{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRY-0038",
  "title": "A Common Positive Point for Braid Varieties",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We compare two positive subsets of a braid variety: the positive locus of its weave cluster structure and the geometric positive locus defined by the joint nonnegative closure in a twisted product of flag varieties. For every positive braid word of finite crystallographic simple type, with a fixed standard pinning, these subsets agree on the same flag-chain variety. We construct a point at which every variable of the left-inductive seed is one. A root-subgroup stabilizer fixes its entire flag chain, allowing a simultaneous rank-one positive limit that puts the point in the geometric locus. An elementary connected-component criterion for locally acyclic cluster algebras with inverted frozen variables, together with Bao-He's geometric component theorem, proves the equality. We also distinguish affine from compact closures and give explicit small-rank examples, including a six-component real braid variety. The comparison concerns positive loci, not an identification of two cluster atlases or a classification of all real components.\n\nThis is a scoped contribution to the positivity question recorded as AIM-GEOMETRY-0038 in UnsolvedMath v1.6.0, not a claim to resolve every interpretation of the broader AIM research agenda. Existing cluster constructions and geometric component theorems retain attribution. The rank-one raw-coordinate example is credited to the dataset's accompanying research attempt. No absolute priority claim is made.\n\nAI-assisted, self-audited and unrefereed preprint. No independent review or formal proof-assistant verification is claimed. Two portable exact-arithmetic checkers accompany the source; their 300,204 checks per execution test specified finite domains and do not replace the general written proof.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AG",
    "math.RT",
    "math.CO"
  ],
  "keywords": [
    "braid varieties",
    "total positivity",
    "cluster algebras",
    "flag varieties",
    "AIM-GEOMETRY-0038"
  ],
  "manuscript_version_date": "2026-10-08",
  "publication_date": "2026-10-08",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-08",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-geometry-0038/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometry-0038/paper.pdf?v=508026ef0a64",
  "doi": "10.5281/zenodo.23246898",
  "zenodo_record_url": "https://zenodo.org/records/23246898",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every positive braid word of finite crystallographic simple type, weave cluster positivity equals Bao-He geometric positivity on the same standard-pinned flag-chain variety. The broader AIM agenda, equality of cluster atlases and classification of all real components are not resolved. AI-assisted, self-audited and unrefereed. No independent review, formal verification or absolute priority is claimed.",
  "files": {
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      "sha256": "508026ef0a64318cc3514f9d0adadb66f9b92b5117eec5e60c5ff4ced254fadb"
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    "source.zip": {
      "sha256": "6030c87540c494ab9e0d452b4b6c137b93978b1a8184b5904e9dedda62694e2e"
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    "verification_report.md": {
      "sha256": "d45552d64b5533dce19a8441378a490922e68bec593dd9d4346955ddd119eaad"
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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."
}