{
  "target_id": "eq:cIQ1def",
  "proof_status": "locally_supported",
  "summary": "The displayed estimate follows conditionally from the supplied dependency statements. The modulus reduction, frequency count, power U^6, and archimedean integral bound are consistent. This does not establish the dependencies: uniform frequency truncation and the modular estimate\u2019s applicability to both diagonal and off-diagonal pairs remain prerequisites requiring context.",
  "used_dependencies": [
    "eq:IQbnd1",
    "eq:Mis",
    "eq:cJfbnd",
    "eq:modBndaNeqAp",
    "eq:nmRest",
    "eq:tilQis"
  ],
  "missing_context": [
    "The definition eq:cSdef and full hypotheses preceding eq:modBndaNeqAp, including applicability when a=a'.",
    "A uniform, summable frequency-tail estimate for eq:nmRest over the entire beta interval, together with the cutoff function\u2019s support and smoothness assumptions.",
    "The upstream form-size and discriminant hypotheses supporting eq:cJfbnd, including uniformity of its implied constant over the family.",
    "If the stationary-phase citation is to establish a prerequisite rather than remain conditional: its exact theorem and hypotheses, available_from, date_evidence, date_basis, and version."
  ],
  "findings": [
    {
      "kind": "context",
      "source_quote": "\\left({q/ q_{0}}\\right)^{2}",
      "explanation": "The modulus simplification is valid: q0=q/(u^2,q), so (q/q0)^2\u2264u^4. Since q0 divides q, each gcd involving q0 is bounded by its counterpart involving q. The four frequency variables each have O(u) possible values under eq:nmRest. Consequently the remaining u-factor is U sum_{1\u2264u<U} u^(-4)u^4 O(u^4)=O(U^6). This checks the claimed power without establishing the modular or truncation premises."
    },
    {
      "kind": "map_correction",
      "source_quote": "The claim then follows immediately from \\eqref{eq:Mis} and $Q<X$.",
      "explanation": "The final integral bound needs neither M=XT nor Q<X once the preceding displayed inequality is accepted. For any b>0, splitting at beta0=1/(TX^2) gives integral_{-b}^b min(X^2,1/(|beta|T))^2 d beta\u22644X^2/T. Thus eq:Mis is an upstream input to the frequency reduction, rather than a necessary direct input to this final integration. Q<X is compatible with the section\u2019s setup and frequency analysis; no further restriction on Q is needed for this integral estimate."
    },
    {
      "kind": "unverified",
      "source_quote": "has negligible contribution unless",
      "explanation": "The proof uses eq:nmRest as a uniform effective cutoff throughout |beta|<1/(qM). The packet does not quantify the summed discarded tail or show that it is absorbed by the claimed bound. In particular, the stationary-frequency condition alone does not justify discarding all nonstationary frequencies near beta=0 without also controlling the Fourier scale 1/X. The exact remaining prerequisite is a uniform truncation or weighted-tail estimate whose four-frequency cost is O(u^4). This is a limitation of the unverified input, not an algebraic error in the target deduction."
    },
    {
      "kind": "context",
      "source_quote": "with the former being exactly equal to $\\cS$ in \\eqref{eq:cSdef}.",
      "explanation": "The source asserts the required identification, but eq:cSdef and the preceding hypotheses of eq:modBndaNeqAp are absent. They are needed to check normalization, coprimality, and uniformity in n,m,n',m'. The quoted modular lemma states no explicit a\u2260a' restriction, so its label alone does not exclude the diagonal. Conditional on applicability to a=a', that case is valid using (0,q)=q."
    },
    {
      "kind": "context",
      "source_quote": "By (non)stationary phase (see, e.g., \\cite[\\S8.3]{IwaniecKowalski}),",
      "explanation": "No external theorem is imported to establish this premise. The cited text\u2019s exact statement, hypotheses, version, and availability by the recorded cutoff 2012-05-24 have not been verified. Later knowledge cannot be excluded from model weights, but none is used as evidence to close this obligation."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:cSdef",
      "role": "proof_input",
      "source_quote": "with the former being exactly equal to $\\cS$ in \\eqref{eq:cSdef}.",
      "explanation": "This definition supplies the normalization bridge between the modular bracket in eq:IQbnd1 and the estimate eq:modBndaNeqAp. Its omission is a mapping/context limitation."
    },
    {
      "label": "eq:Mis",
      "role": "background",
      "source_quote": "Here we used $|\\gb|\\ll (qM)^{-1}$ with $M$ given by \\eqref{eq:Mis}.",
      "explanation": "Retain its upstream relevance to eq:nmRest, but distinguish that role from the final integral bound, which holds for every integration radius."
    },
    {
      "label": "IwaniecKowalski, \u00a78.3",
      "role": "excluded",
      "source_quote": "\\cite[\\S8.3]{IwaniecKowalski}",
      "explanation": "Not used as an independently verified external proof input. Historical admissibility and the exact applicable result remain unverified."
    }
  ],
  "_meta": {
    "pipeline": "claim_review",
    "model": "gpt-6.1-sol",
    "reasoning_effort": "high",
    "provider": "openai",
    "batch_job_id": "batch_6ad4023e64a48190a34e245075554b92",
    "custom_id": "claim-5ec2b6efbf100c7c8a3610ff",
    "source_sha256": "716b8c7160c2eafe66c42c6c929b60ed0cce96976e116b6a9c3edc2bf6bda73f",
    "usage": {
      "input_tokens": 13816,
      "input_tokens_details": {
        "cache_write_tokens": 13813,
        "cached_tokens": 0
      },
      "output_tokens": 3350,
      "output_tokens_details": {
        "reasoning_tokens": 2070
      },
      "total_tokens": 17166
    },
    "max_output_tokens": 24576,
    "prompt_profile": "claim_review-v2",
    "prompt_profile_sha256": "4aa7aa2ef9a18c790385b9e0e0f7ad6c44ff664be253c2d9f790a1469765c3a4",
    "review_policy": {
      "schema": 1,
      "cutoff_date": "2012-05-24",
      "basis": "journal_received",
      "evidence": "https://link.springer.com/article/10.1007/s00222-013-0475-y : Received 24 May 2012; publisher.html retained locally",
      "status": "recorded",
      "rule": "external-results-available-by-cutoff"
    },
    "chronology_prompted": true
  }
}