{
  "target_id": "theorem:superrigiditydichotomy",
  "proof_status": "needs_context",
  "summary": "The extension argument is coherent conditional on the stated internal premises and the cited external results. Compatibility is used at the correct point, and the fixed-point contradiction does not require an adjoint target. However, the external inputs establishing the proper-subgroup map and the final continuous extension have not been supplied or historically verified. One normalizer-lemma application also needs an additional initiality argument. This is a bounded assessment, not a counterexample or a full-theorem verdict. No later theorem is used; because model knowledge can include later developments, external claims remain conditional on dated source verification.",
  "used_dependencies": [
    "lemma:properness",
    "ssec:AlgRep",
    "thm:BF",
    "lem:initobj",
    "ssec:Compatible",
    "external:BaderFurman",
    "external:Zimmer"
  ],
  "missing_context": [
    "A pre-2019-06-20 version of Bader\u2013Furman containing Theorems 4.3 and 4.6, Proposition 5.1, and the cited end of Theorem 1.3, with exact hypotheses.",
    "The Bader\u2013Gelander bibliographic record, dated applicable version, and Corollary 6.7 with Definition 6.5.",
    "Zimmer's cited orbit and stabilizer statements. Available_from: 1984 according to the supplied bibliography; date_basis: publication year; version: cited 1984 monograph. Their contents and applicability remain uninspected.",
    "The exact Howe\u2013Moore input used for mixing, with hypotheses and dated source/version evidence; these are not supplied.",
    "Continuity conventions for homomorphisms in algebraic representations: the supplied definition says homomorphism, whereas compatibility requires a continuous homomorphism.",
    "An explicit initiality argument for the restriction to A and for intermediate normalizing subgroups."
  ],
  "findings": [
    {
      "kind": "unverified",
      "source_quote": "by \\cite[Cor.\\ 6.7]{Bader-Gelander} the action of $\\Gamma$ on $W\\backslash G$ is metrically ergodic",
      "explanation": "This essential input is absent from the dependency list and has no bibliographic or version record in the packet. Its exact hypotheses must cover the stated noncompact simple W and arbitrary lattice, including nonuniform lattices. The compact-stabilizer contradiction is sound conditional on this input: a constant equivariant map into a quotient by a compact subgroup would make the image of rho relatively compact."
    },
    {
      "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": "The proof of lemma:properness also needs the orbit-reduction and stabilizer conclusions cited here, followed by Zimmer Proposition 3.2.15. Their precise scope must include every local field allowed by the theorem, not merely characteristic-zero fields, and the stated connected target with a faithful irreducible representation. The preceding ergodicity and measurable-orbit prerequisites should be checked. These citations are not verified by the supplied bibliography."
    },
    {
      "kind": "unverified",
      "source_quote": "and therefore must be an initial object for the category of $A$-algebraic representations by Lemma \\ref{lem:initobj}.",
      "explanation": "The stated lemma compares a subgroup with its normalizer or iterated normalizers. It does not directly compare A with P: the normalizer iteration starting from A does not give P. The remaining local obligation is to prove that restricting the H-valued P-representation to A is initial, using the coset initial-object theorem and its universal property. Likewise, the earlier passage through U' normal in U normal in P requires explaining initiality under restriction to intermediate normalizing subgroups."
    },
    {
      "kind": "unverified",
      "source_quote": "using \\cite[Prop.\\ 5.1]{BaderFurman} and following the end of the proof of \\cite[Thm.\\ 1.3]{BaderFurman}",
      "explanation": "The final measurable-to-continuous extension step is delegated to external statements not supplied here. Check their exact assumptions, the agreement of the P and N_G(A) actions on overlaps, and recovery of rho itself after any conjugation. For the needed statements, available_from is unresolved; date_evidence is the bibliography identifier arXiv:1810.01608; date_basis is an unversioned identifier; version is unspecified. A version containing these results by 2019-06-20 is required."
    },
    {
      "kind": "context",
      "source_quote": "Since $\\phi$ is also $W$-invariant and $\\langle U, W \\rangle = G$",
      "explanation": "This central transition is appropriate for the stated real hyperbolic groups, including n=3: a Weyl element in W exchanges U with the opposite maximal unipotent subgroup, and these generate G. Compatibility applies only under the temporary assumption that J is nontrivial. Algebraic equivariance extends invariance from tau(U') to its Zariski closure, giving U-invariance of phi. Zariski density then contradicts a fixed coset of proper L. The proof calls H adjoint later, although that is not a hypothesis; adjointness is unnecessary for this contradiction."
    },
    {
      "kind": "map_correction",
      "source_quote": "In particular, this section reduces Theorem \\ref{thm:main} to Theorem \\ref{thm:main:homogeneous} and Theorem \\ref{thm:main:homogeneous} to Theorem \\ref{theorem:superrigiditydichotomy}.",
      "explanation": "The arithmeticity theorems, prop:homogen, measure construction, trace-field results, and compatibility lemma belong to applications of the target, not its proof inputs. In particular, the target assumes compatibility; its proof does not establish that assumption through lemma:compatible. Exceptional-dimension arguments in those application sections cannot be counted as support for the extension theorem."
    }
  ],
  "dependency_proposals": [
    {
      "label": "external:Bader-Gelander",
      "role": "conditional",
      "source_quote": "\\cite[Cor.\\ 6.7]{Bader-Gelander}",
      "explanation": "Add the metric-ergodicity input. Available_from, date_evidence, date_basis, and version are all unresolved in this packet."
    },
    {
      "label": "external:Howe--Moore",
      "role": "conditional",
      "source_quote": "Observe that the action of $G$ on $G/\\Gamma$ is mixing by the Howe--Moore theorem.",
      "explanation": "Add the mixing input needed to apply thm:BF and lem:initobj to noncompact subgroups. Available_from, dated evidence, and version are not supplied; no external verification is claimed."
    },
    {
      "label": "external:BaderFurman",
      "role": "conditional",
      "source_quote": "\\begin{thm}[Thm.\\ 4.3 \\cite{BaderFurman}]\\label{thm:BF}",
      "explanation": "Track the initial-object theorem, normalizer theorem, and final extension proposition separately, with cutoff-admissible versions."
    },
    {
      "label": "Ssec:buildingmeasures",
      "role": "excluded",
      "source_quote": "Under the hypotheses of Theorem \\ref{thm:main:homogeneous} or Theorem \\ref{thm:main},",
      "explanation": "Constructs the target's assumed measure in arithmeticity applications; not needed to prove the target."
    },
    {
      "label": "ssec:Compatible",
      "role": "background",
      "source_quote": "Recall that the pair consisting of $k$ and $\\bH$ is \\emph{compatible} with $G$",
      "explanation": "The definition is a proof input. The later assertion establishing compatibility for particular arithmetic targets is an application-side obligation, not a premise proved here."
    }
  ],
  "_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": "bca048910615ca325447d5df8e17cebea8d623571574b64dedaef7773a8cb0e5",
    "original_assessment_sha256": "a8a5b4333d70c5f4b52199daa4945f94fef3f16afdc8eb623141b6e10878f53b",
    "note": "Original final assessment fields retained; embedded roadmap and private provider/run metadata omitted. No raw reasoning stream is exported."
  }
}