{
  "schema_version": 1,
  "problem_number": "AIM-SEVERAL_COMPLEX_VARIABLES-0020",
  "title": "Infinite Boundary-Jet Ambiguity for Proper Embeddings into a Smooth Strongly Convex Domain",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For every n >= 2, we construct one bounded domain D in C^(n+1) with smooth strongly convex boundary and a circle of distinct proper holomorphic embeddings of B^n into D. All embeddings extend smoothly to the closed ball and have the same infinite jet at one fixed boundary point. The domain can be chosen arbitrarily close to the unit ball in any prescribed finite smoothness norm on its defining function. A holomorphic function flat at the boundary point supplies the graphs, and a compactly supported perturbation of the quadratic defining function places all graphs in the same target. For comparison, an elementary orthogonality argument shows that a proper rational ball map of degree at most d is determined by its d-jet at the interior origin, even among all holomorphic maps to the target ball. The boundary example concerns a prescribed boundary point; it does not resolve the general optimal dimension-dependent interior jet order or close the full two-part AIM question. This is an unrefereed preprint; no absolute priority claim is made.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CV"
  ],
  "keywords": [
    "proper holomorphic maps",
    "finite jet determination",
    "strongly convex domain",
    "boundary flatness",
    "rational ball maps",
    "AIM-SEVERAL_COMPLEX_VARIABLES-0020"
  ],
  "manuscript_version_date": "2026-10-18",
  "publication_date": "2026-10-18",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-18",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-several-complex-variables-0020/",
  "pdf_url": "https://eulersolve.org/papers/aim-several-complex-variables-0020/paper.pdf?v=fd9f4c65d377",
  "doi": "10.5281/zenodo.23443122",
  "zenodo_record_url": "https://zenodo.org/records/23443122",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete counterexample at a prescribed boundary point in one fixed smooth strongly convex target, plus an interior degree-jet theorem. General optimal interior orders and the full two-part AIM source remain unresolved (PARTIALLY_SOLVED). AI-assisted, self-audited and unrefereed; no independent review, formal certification or absolute-priority claim.",
  "files": {
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      "sha256": "fd9f4c65d3775c034349df2b2fa0c005ad30bc8412beb687f9c90bba0dfd3ead"
    },
    "source.zip": {
      "sha256": "79c152c25b88842654447af9a383d7754f00b4ad4e50eb3a467e5a8372091b2e"
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    "verification_report.md": {
      "sha256": "238c510a4b61e6bae46feda3fe5b542ef9b6b9cabb39dd0a20e271d95890a4c1"
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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."
}