{
  "target_id": "cor:curve",
  "proof_status": "defective",
  "summary": "Conditional on Theorem thm:main-thm, the diagonal parameters and compactness thresholds are consistent: at time t/2 the required ell-range lies inside the displayed wider range, and X_{e^{-t/10}} supplies the required vector and exterior-square bounds. However, the stated Taylor approximation has an order-one, rather than exponentially small, remainder. The graph reduction and passage from fixed-ell estimates to simultaneous control also require justification. These are defects in the supplied argument, not counterexamples to the corollary. No external theorem is treated as verified.",
  "used_dependencies": [
    "cor:straight-line"
  ],
  "missing_context": [
    "The exact cited Bernik\u2013Kleinbock\u2013Margulis Theorem 1.4, its hypotheses, and dated evidence identifying an admissible pre-cutoff version.",
    "The convention for [1], the norms on R^3 and its exterior square, and Sobolev controls sufficient for approximation errors, discarded parameter sets, and translated test functions.",
    "A justified reduction from the stated smooth non-degenerate curves to the graph form used in the proof.",
    "A finite-parameter argument establishing simultaneous compactness, followed by a justified passage from good centers to the full parameter integral, including endpoints.",
    "The supplied text is the September 2024 v6 edition; its identity with the September 2022 submission is not established."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "with exponentially small error",
      "explanation": "At the chosen interval length h=e^{-t/2}s, a quadratic Taylor remainder is multiplied by e^t under conjugation by a_1(t), leaving size O(1). Explicitly, take the admissible non-degenerate curve psi(r)=(r,r^2), and denote the displayed approximating point by P_t(s)=z(2r_0)a(t/2)u(s)x_0. Direct matrix multiplication gives a_1(t)n(psi(r_0+e^{-t/2}s))Gamma=(I+s^2 E_{13})P_t(s), where E_{13} is the elementary matrix. This is a fixed nontrivial left shear, not an exponentially vanishing group error. The asserted approximation therefore does not provide the needed Sobolev error estimate. This calculation diagnoses the proof step; it does not disprove the final equidistribution statement."
    },
    {
      "kind": "error",
      "source_quote": "\\par Let us denote $\\psi(r) = (r, \\psi_2(r))$.",
      "explanation": "This is a substantive restriction, not merely notation. The hypotheses do not require a nonvanishing first coordinate derivative, or even a nonvanishing first derivative. For example, psi(r)=(r^2,r^3) on [0,1] satisfies the stated derivative-spanning condition, including at zero, but is stationary there. A reduction to graph charts must address exceptional parameters, parameter densities, and uniform quantitative bounds. None is supplied."
    },
    {
      "kind": "error",
      "source_quote": "Let us remove all \n  $\\fm_{\\ell}$ from $[1]$ and get a subset $\\fm \\subset [1]$. Then we have that \n  $$\\Leb \\left([1]\\setminus \\fm\\right) = O(t e^{-\\eta_2 t}),$$",
      "explanation": "The preceding estimate is for each fixed real ell. Even a uniform bound on individual measures does not bound their uncountable union. The O(t) factor requires an explicit finite discretization and control between grid parameters, including changes to the compactness threshold in both the vector and exterior-square representations. This is a missing inference, not a demonstrated counterexample to the claimed union bound."
    },
    {
      "kind": "unverified",
      "source_quote": "By \\cite[Theorem 1.4]{Bernik-Kleinbock-Margulis}, we have that \n  $$\\Leb\\left(\\fm_{\\ell}\\right) = O(e^{-\\eta_2 t})$$",
      "explanation": "The cited theorem and its hypotheses are absent. The remaining obligation is to verify its applicability to the t- and ell-dependent family a_0(ell)a_0(t/2)a_1(t/2)z(-psi_2'(r))n(psi(r)), uniformly over both signs of ell, with all lattice and exterior-square conditions required by X_{e^{-t/10}}. Non-degeneracy alone is not a supplied verification of that application. Historical record for this unverified input: available_from unresolved; date_evidence absent; date_basis unresolved; version unresolved; exact theorem and hypotheses unavailable. A version available by 2022-09-26 is required. Model knowledge cannot itself certify historical availability, and none is used to close this step."
    },
    {
      "kind": "map_correction",
      "source_quote": "\\begin{proof}[Proof assuming Theorem \\ref{thm:main-thm}]",
      "explanation": "The excerpt contains an explicit conditional proof despite the missing-anchor metadata; that metadata is a mapping limitation, not another proof defect. The actual proof inputs include thm:main-thm and the cited Bernik\u2013Kleinbock\u2013Margulis theorem. Corollary cor:straight-line principally supplies notation here, while Shah_2 supports only the historical remark about an ineffective result."
    }
  ],
  "dependency_proposals": [
    {
      "label": "thm:main-thm",
      "role": "conditional",
      "source_quote": "\\begin{proof}[Proof assuming Theorem \\ref{thm:main-thm}]",
      "explanation": "Explicitly assumed proof input. Its statement is supplied, but its proof is outside this bounded review and is not certified here."
    },
    {
      "label": "external:Bernik-Kleinbock-Margulis:Theorem-1.4",
      "role": "conditional",
      "source_quote": "By \\cite[Theorem 1.4]{Bernik-Kleinbock-Margulis}",
      "explanation": "Proposed quantitative proof input absent from the frozen dependency list. Its statement, applicability, uniformity, and pre-cutoff availability remain unverified."
    },
    {
      "label": "cor:straight-line",
      "role": "background",
      "source_quote": "Let us use the same notation as in Corollary \\ref{cor:straight-line}.",
      "explanation": "Supplies a_1 and n. The straight-line equidistribution conclusion is not invoked as a proof input for the displayed curved-arc argument."
    },
    {
      "label": "external:Shah_2",
      "role": "background",
      "source_quote": "The ineffective version of Corollary \\ref{cor:curve} was proved in \\cite{Shah_2} using Ratner's theorem.",
      "explanation": "Historical background, not support for the exponential rate. The bibliography identifies a 2009 journal article, predating the cutoff, but no exact theorem or hypotheses were inspected or imported."
    }
  ],
  "_meta": {
    "model": "gpt-6.1-sol",
    "effort": "high",
    "prompt_profile": "claim_review-v2",
    "prompt_profile_sha256": "a41def3eca770d93f8dbccbdeb9898d6be192d07d7c119138273122fd96954fd"
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}