{
  "target_id": "thm:main-thm",
  "proof_status": "defective",
  "summary": "The printed theorem has a direct boundary counterexample under the standard Euclidean or coordinate-maximum lattice norms. Independently, the retained-set construction invokes a quantitative nondivergence statement that is false as printed and whose hypotheses do not match that application. The diagonal conjugacy is correct, and the structured-component stopping restriction permits the proposed terminal dimension. These checks do not validate the remaining dependencies or other editions.",
  "used_dependencies": [
    "eq:define-a",
    "eq:define-a0",
    "prop:quantitative-nondivergence",
    "prop:initial-bound-dimension",
    "prop:dimension-improvement-structured-component",
    "prop:high-dim-decomposition",
    "prop:high-dimension-to-equidistribution-a"
  ],
  "missing_context": [
    "Exact lattice and exterior-square norm definitions and normalizations.",
    "A valid retained-set estimate under the actual negation of alternative (2), including uniform control over the two continuous parameter ranges.",
    "An all-centers initial-goodness estimate and a probability-consistent terminal decomposition, including the basepoints omitted from the displayed integrals defining \u03bc_k.",
    "Sobolev norm specifications sufficient to control the C^1 approximation errors and the conventions for normalized restrictions and orbit pieces.",
    "For any external result actually used: exact theorem and hypotheses, available_from, date_evidence, date_basis, and the version containing it. Pretrained knowledge is not dated source evidence."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "\\par (2) $ a_0 (\\ell )x_0 \\not\\in X_{e^{-|\\ell|/3}}$ for some  $ 0.99 t \\le  |\\ell| \\le 1.01 t$.",
      "explanation": "Statement counterexample under standard Euclidean or coordinate-maximum norms: take x_0=\u0393. For every real \u2113, the minimum over nonzero integral vectors and exterior-square vectors after applying a_0(\u2113) is exactly exp(-|\u2113|/3). For \u2113>0 the exterior vector e_2\u2227e_3 attains it; for \u2113<0 the vector e_1 attains it. The defining non-strict inequality therefore puts a_0(\u2113)\u0393 inside the required set, so (2) never holds. But every a(t)u(r)\u0393 contains e_1. The lattices containing e_1 form a closed proper subset Z of X. Choose a nonnegative, nonzero smooth compactly supported f in an open set disjoint from Z. Its orbit integral is zero for every t, whereas its Haar integral is positive, contradicting (1) as t tends to infinity. This is a counterexample to the printed statement with these norm conventions, not merely a proof gap."
    },
    {
      "kind": "context",
      "source_quote": "\\|g \\vv v\\| \\ge \\varpi",
      "explanation": "The supplied text does not specify the norms on R^3 and its exterior square. Their normalization matters at the exact boundary used above; it cannot be dismissed as equivalence of norms. The counterexample explicitly uses conventional norms with unit coordinate and exterior-coordinate basis vectors. A different intended normalization requires a separate statement check."
    },
    {
      "kind": "error",
      "source_quote": "$ a_0(-s)x ,  a_0(s)x \\not\\in X_{e^{- s/3 }}$",
      "explanation": "The nondivergence proposition is false as printed. Fix s=1 and let x=diag(\u03b4,\u03b4^{-1/2},\u03b4^{-1/2})\u0393. Choose \u03b4 sufficiently smaller than \u03b2. Both endpoint lattices lie outside X_{exp(-1/3)}, while a(1)u(r)x contains the vector \u03b4e_1 for every r. Thus the bad set has full interval measure for arbitrarily small \u03b2, contradicting a uniform O(\u03b2^\u03b1) estimate. Moreover, negating main-theorem alternative (2) gives endpoint membership, not the printed nonmembership hypotheses. Consequently this proposition cannot justify the retained-set estimate even conditional on its being accepted as a premise. Changing the hypothesis would be a proposed correction, not a proof of the printed claim."
    },
    {
      "kind": "unverified",
      "source_quote": "According to this definition, Proposition \\ref{prop:initial-bound-dimension} says that",
      "explanation": "The initial-dimension proposition controls neighborhoods centered at x\u2208F, whereas the definition of goodness quantifies over every center in X. The stated implication needs a recentering/enlargement argument with the same required bound, or an explicitly quantified stronger estimate. It is not immediate from the supplied proposition."
    },
    {
      "kind": "error",
      "source_quote": "\\left| \\int f \\dd a(2s')_\\ast\\mu_k - \\int f \\dd a(2s')_\\ast \\bar{\\mu}_k \\right| \\ll \\beta \\|f\\|_{C^1}.",
      "explanation": "The high-dimensional decomposition proof establishes smooth-test-function proximity after replacing orbit pieces. This does not establish its stated exact positive-measure decomposition with an O(\u03b2)-weight exceptional probability measure. Also, summing normalized restrictions over a cover requires disjointization or overlap weights. An approximate decomposition could support a different terminal inference, but the printed exact identity is not obtained here."
    },
    {
      "kind": "unverified",
      "source_quote": "The statement can be proved by following the proof of \\cite[Proposition 4.2]{LM-preprint2021} step by step.",
      "explanation": "The terminal equidistribution input is not locally proved. Its applicability requires verification of the cited proposition and the adaptation to this group, scale-dependent \u03b2, thickness condition, and stated dimension bound. Chronology is unresolved: the supplied bibliography identifies a 2023 journal publication, later than the 2022-09-26 cutoff; the citation key alone does not establish an admissible earlier version. No external theorem is imported in this assessment."
    },
    {
      "kind": "map_correction",
      "source_quote": "\\begin{proof}[Proof assuming Theorem \\ref{thm:main-thm}]",
      "explanation": "The straight-line corollary is downstream of the target, not a proof input. Likewise, the BKM citation occurs in the expanding-curve corollary proof. Their inclusion in the broad source scope does not create dependencies proving the main theorem."
    }
  ],
  "dependency_proposals": [
    {
      "label": "cor:straight-line",
      "role": "excluded",
      "source_quote": "\\begin{proof}[Proof assuming Theorem \\ref{thm:main-thm}]",
      "explanation": "Reverse the proposed proof direction: the target proves this corollary."
    },
    {
      "label": "external:Bernik-Kleinbock-Margulis",
      "role": "excluded",
      "source_quote": "By \\cite[Theorem 1.4]{Bernik-Kleinbock-Margulis}",
      "explanation": "This invocation belongs to the downstream curve corollary, not the target proof."
    },
    {
      "label": "external:LM-preprint2021",
      "role": "conditional",
      "source_quote": "\\cite[Proposition 4.2]{LM-preprint2021}",
      "explanation": "Potential input to the terminal equidistribution proposition. available_from: unresolved for the needed preprint; date_evidence: supplied bibliography gives 2023; date_basis: journal publication year only; version: pre-cutoff version unresolved. The exact proposition and its hypotheses remain uninspected."
    }
  ],
  "_meta": {
    "model": "gpt-6.1-sol",
    "effort": "high",
    "prompt_profile": "claim_review-v2",
    "prompt_profile_sha256": "a41def3eca770d93f8dbccbdeb9898d6be192d07d7c119138273122fd96954fd"
  }
}