{
  "target_id": "prop:homogen",
  "proof_status": "defective",
  "summary": "Condition (3) omits proper support, so the asserted immediate implication (3)\u21d2(2) and the construction of proper maximal subspaces from (3) are unjustified. The (1)\u21d2(3) argument is locally coherent conditional on the supplied structural and limit-measure premises, which are not certified here. The attached auxiliary proofs contain another false assertion and a degenerate-group issue. For n=2, no admissible m exists and neither geometric nor measure condition can hold. No later external result is used.",
  "used_dependencies": [
    "lem:clarifying1",
    "lem:clarifying2",
    "lem:clarifying3",
    "thm:MS"
  ],
  "missing_context": [
    "Admissible external theorem texts, with theorem numbers and version records, establishing the required measure classification, nonescape, limit homogeneity, and translated-support assertions.",
    "FLMS Lemma 5.13: the compact set meeting every closed totally geodesic subspace of dimension at least two; available_from, date_evidence, date_basis, and version are unresolved in this packet.",
    "Gelander-Levit Proposition 3.4: finite volume for every properly immersed closed totally geodesic subspace in the stated orbifold setting; availability and version are unresolved.",
    "Exact Howe\u2013Moore hypotheses supporting restriction ergodicity for W_m, its unipotent subgroups, and the larger stabilizer-generated semisimple groups; dated source records are absent."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "\\item for some $1<m<n$, there exists an infinite sequence of homogeneous, $W_m$-ergodic measures $\\{\\mu_i\\}$ for which Haar measure on $G/\\Gamma$ is a weak-$*$ limit of the $\\mu_i$.",
      "explanation": "Unlike (2), this imposes no proper-support condition. Nothing excludes Haar measure itself: under the Haar ergodicity invoked elsewhere in the proof, its constant sequence satisfies (3). An infinite sequence need not have distinct terms. Even distinctness would not address the missing restriction when arbitrary finite scalar multiples are allowed. Thus homogeneity does not supply the properness needed for the claimed (3)\u21d2(2), or for choosing proper maximal subspaces in (3)\u21d2(1)."
    },
    {
      "kind": "error",
      "source_quote": "thus if $K\\backslash KS_1g$ is an $m^\\prime$-dimensional closed totally geodesic subspace of $K\\backslash G$ then $W_{m^\\prime} \\le S_1 \\le N_{m^\\prime}$.",
      "explanation": "The lower inclusion is false without an additional hypothesis. Direct countercheck: in the upper-half-space model of the standard hyperbolic m'-plane, translations and positive dilations act transitively. Their proper solvable subgroup S_1 of W_{m'} therefore has exactly the asserted projected orbit, but cannot contain W_{m'}. This false converse appears in the auxiliary proof; subsequent applications already assume containment of W_m and do not require this converse."
    },
    {
      "kind": "error",
      "source_quote": "Suppose that $W< G$ is a closed connected semisimple subgroup that is generated by unipotent elements.",
      "explanation": "Under the usual convention allowing the trivial semisimple group, the auxiliary theorem admits W={e}. In a noncompact quotient, Dirac probabilities at points escaping every compact set are homogeneous and W-ergodic, but converge weak-* to zero. This contradicts its probability-limit conclusion. Nontriviality, or an explicit convention excluding this case, is required. The proposition's applications W=W_m with m>1 avoid this issue."
    },
    {
      "kind": "map_correction",
      "source_quote": "Applying Lemma~\\ref{lem:clarifying}\\eqref{lem:clarifying1} to the closed $S^{g_i}$-orbit",
      "explanation": "The following conclusion that the enlarged N_m-orbit is closed and has the same projection is supplied by part \\ref{lem:clarifying2}, not part \\ref{lem:clarifying1}. The map already includes part 2 elsewhere; this particular source-reference occurrence is incorrect."
    },
    {
      "kind": "unverified",
      "source_quote": "We will use the following theorem which follows from combining the existence of compact cores for hyperbolic manifolds with works of Dani--Margulis\n\\cite[Thm.~6.1]{DaniMargulis} and Mozes--Shah \\cite{MS}.",
      "explanation": "The nonescape, probability-limit, and translated-support conclusions require the exact external statements and hypotheses. Their cited bibliography years precede 2019-06-20, but the external texts were not inspected. FLMS, Gelander\u2013Levit, and the Howe\u2013Moore applications also remain external-input obligations. Model knowledge cannot establish dated availability; no later knowledge is treated as evidence."
    }
  ],
  "dependency_proposals": [
    {
      "label": "external:RatnerOrbitClosure",
      "role": "conditional",
      "source_quote": "the reverse implication is a theorem of Ratner \\cite{RatnerOrbitClosure}",
      "explanation": "Required input: every finite W_m-invariant ergodic measure on G/\u0393 is homogeneous, for \u0393 a lattice and m>1. Exact cited theorem unverified. available_from: 1991, bibliographic year only; date_evidence: supplied Duke citation; date_basis: journal publication; version: cited journal article, theorem text uninspected."
    },
    {
      "label": "external:Einsiedler",
      "role": "background",
      "source_quote": "(see also Einsiedler \\cite{Einsiedler})",
      "explanation": "Alternative exposition, not a separate established premise. The required classification and its applicability to every stated m need checking. available_from: 2006, bibliographic year only; date_evidence: supplied journal citation; date_basis: journal publication; version: cited journal article, exact theorem uninspected."
    },
    {
      "label": "external:DaniMargulis",
      "role": "conditional",
      "source_quote": "\\cite[Thm.~6.1]{DaniMargulis}",
      "explanation": "Indirect input through thm:MS. Required assertion: for fixed nontrivial unipotent flow and compact F, some compact F' captures at least half of every positive-time orbit segment starting in F. Exact theorem hypotheses unverified. available_from: 1993, bibliographic year only; date_evidence: supplied proceedings citation; date_basis: publication; version: cited proceedings article."
    },
    {
      "label": "external:MS",
      "role": "conditional",
      "source_quote": "Using \\cite[Cor.~1.3]{MS}",
      "explanation": "Indirect input through thm:MS. Required Corollary 1.3: nonzero limits in this setting are homogeneous probabilities. Required Theorem 1.1: support containment and conjugated invariance for the selected convergent generic points and unipotent flows. Exact hypotheses unverified. available_from: 1995, bibliographic year only; date_evidence: supplied ETDS citation; date_basis: journal publication; version: cited journal article."
    },
    {
      "label": "FLMS:Lem.5.13",
      "role": "conditional",
      "source_quote": "Using \\cite[Lem.~5.13]{FLMS}",
      "explanation": "Missing proof input for nonescape: a compact core meeting all closed totally geodesic subspaces of dimension at least two. available_from: unknown; date_evidence: absent; date_basis: unresolved; version: unspecified."
    },
    {
      "label": "Gelander-Levit:Prop.3.4",
      "role": "conditional",
      "source_quote": "see \\cite[Prop.~3.4]{Gelander-Levit} for a recent reference.",
      "explanation": "Missing proof input for lem:clarifying3: finite volume under the manuscript's closed, properly immersed orbifold hypotheses. available_from: unknown; date_evidence: absent; date_basis: unresolved; version: unspecified."
    }
  ],
  "_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": "91099ec6c1da22f29d2a5b2066b858a6d96ae3a7856a946eeeba2725e02b6193",
    "original_assessment_sha256": "1aa70a547fb64339bebf9b37ceb58144db6f562736daba40182b357761ce0f37",
    "note": "Original final assessment fields retained; embedded roadmap and private provider/run metadata omitted. No raw reasoning stream is exported."
  }
}