{
  "target_id": "cor:knots",
  "proof_status": "needs_context",
  "summary": "Conditional on thm:main and the finite-volume lattice prerequisite, infinitely many distinct totally geodesic surfaces imply arithmeticity in dimension 3. Identifying the knot as the figure-eight knot requires an additional arithmetic-knot classification result. The converse also requires a figure-eight-specific existence argument. Neither input is supplied. The citations following the corollary introduce a subsequent statement and are not established proof dependencies of this target. This is a bounded assessment, not a verdict that the corollary is false.",
  "used_dependencies": [
    "thm:main"
  ],
  "missing_context": [
    "A proof anchor or preceding argument for cor:knots.",
    "The finite-volume prerequisite for the complete hyperbolic knot complement and its lattice holonomy.",
    "The exact arithmetic-knot classification theorem, including all hypotheses and the passage from complement identification to knot identification.",
    "A proof or precise external theorem giving infinitely many distinct properly immersed, topologically closed totally geodesic surfaces in the figure-eight complement.",
    "For the missing external inputs: available_from=unknown; date_evidence=none supplied; date_basis=unresolved; version=unknown. Supply dated versions containing the required statements and verify availability by 2019-06-20. Bibliographic entries alone do not verify theorem content or the relevant preprint version.",
    "No external result is imported here to close an obligation. Model knowledge may include later developments; it is not used as evidence for this historical assessment."
  ],
  "findings": [
    {
      "kind": "context",
      "source_quote": "Let $\\Gam$ be a lattice in $\\SO_0(n, 1)$. If the associated locally symmetric space contains infinitely many maximal totally geodesic subspaces of dimension at least $2$, then $\\Gam$ is arithmetic.",
      "explanation": "Conditional on this theorem, the forward implication reduces to arithmetic-knot classification. In dimension 3, a proper totally geodesic surface is maximal: a strictly larger complete connected totally geodesic subspace would have dimension 3 and would not be proper. Thus dimension 2 meets the theorem's threshold, with no exceptional dimension remaining. The prerequisite that the complete hyperbolic knot complement has finite volume, hence lattice holonomy, is not established in this packet."
    },
    {
      "kind": "context",
      "source_quote": "In this paper, a totally geodesic subspace of a finite volume hyperbolic manifold or orbifold will always mean a properly immersed, topologically closed, totally geodesic subspace.",
      "explanation": "This convention is essential when applying thm:main. The surfaces must be understood as distinct subspaces satisfying this convention, not merely different covers or parametrizations of one surface. Without that interpretation, infinitely many immersed surfaces would not provide infinitely many maximal subspaces."
    },
    {
      "kind": "unverified",
      "source_quote": "Then $S^3 \\ssm K$ contains infinitely many immersed totally geodesic surfaces if and only if $K$ is the figure-eight knot.",
      "explanation": "Arithmeticity alone does not identify the knot within the supplied argument. The exact remaining forward obligation is a theorem asserting that an arithmetic complete finite-volume hyperbolic knot complement in S^3 is the figure-eight knot complement, with the corresponding knot identification justified. No statement or proof of that classification is supplied."
    },
    {
      "kind": "unverified",
      "source_quote": "if and only if $K$ is the figure-eight knot.",
      "explanation": "The reverse implication requires proving that the figure-eight complement contains infinitely many distinct properly immersed, topologically closed totally geodesic surfaces. thm:main is one-way and cannot establish this. An explicit construction or a suitable existence-and-infinitude theorem is missing; arithmeticity by itself is not a supplied converse."
    },
    {
      "kind": "map_correction",
      "source_quote": "Combining Theorem \\ref{thm:main} with results of Benoist--Oh \\cite[Thm. 10.1]{BO}, Lee--Oh \\cite[Thm. 1.9(3)]{LO}, and the classification of arithmetic hyperbolic $n$-manifolds (e.g., see \\cite{Meyer}), we also obtain the following.",
      "explanation": "This sentence follows the completed corollary and explicitly introduces what follows. Its citation edges to cor:knots are therefore not supported as proof-reference edges by this excerpt. The missing proof anchor is a mapping limitation, not itself a manuscript proof defect. The external theorem texts have not been supplied or checked."
    }
  ],
  "dependency_proposals": [
    {
      "label": "thm:main",
      "role": "conditional",
      "source_quote": "If the associated locally symmetric space contains infinitely many maximal totally geodesic subspaces of dimension at least $2$, then $\\Gam$ is arithmetic.",
      "explanation": "A genuine conditional input to the forward implication, once lattice holonomy and the distinct-subspace interpretation are supplied. Its proof is not reviewed here."
    },
    {
      "label": "external:BO",
      "role": "excluded",
      "source_quote": "Benoist--Oh \\cite[Thm. 10.1]{BO}",
      "explanation": "The citation belongs to prose introducing the following statement, not an identified proof of cor:knots. Exclusion is from the demonstrated dependency chain, not a judgment about the external theorem's truth."
    },
    {
      "label": "external:LO",
      "role": "excluded",
      "source_quote": "Lee--Oh \\cite[Thm. 1.9(3)]{LO}",
      "explanation": "The supplied placement does not establish this as an input to the knot corollary."
    },
    {
      "label": "external:Meyer",
      "role": "excluded",
      "source_quote": "the classification of arithmetic hyperbolic $n$-manifolds (e.g., see \\cite{Meyer})",
      "explanation": "This citation introduces the next statement. General arithmetic-manifold classification must not silently substitute for the specific arithmetic-knot classification needed here."
    },
    {
      "label": "missing:arithmetic-knot-classification",
      "role": "proof_input",
      "source_quote": "$K$ is the figure-eight knot.",
      "explanation": "Needed to pass from arithmetic lattice holonomy to the asserted knot identification. Exact theorem, hypotheses, source version, and pre-cutoff availability remain unverified."
    },
    {
      "label": "missing:figure-eight-surface-infinitude",
      "role": "proof_input",
      "source_quote": "$S^3 \\ssm K$ contains infinitely many immersed totally geodesic surfaces",
      "explanation": "Needed for the reverse implication, specifically for the figure-eight knot and with the manuscript's properness and closedness convention. No proof or dated external input is supplied."
    }
  ],
  "_public_provenance": {
    "origin": "returned_native_final_assessment_not_campaign_verdict",
    "model": "gpt-6.1-sol",
    "effort": "high",
    "provider": "openai",
    "manuscript_sha256": "6776902687fa24498e9bc0084edb02b1630009b90b7b06a7e2fba0e6d697397c",
    "review_input_sha256": "3b43ae52e868f7c23fe62e0576bb627ea6774776440ec4010429ae294bcc1421",
    "original_assessment_sha256": "0076a96b595d0bbbdecf2568fbc45a9c7a63059a8045049b43614d6373883038",
    "note": "Original final assessment fields retained; embedded roadmap and private provider/run metadata omitted. No raw reasoning stream is exported."
  }
}