{
  "target_id": "lemma:properness",
  "proof_status": "needs_context",
  "summary": "The proof has a coherent conditional structure, and the compact-stabilizer contradiction and final quotient composition are locally valid. However, the supplied packet omits the ambient group and representation hypotheses and does not provide the cited external statements. Consequently, the single-orbit reduction, metric ergodicity, and properness of the stabilizer's Zariski closure remain unverified. This is a bounded context limitation, not an established manuscript defect.",
  "used_dependencies": [
    "external:Bader-Gelander",
    "external:Zimmer"
  ],
  "missing_context": [
    "The full ambient hypotheses of Theorem \\ref{theorem:superrigiditydichotomy}, including definitions and restrictions on G, W, k, H, V, rho, and all dimension ranges.",
    "The precise bundle action and measurable conventions needed for disintegration and almost-everywhere equivariance.",
    "Zimmer's Corollary 3.2.12, Theorem 3.2.4, and Proposition 3.2.15, with their exact hypotheses and the applicable edition.",
    "Bader\u2013Gelander's Definition 6.5 and Corollary 6.7, with their exact hypotheses; verify applicability also in every permitted low-dimensional case.",
    "The recorded external-results cutoff is 2019-06-20. Bibliographic years precede it, but the exact cited theorem contents have not been inspected. Model recall cannot certify dated theorem content; no later result is used to close these obligations."
  ],
  "findings": [
    {
      "kind": "context",
      "source_quote": "Throughout this section, we assume that we have a $W$-invariant measure $\\nu$ on the bundle $(G\\times \\mathbb{P}(V))/\\Gamma$ that projects to Haar measure on $G / \\Gam$.",
      "explanation": "The measure hypothesis needed for disintegration is supplied. Under the usual standard-Borel bundle assumptions, disintegration gives probability-valued conditional measures with the claimed invariance and equivariance, interpreted almost everywhere. The packet does not specify the full definitions and hypotheses for G, W, k, H, V, rho, or the diagonal Gamma-action, so those prerequisites cannot all be checked."
    },
    {
      "kind": "unverified",
      "source_quote": "By \\cite[Cor.\\ 3.2.12 and Thm.\\ 3.2.4]{Zimmer}, the image of this map lies in a single $\\bH(k)$-orbit",
      "explanation": "This reduction requires more than disintegration alone: an appropriate regularity or smoothness result for the action on probability measures and ergodicity of the source action are needed. Neither the exact cited statements nor their hypotheses are supplied. Metric ergodicity invoked subsequently could supply ordinary ergodicity if applicable, but its applicability also remains conditional. The asserted compact-extension description of the stabilizer must likewise be checked against the actual Zimmer statements."
    },
    {
      "kind": "unverified",
      "source_quote": "by \\cite[Cor.\\ 6.7]{Bader-Gelander} the action of $\\Gamma$ on $W\\backslash G$ is metrically ergodic",
      "explanation": "The packet does not provide Corollary 6.7 or enough ambient assumptions to verify its application, including the permitted groups G and W and any noncompactness, simplicity, or dimension restrictions. Conditional on metric ergodicity and the usual separable invariant metric on the quotient by a compact subgroup, the next inference is correct: an essentially constant equivariant map has a fixed coset, so rho(Gamma) lies in a conjugate of the compact stabilizer, contradicting unboundedness."
    },
    {
      "kind": "unverified",
      "source_quote": "Let $\\bfL$ be the Zariski closure of $\\tilde L$. Then \\cite[Prop.\\ 3.2.15]{Zimmer} implies that $\\bfL$ is a proper $k$-subgroup of $\\bH$.",
      "explanation": "This is the decisive remaining external-input obligation. Noncompactness of the stabilizer alone does not imply properness of its Zariski closure; for example, a trivial projective action has the whole group as every measure stabilizer. The representation and algebraic-group hypotheses that exclude such behavior are absent from the packet, and Proposition 3.2.15 has not been inspected. This illustrates why those hypotheses must be checked, not a counterexample under the manuscript's omitted assumptions."
    },
    {
      "kind": "context",
      "source_quote": "We are then done by composing $G\\to \\bH(k)/\\tilde{L}$ with the natural map $\\bH(k)/\\tilde{L} \\to \\bH/\\bfL(k)$.",
      "explanation": "Once a proper k-defined Zariski closure is obtained, this step is valid: inclusion of the stabilizer in L(k) makes the natural map well-defined, and composition preserves measurability, W-invariance, and Gamma-equivariance. There is no separate local defect in this final transition."
    }
  ],
  "dependency_proposals": [
    {
      "label": "external:Zimmer",
      "role": "conditional",
      "source_quote": "\\newblock Birkh\\\"auser Verlag, Basel, 1984.",
      "explanation": "Required inputs: single-orbit reduction, compact-extension structure of the probability-measure stabilizer, and properness of its Zariski closure under the relevant representation hypotheses. Exact original theorem statements and hypotheses: unavailable in this packet. available_from: 1984, at year precision; date_evidence: supplied bibliography; date_basis: bibliographic publication year; version: cited 1984 monograph, edition details unverified. These remain proposed premises rather than established inputs."
    },
    {
      "label": "external:Bader-Gelander",
      "role": "conditional",
      "source_quote": "\\newblock {\\em Groups Geom. Dyn.}, 11(3):1003--1039, 2017.",
      "explanation": "Required input: Corollary 6.7 must establish metric ergodicity of Gamma acting on W\\G for the actual ambient data, with Definition 6.5 matching the target metric-space setting. Exact original hypotheses: unavailable in this packet. available_from: 2017, at year precision; date_evidence: supplied bibliography; date_basis: bibliographic publication year; version: cited journal article, theorem text uninspected."
    },
    {
      "label": "theorem:superrigiditydichotomy",
      "role": "proof_input",
      "source_quote": "\\section{Proof of Theorem \\ref{theorem:superrigiditydichotomy}}",
      "explanation": "The theorem's ambient assumptions are an essential missing dependency for checking both external applications and all dimension restrictions. The section heading identifies the relevant source location but does not supply those assumptions. Existing external-reference edges are supported by the explicit citations; this proposal adds context rather than correcting those edges."
    }
  ],
  "_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": "23e968879120cb937a349c4725eeb339ebc37be29b50437ba8558abb456f1d9d",
    "original_assessment_sha256": "dc69be65b38cefc41527cf6b3db10b220e27caa40f95f75471eef7ec915dfcb0",
    "note": "Original final assessment fields retained; embedded roadmap and private provider/run metadata omitted. No raw reasoning stream is exported."
  }
}