{
  "target_id": "thm:main:homogeneous",
  "proof_status": "needs_context",
  "summary": "The supplied material states the target and sketches a route through an invariant bundle measure and superrigidity, but does not supply the arguments completing that route. Conditional on Proposition \\ref{prop:homogen} and matching standing hypotheses, the target implies Theorem \\ref{thm:main}; this does not prove the target. No local contradiction is established. The missing proof anchor is a mapping limitation, not itself a manuscript defect.",
  "used_dependencies": [
    "prop:homogen",
    "ssec:Compatible",
    "thm:main"
  ],
  "missing_context": [
    "Standing definitions of G, Gamma, W, the permitted dimensions and subgroup embeddings, the measure class and normalization, and the meaning of weak-* convergence on the quotient.",
    "The complete proof or correctly anchored proof scope for thm:main:homogeneous, including the bundle-measure construction and its n=3 case.",
    "The exact superrigidity statement, its required conditions on rho and V, and the verification of every hypothesis for the representations used.",
    "The compatibility results and arithmeticity criterion actually invoked, with exact statements and hypotheses. For each external input: available_from, date_evidence, date_basis, and the version containing the needed result, demonstrating availability by 2019-06-20."
  ],
  "findings": [
    {
      "kind": "context",
      "source_quote": "If there exists an infinite sequence $\\{\\mu_i\\}$ of $W$-invariant, ergodic measures with proper support for which Haar measure on $G/\\Gamma$ is a weak-$*$ limit of the $\\mu_i$, then $\\Gamma$ is arithmetic.",
      "explanation": "The packet does not supply the standing definitions and restrictions on G, W, and Gamma. In particular, the lattice assumption, permitted dimensions and subgroup types, and measure normalization must be recovered before the hypotheses can be checked. Proposition prop:homogen concerns W_m with 1<m<n; its application requires that these groups are allowed in the target. That range is empty when n=2. This is missing packet context, not evidence that the manuscript omitted these assumptions."
    },
    {
      "kind": "map_correction",
      "source_quote": "We show in Proposition \\ref{prop:homogen} that Theorem \\ref{thm:main:homogeneous} implies Theorem \\ref{thm:main}.",
      "explanation": "The stated implication is supported conditionally by item (1) implying item (2) of prop:homogen, followed by the target for W=W_m. Thus thm:main is a downstream conclusion, not a proof premise for the target. The source-reference edges from thm:main must not be interpreted as establishing the target. The proposition itself remains an unverified premise."
    },
    {
      "kind": "unverified",
      "source_quote": "We then use the hypotheses of either Theorem \\ref{thm:main} or Theorem \\ref{thm:main:homogeneous} to build a $W$-invariant ergodic measure on the bundle $(G\\times\\bbP(V))/\\Gam$ that projects to Haar measure on $G/\\Gamma$.",
      "explanation": "This announces, rather than proves, the essential measure construction. The remaining obligation is to specify V and the bundle action, construct the measure from the proper-support approximating measures, and justify invariance, projection, and ergodicity. The detached fragment mentioning V(3,3) and G=SO_0(3,1) does not provide the complete construction or its hypotheses, so the exceptional dimension n=3 cannot be checked here."
    },
    {
      "kind": "unverified",
      "source_quote": "We now state the superrigidity theorem that finishes the proof from the existence of such a measure.",
      "explanation": "Neither that theorem nor its application is supplied. A complete local verification needs its exact representation and measure hypotheses, verification of compatibility for the pairs actually used, and the argument turning the resulting representation extension into arithmeticity. Characteristic zero and a connected semisimple adjoint target alone do not verify these omitted steps."
    },
    {
      "kind": "context",
      "source_quote": "This requires an additional technical assumption on the pair $k$ and $\\bfH$.",
      "explanation": "The Compatibility excerpt supplies a matching definition, not a result establishing compatibility for the arithmeticity argument. Any external inputs must be documented in versions available by 2019-06-20; none are inspected or imported here. The supplied manuscript edition is dated 2020, which by itself establishes no external result's availability before the cutoff. Model knowledge may include later developments, but it is not used as evidence to close these obligations."
    }
  ],
  "dependency_proposals": [
    {
      "label": "prop:homogen",
      "role": "conditional",
      "source_quote": "\\item for some $1<m<n$, there exists an infinite sequence $\\{\\mu_i\\}$ of $W_m$-invariant, ergodic measures with proper support for which Haar measure on $G/\\Gamma$ is a weak-$*$ limit of the $\\mu_i$",
      "explanation": "Conditional bridge from the geometric hypothesis to the target's measure hypothesis, subject to matching standing definitions. It supports the reduction to the target, not a proof of the target."
    },
    {
      "label": "thm:main",
      "role": "excluded",
      "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": "Exclude as a proof input for this target: the supplied prose explicitly presents it as a consequence of the target and prop:homogen."
    },
    {
      "label": "ssec:Compatible",
      "role": "background",
      "source_quote": "Recall that the pair consisting of $k$ and $\\bH$ is \\emph{compatible} with $G$",
      "explanation": "Provides the definition used by the announced superrigidity approach; it does not establish compatibility of the required pairs."
    },
    {
      "label": "unsupplied:bundle-measure-construction",
      "role": "proof_input",
      "source_quote": "to build a $W$-invariant ergodic measure on the bundle $(G\\times\\bbP(V))/\\Gam$ that projects to Haar measure on $G/\\Gamma$.",
      "explanation": "A proof anchor for this construction is required, including its exceptional-dimensional treatment."
    },
    {
      "label": "unsupplied:superrigidity-and-arithmeticity",
      "role": "proof_input",
      "source_quote": "the superrigidity theorem that finishes the proof from the existence of such a measure",
      "explanation": "The announced theorem and the subsequent arithmeticity deduction are necessary but absent from the supplied proof scope."
    }
  ],
  "_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": "93389dc78df93653ebc60630853ab6d59a16802ce42a81efc01ab5f991bfb316",
    "original_assessment_sha256": "040a564e5d14865e089b482aed852f1a356e7a9a6d17440c2a5b1def58ab016c",
    "note": "Original final assessment fields retained; embedded roadmap and private provider/run metadata omitted. No raw reasoning stream is exported."
  }
}