{
  "target_id": "thm:MS",
  "proof_status": "defective",
  "summary": "As literally stated, the theorem permits the trivial subgroup W, for which escaping point masses give a counterexample on a noncompact quotient. If nontriviality is an implicit standing convention, that defect disappears, but the supplied proof still requires rank-one subgroup classification and verification of its external inputs. Conditional on those inputs, the nonzero-mass argument and the final support-containment argument are coherent. This is a local assessment, not a verdict on the intended theorem.",
  "used_dependencies": [
    "eq:DM",
    "external:DaniMargulis",
    "external:FLMS",
    "external:MS",
    "lem:clarifying1"
  ],
  "missing_context": [
    "Standing definitions of G, K, Gamma and pi, including the permitted dimensions and whether nontriviality of W is stipulated.",
    "The rank-one subgroup results supporting both classification steps.",
    "Exact cited external theorem texts and the pre-cutoff version of FLMS containing Lemma 5.13.",
    "A dated source for the Howe--Moore and Birkhoff formulations used here. No external theorem has been independently verified or imported in this assessment; model knowledge cannot establish historical availability."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "Suppose that $W< G$ is a closed connected semisimple subgroup that is generated by unipotent elements.",
      "explanation": "Under the usual literal definitions, W={e} satisfies these conditions. On a permitted noncompact quotient, choose x_i escaping every compact subset. Then mu_i=delta_{x_i} are homogeneous, W-ergodic probability measures converging weak-* to zero as finite Radon measures. Thus the stated conclusion fails unless nontriviality of W is imposed elsewhere."
    },
    {
      "kind": "context",
      "source_quote": "As this claim is conjugation invariant, we will assume as we may that $W=W_m$ for some $m>1$.",
      "explanation": "Conjugation invariance alone does not justify this reduction. It requires the classification of nontrivial connected semisimple subgroups of the ambient real hyperbolic group generated by unipotents. The supplied packet does not establish that classification. It also cannot cover W={e}. The m=2 case needs to be included explicitly in the prerequisite."
    },
    {
      "kind": "context",
      "source_quote": "The group $S$ is not unipotent, as it contains $W$, therefore it must be semisimple.",
      "explanation": "This implication is not valid for general Lie subgroups. Here it needs a rank-one classification for the subgroup generated by all unipotents in the stabilizer, including its closedness and connectedness. The subsequent Howe--Moore applications need the appropriate noncompact simple-group setting; mere semisimplicity would not suffice in general."
    },
    {
      "kind": "unverified",
      "source_quote": "We note that $\\mu_i$ is $\\{u_t\\}$-ergodic by the Howe--Moore theorem",
      "explanation": "This is a substantive, unmapped input. It requires a nontrivial, hence unbounded, unipotent subgroup and an appropriate simple ambient acting group. The same issue occurs for every U_j and for the final restriction from S to the conjugate of W."
    },
    {
      "kind": "context",
      "source_quote": "The claim will follow once we show that $\\mu_i(F^\\prime) \\geq 1/2$ for every $i$, thus $\\mu(F^\\prime)\\geq 1/2$.",
      "explanation": "This passage is correct conditional on the preceding inputs: positive mu_i(F) permits selection of a generic point in F; the uniform occupation bound then yields mu_i(F')>=1/2. Compactness of F' gives the asserted limit inequality for vague/weak-* convergence. Likewise, the later selection of nearby support points and their small lifts in G is valid for a discrete lattice quotient."
    },
    {
      "kind": "error",
      "source_quote": "for every $y\\in F$ and every $T\\ge 0$",
      "explanation": "The displayed averages contain 1/T and are undefined at T=0. The required range is T>0. This endpoint error does not affect the limiting argument."
    },
    {
      "kind": "unverified",
      "source_quote": "By \\cite[Thm.~1.1]{MS}, we conclude that for every $1\\le j \\le k$ there exists an $i_j$ such that $Y_i\\subset g_iY$ and $\\mu$ is $U_j^{g_i^{-1}}$-invariant for every $i\\geq i_j$.",
      "explanation": "The cited theorem and Cor. 1.3 are not supplied. Their exact hypotheses and conclusions must be checked, especially probability of the nonzero limit, the description of S, translated support inclusion, and conjugated-flow invariance. Conditional on these conclusions and the subgroup prerequisites, generation by the U_j gives the claimed W-invariance, and the remaining translation argument is sound."
    },
    {
      "kind": "map_correction",
      "source_quote": "By Lemma~\\ref{lem:clarifying}\\eqref{lem:clarifying1}",
      "explanation": "Only item lem:clarifying1 is used here; the whole-lemma edge duplicates that specific input. Equation eq:DM is an internal intermediate estimate, not an independent external premise. Howe--Moore, Birkhoff genericity, and the subgroup-classification prerequisites are missing from the proposed dependency map."
    }
  ],
  "dependency_proposals": [
    {
      "label": "external:FLMS, Lemma 5.13",
      "role": "conditional",
      "source_quote": "Using \\cite[Lem.~5.13]{FLMS} we fix a compact set $C_1$",
      "explanation": "Required input: for the finite-volume hyperbolic quotient, one compact set meets every closed totally geodesic subspace of dimension at least 2. available_from: unresolved for this lemma; date_evidence: arXiv:1802.04619; date_basis: preprint identifier; version: unspecified. The identifier alone does not verify the lemma's availability by 2019-06-20."
    },
    {
      "label": "external:DaniMargulis, Theorem 6.1",
      "role": "conditional",
      "source_quote": "Applying \\cite[Thm.~6.1]{DaniMargulis}",
      "explanation": "Required input: for compact F in the lattice quotient and the chosen nontrivial unipotent flow, a compact F' gives the displayed occupation bound uniformly for y in F and T>0. available_from: 1993 bibliographically; date_evidence: supplied bibliography; date_basis: publication year; version: cited Adv. Soviet Math. 16 article. Exact hypotheses remain uninspected."
    },
    {
      "label": "external:MS, Corollary 1.3 and Theorem 1.1",
      "role": "conditional",
      "source_quote": "Using \\cite[Cor.~1.3]{MS}",
      "explanation": "Required inputs are the nonzero-limit classification and translated support/invariance conclusions identified above, for the supplied unipotent-ergodic probability measures and generic points. available_from: 1995 bibliographically; date_evidence: supplied bibliography; date_basis: publication year; version: cited ETDS 15(1):149--159 article. Exact statements and hypotheses remain uninspected."
    },
    {
      "label": "Howe--Moore restriction ergodicity",
      "role": "proof_input",
      "source_quote": "Then $W^{g_i^{-1}}$-ergodicity follows from the Howe--Moore theorem and $S$-ergodicity.",
      "explanation": "Needed formulation: an ergodic probability-preserving action of the relevant noncompact simple group remains ergodic on restriction to an unbounded subgroup. available_from: unresolved; date_evidence: none supplied; date_basis: unresolved; version: unspecified."
    },
    {
      "label": "Rank-one subgroup classification",
      "role": "proof_input",
      "source_quote": "therefore it must be semisimple.",
      "explanation": "Supply the internal classification prerequisite for W and S, including closedness, connectedness, simplicity and the dimension-2 case. No external classification result is being imported 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": "bdf6d19e01a478eae2b02e10aff4af2aabe9f0f8bb3f54f5a3efa5523b93a1d8",
    "original_assessment_sha256": "271fdeb00a2d53a16f3dcb04d74af4b68be1ecc907d74b28cb53e2e0f075a339",
    "note": "Original final assessment fields retained; embedded roadmap and private provider/run metadata omitted. No raw reasoning stream is exported."
  }
}