",
    "fixed points",
    "convex potentials",
    "counterexample",
    "Nielsen realization",
    "math.CO",
    "math.GR",
    "math.GT"
  ],
  "manuscript_version_date": "2026-10-02",
  "publication_date": "2026-10-02",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-02",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-geometry-0036/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometry-0036/paper.pdf?v=0ed8e2898250",
  "doi": "10.5281/zenodo.23110743",
  "zenodo_record_url": "https://zenodo.org/records/23110743",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Explicit rank-one and full-rank A2 filling sets refute uniqueness of the maximum-potential minimizer in Ishibashi arXiv:2603.14338v2 Theorem 4.7. Adjoining an integral positive constant can make any prescribed compact set minimize that maximum. For a finite cluster group and a finite universally positive Laurent collection with balanced full-dimensional support, its orbit-sum and the logarithm of that sum are coercive and strictly convex in one global logarithmic chart, with a unique common minimizer fixed by the group. This repairs the credited finite-subgroup fixed-point conclusion under balanced support alone; it does not resolve the general properness or centralizer questions in AIM 8.1. AI-assisted, self-audited and unrefereed. Novelty remains undetermined.",
  "files": {
    "paper.pdf": {
      "sha256": "0ed8e2898250d9979e76ca39edb2b5d4355343bbcae5f484c80fa7322fff41d2"
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    "source.zip": {
      "sha256": "4b7edd427e878fdf16e4fabe4160e0db75fc444b21046614d403cb783ff00410"
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    "verification_report.md": {
      "sha256": "64789a0498007292d72a196f451d86cc0f3340e576bdb00d838fdf9bad81f517"
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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."
}