{
  "schema_version": 1,
  "problem_number": "AIM-LOGIC-0084",
  "title": "Sharp Computability Bounds for Nonforking Sections",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We study computability of sections of the restriction map S_1(N) to S_1(M) for countable stable models M elementary in N. Types are represented by characteristic functions on formulas with named parameters, and the two elementary diagrams are separately decidable. Classical stable definability gives a continuous section computable from the input type together with the fixed oracle 0'. This bound is sharp, already for the decidable, omega-stable, omega-categorical theory of infinitely many infinite equivalence classes. For a fixed decidable-range elementary inclusion in this theory, an oracle X computes a section exactly when it computes the set of classes of N that meet M. Explicit inclusions realize every computably enumerable degree as this least auxiliary degree. Thus a computable section need not exist, although every individual type in the example is computable. In contrast, a decidable full diagram of the predicate expansion (N,M) gives a computable section for every stable theory. The result addresses a Type-2 formulation of an AIM question whose printed statement leaves the effective presentation unspecified.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.LO"
  ],
  "keywords": [
    "computable model theory",
    "stability",
    "nonforking",
    "type spaces",
    "Turing degrees",
    "equivalence relations",
    "AIM-LOGIC-0084",
    "math.LO"
  ],
  "manuscript_version_date": "2026-09-28",
  "publication_date": "2026-09-28",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-28",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-logic-0084/",
  "pdf_url": "https://eulersolve.org/papers/aim-logic-0084/paper.pdf?v=d33d8af66b40",
  "doi": "10.5281/zenodo.23009494",
  "zenodo_record_url": "https://zenodo.org/records/23009494",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The counterexample uses a specified Type-2 representation with separately decidable elementary diagrams; it does not assert a presentation-independent negative answer to the original AIM question. A decidable full predicate-pair diagram instead yields a computable section. The paper is self-audited, not independently refereed or formally verified, and the bounded literature search does not certify priority.",
  "files": {
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      "sha256": "d33d8af66b406c8bfdc76a1a707d7b8e44b669102c11b00a0c9afff305cc20e9"
    },
    "source.zip": {
      "sha256": "097f76f2f2211529c74995c65133a1a41603ccab5a11fc5c585b4451f0700567"
    },
    "verification_report.md": {
      "sha256": "bff78892081927d1ec28fe4eab14b3931031ee255e18ee79ddb395d4fee122d0"
    }
  },
  "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."
}