{
  "target_id": "emtheorem",
  "proof_status": "defective",
  "summary": "The counting statement is false under the hypotheses displayed: well-roundedness alone does not prevent concentration in a bounded region of one factor of a reducible symmetric space. The normalization also lacks stated finite-covolume prerequisites. This concerns the manuscript's formulation, not the correctness of Eskin\u2013McMullen's original theorem, whose text has not been supplied or verified. The missing proof anchor is only a mapping limitation.",
  "used_dependencies": [
    "external:EM"
  ],
  "missing_context": [
    "The original journal text of EM Theorem 1.4, including its standing assumptions, definition of well-roundedness, and measure conventions.",
    "The precise additional hypotheses needed to exclude the explicit product counterexample; these must be checked against EM rather than reconstructed from later literature.",
    "For the intended lattice-counting application, the symmetric-space realization and verification of the external theorem's lattice, stabilizer, orbit, and normalization hypotheses.",
    "External texts were not inspected. No later theorem is used to close the obligation; possible later knowledge in model weights is not treated as historical evidence."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "For any well-rounded sequence, the cardinality of the number of points of $\\Gamma v$ which lie in $B_n$, grows like the volume of $B_n$: asymptotically,",
      "explanation": "Elementary counterexample to the stated generality: take G=SL_2(R)\u00d7SL_2(R), H=SO(2)\u00d7SO(2), and \u0393=SL_2(Z)\u00d7SL_2(Z). These satisfy the displayed structural hypotheses, and G/H is the product of two hyperbolic planes. The modular domain |x|\u22641/2, |z|\u22651 has hyperbolic area \u03c0/3, giving the lattice property. Let D be the annulus 1/10<d(i,z)<1/5 in the second plane, and let B_j be the radius-j disk in the first plane times D. No modular translate of i lies in D: for an integral determinant-one matrix \u03b3, cosh d(i,\u03b3i)=(a\u00b2+b\u00b2+c\u00b2+d\u00b2)/2, which is either 1 or at least 3/2. Thus |\u0393v\u2229B_j|=0, although vol(B_j) tends to infinity and the proposed covolume coefficient is finite and positive. These sets are well-rounded: small group elements change radial distances by uniformly small amounts, and the area formula 2\u03c0(cosh r\u22121) makes the resulting relative boundary volumes tend uniformly to zero for j\u22651. Additional hypotheses excluding this product obstruction are therefore necessary. This is a direct diagnostic, not an imported theorem or replacement proof."
    },
    {
      "kind": "context",
      "source_quote": "$$|\\Gamma v\\cap B_n|\\sim \\frac{m((\\Gamma\\cap H)\\backslash H}{\\Gamma\\backslash G}m(B_n).$$",
      "explanation": "The displayed coefficient is malformed: its numerator lacks a closing parenthesis and its denominator is a quotient space rather than its measured volume. Even under the natural intended interpretation as a ratio of covolumes, the hypotheses do not state that \u0393\u2229H is a lattice in H, so finiteness of the numerator is not supplied. Closedness/local finiteness of the orbit and compatible Haar and quotient-measure normalizations also need to be specified or obtained from the actual external theorem."
    },
    {
      "kind": "unverified",
      "source_quote": "\\begin{thm}[Theorem~1.4, Eskin-Mcmullen \\cite{EM}]\\label{emtheorem}",
      "explanation": "Neither the bibliography nor the surrounding well-roundedness discussion verifies the exact statement of EM Theorem 1.4. Its original hypotheses must be compared with this transcription, particularly those controlling the lattice, stabilizer covolume, and escape in semisimple factors. The subsequent definitions and region arguments concern applicability; they are not a proof of this general external counting theorem."
    },
    {
      "kind": "map_correction",
      "source_quote": "It is also necessary in using the equidistribution result of Gorodnick-Oh \\cite{GO} in proving Lemma~\\ref{prox}.",
      "explanation": "This passage makes well-roundedness an input to the GO-based proof of prox. It does not make prox or GO a premise proving emtheorem. Those incoming edges are incidental source-reference associations and should not be treated as proof dependencies. Likewise, EM's asserted well-roundedness of another family is an application/background reference, not a separate proof of the target theorem."
    }
  ],
  "dependency_proposals": [
    {
      "label": "external:EM",
      "role": "conditional",
      "source_quote": "\\newblock {\\em Duke Math. J.}, 71(1):181--209, 1993.",
      "explanation": "This is the intended external proof input, conditional on checking its exact statement and all hypotheses. Availability record: available_from=1993 at year resolution; date_evidence=supplied journal bibliography; date_basis=journal-publication year; version=Duke Math. J. 71(1):181\u2013209, journal version, not inspected. The recorded year precedes the 2015-07-28 cutoff. Exact theorem and hypotheses: identified only as Theorem 1.4; their original wording and agreement with the manuscript remain unresolved. No verified external result is imported here."
    },
    {
      "label": "external:GO",
      "role": "excluded",
      "source_quote": "It is also necessary in using the equidistribution result of Gorodnick-Oh \\cite{GO} in proving Lemma~\\ref{prox}.",
      "explanation": "The supplied citation concerns the separate equidistribution argument for prox, not proof of the target counting theorem. No GO theorem is imported."
    },
    {
      "label": "prox",
      "role": "excluded",
      "source_quote": "In the preceding section, we use heavily the following result from \\cite{EM}.",
      "explanation": "The narrative presents EM as an earlier-used input, not as a consequence of prox. The supplied prox statement provides no proof of emtheorem and is not used in this assessment."
    }
  ],
  "_public_provenance": {
    "origin": "returned_native_final_assessment_not_campaign_verdict",
    "model": "gpt-6.1-sol",
    "effort": "high",
    "provider": "openai",
    "manuscript_sha256": "8adb9681e2cd5e4bdcb292162a32796b00ae7b757a5a900fbd580d0bc939e34e",
    "review_input_sha256": "8fce1933e6ec1e6c61f8fd00a30cbc3090427b01630c7d752730bca73b928067",
    "original_assessment_sha256": "34ef40a9a5315b73f1c664a5914231386ae37f4ff0189ef0fb238c86190071f5",
    "note": "Original final assessment fields retained; embedded roadmap and private provider/run metadata omitted. No raw reasoning stream is exported."
  }
}