{
  "target_id": "cor:kleinian",
  "proof_status": "needs_context",
  "summary": "The supplied material does not establish the full corollary, but it reveals no definite contradiction. Conditional on thm:main, arithmeticity follows in (1) after finite volume is established, in (2) after compactness is established, and in the infinite-hypersurface direction of (3). The geometric finiteness/compactness bridges and the arithmetic-to-infinitely-many-hypersurfaces direction are not supplied. The missing proof anchor is a mapping limitation, not itself a manuscript defect.",
  "used_dependencies": [
    "thm:main"
  ],
  "missing_context": [
    "A proof or precise pre-cutoff input establishing the finite-volume conclusion in (1) under geometric finiteness, including possible cusps.",
    "A proof or precise pre-cutoff input establishing compactness in (2), with maximality used and without extra acylindricity or dimension restrictions.",
    "The arithmetic structural input and infinitude argument for the converse in (3), including distinctness, properness, closedness, and applicability to every stated even dimension.",
    "Any required finite-cover reductions for manifolds whose fundamental groups are not initially lattices in SO_0(n,1).",
    "External evidence record for MMO: available_from = 2017 according to the supplied bibliography; date_evidence = Invent. Math., 209(2):425--461, 2017; date_basis = uninspected bibliographic publication year; version = cited journal article, text uninspected; exact theorem and hypotheses = not supplied.",
    "External evidence record for MMO2: available_from = unresolved for the needed theorem; date_evidence = supplied arXiv:1802.03853, 2018 citation; date_basis = uninspected preprint citation; version = unspecified; exact theorem and hypotheses = not supplied. A dated version containing the needed result must be checked against 2019-06-20."
  ],
  "findings": [
    {
      "kind": "unverified",
      "source_quote": "\\item If $M$ is a geometrically finite hyperbolic $3$-manifold containing infinitely many totally geodesic surfaces with finite area, then $M$ has finite volume and $\\pi_1(M)$ is arithmetic.",
      "explanation": "The introductory convention supplies proper immersion and topological closedness, and finite area is explicit. In dimension 3 these proper surfaces are maximal by dimension. However, thm:main assumes a lattice and therefore cannot establish the asserted finite volume. The remaining geometric obligation is to show that an infinite-volume geometrically finite hyperbolic 3-manifold has only finitely many such surfaces. Once finite volume is known, the arithmeticity conclusion follows conditionally from thm:main."
    },
    {
      "kind": "unverified",
      "source_quote": "\\item If $M$ is a convex cocompact hyperbolic $n$-manifold containing infinitely many maximal totally geodesic surfaces with finite area, then $M$ is compact and $\\pi_1(M)$ is arithmetic.",
      "explanation": "Maximality and dimension at least 2 match thm:main, but convex cocompactness does not supply its lattice hypothesis. The missing obligation is the compactness conclusion under the stated infinitude assumption, in every applicable dimension, without adding acylindricity. Once compactness is established, thm:main conditionally gives arithmeticity. For n below 3, infinitude of distinct two-dimensional subspaces is vacuous under the usual subspace convention."
    },
    {
      "kind": "unverified",
      "source_quote": "\\item If $n \\ge 4$ is even and $M$ is a finite volume hyperbolic $n$-manifold, then $M$ is arithmetic if and only if it contains infinitely many totally geodesic hypersurfaces.",
      "explanation": "The infinite-hypersurface-to-arithmetic direction is conditionally supported by thm:main: hypersurfaces are maximal, and their dimension n-1 is at least 3. Evenness is unnecessary for that direction. The converse is not an implication of thm:main. It needs an arithmetic structural result valid for every even n >= 4, together with production of infinitely many distinct hypersurfaces satisfying the manuscript's properness and closedness convention. No such input or argument is supplied."
    },
    {
      "kind": "context",
      "source_quote": "Let $\\Gam$ be a lattice in $\\SO_0(n, 1)$.",
      "explanation": "The corollary does not explicitly assume orientability, whereas the supplied theorem uses this identity component. If nonorientable manifolds are included, its application requires passage to an orientable finite cover and transfer of the relevant infinitude, maximality, and arithmeticity conclusions. This is a prerequisite to make explicit, not a demonstrated obstruction."
    },
    {
      "kind": "map_correction",
      "source_quote": "For convex cocompact acylindrical $3$-manifolds, this result already follows from work of McMullen--Mohammadi--Oh \\cite{MMO, MMO2} and Theorem \\ref{thm:main}. See \\S\\ref{ssec:FinalB} for discussion of (3) in odd dimensions.",
      "explanation": "The external citations are asserted support only for the convex cocompact acylindrical 3-dimensional special case, not for all three clauses. The FinalB reference is explicitly a discussion reference concerning odd dimensions, not a proof input for the even-dimensional assertion. Moreover, the supplied FinalB excerpt contains no such odd-dimensional discussion, so its anchor is incomplete."
    },
    {
      "kind": "map_correction",
      "source_quote": "Methods analogous to those used in the proof of Theorem \\ref{thm:main} can also be used to show the following.",
      "explanation": "This sentence introduces the following result rather than proving cor:kleinian. The thm:main edge generated from this occurrence should not be treated as proof evidence for the corollary. Its separate relevance to the arithmeticity implications remains valid conditionally."
    },
    {
      "kind": "context",
      "source_quote": "\\newblock Preprint: arXiv:1802.03853, 2018.",
      "explanation": "The recorded cutoff is 2019-06-20. The bibliography suggests a pre-cutoff source, but supplies neither its precise version nor the exact theorem needed. External texts were not inspected, and no external theorem is imported here. Model familiarity is not dated evidence and cannot resolve this historical obligation."
    }
  ],
  "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": "Supports the arithmeticity conclusions only once the lattice hypothesis and any finite-cover reduction are available; its own proof is not assessed here."
    },
    {
      "label": "external:MMO",
      "role": "background",
      "source_quote": "\\newblock Geodesic planes in hyperbolic 3-manifolds.",
      "explanation": "Cited for the acylindrical special case; promote to a proof input only after identifying the exact admissible theorem and checking its hypotheses."
    },
    {
      "label": "external:MMO2",
      "role": "background",
      "source_quote": "\\newblock Geodesic planes in the convex core of an acylindrical 3-manifold.",
      "explanation": "Special-case context only on the supplied evidence; the needed theorem and pre-cutoff version remain unverified."
    },
    {
      "label": "ssec:FinalB",
      "role": "background",
      "source_quote": "\\subsection{Final remarks and questions}\\label{ssec:FinalB}",
      "explanation": "A discussion reference, with the relevant odd-dimensional passage absent from this packet; not an established proof dependency."
    }
  ],
  "_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": "6e1bd12f28170c49732e4d48e06498f64c36d16a51e1ff4225659add3c1c4343",
    "original_assessment_sha256": "d000e2dfdeba8143b7317c1a3cb9fc6ac34a52b3abe5f5c3152e4e5a1806ff44",
    "note": "Original final assessment fields retained; embedded roadmap and private provider/run metadata omitted. No raw reasoning stream is exported."
  }
}