{
  "target_id": "narrative:92045",
  "proof_status": "needs_context",
  "summary": "The sentence announces two transitions: combining preceding lemmata into an estimate, and using that estimate in the proof of eq:cI41bnd. Neither transition can be checked because the lemmata, collected estimate, and application are absent. The supplied proposition identifies the intended downstream claim but is not a verified premise. This is a source-coverage limitation, not an identified mathematical defect. No external results are imported; later model knowledge is not evidence for this assessment under the 2012-05-24 cutoff.",
  "used_dependencies": [],
  "missing_context": [
    "Statements of the 'above lemmata,' including every hypothesis needed to combine them.",
    "The precise 'desired estimate' and its proof, with definitions of its objects and parameter ranges.",
    "The relevant portion of the proof of eq:cI41bnd showing how that estimate yields the stated bound under Q<XT, including the parameter relationships ensuring a fixed eta>0."
  ],
  "findings": [
    {
      "kind": "unverified",
      "source_quote": "Finally we collect the above lemmata into  our desired estimate",
      "explanation": "The packet supplies neither the referenced lemmata nor the statement and proof of the collected estimate. Consequently, their hypotheses, compatibility, and sufficiency for the announced aggregation cannot be checked."
    },
    {
      "kind": "unverified",
      "source_quote": "essential in the proof of \\eqref{eq:cI41bnd}",
      "explanation": "This identifies an intended downstream application, not a demonstrated inference. The exact remaining obligation is to exhibit the collected estimate and the step in the proposition's proof where it supplies the required bound."
    },
    {
      "kind": "context",
      "source_quote": "Assuming $Q<XT$, there is some $\\eta>0$ such that",
      "explanation": "The downstream proposition explicitly assumes Q<XT and asserts a power saving as N tends to infinity. Definitions of the integral and parameters, their admissible ranges and relationships, and the permitted dependence of eta and the implied constant are absent. Thus applicability under Q<XT and the claimed positive power saving cannot be checked. The introductory fixed primitive integral gasket does not supply these missing analytic hypotheses."
    },
    {
      "kind": "map_correction",
      "source_quote": "essential in the proof of \\eqref{eq:cI41bnd}",
      "explanation": "The dependency-list entry eq:cI41bnd should not be treated as a proof input to this narrative. The sentence presents that proposition as a downstream beneficiary of the missing estimate. The reference edge records an asserted application; it does not establish the aggregation or the proposition. If the graph convention is 'claim depends on input,' an edge from the proposition toward the estimate may be appropriate, but this prose alone is not the estimate or its proof."
    }
  ],
  "dependency_proposals": [
    {
      "label": "Preceding lemmata and collected estimate",
      "role": "proof_input",
      "source_quote": "Finally we collect the above lemmata into  our desired estimate",
      "explanation": "These are the announced mathematical inputs and intermediate conclusion. Their source statements and proof locations are needed; their validity is not established by the narrative."
    },
    {
      "label": "eq:cI41bnd",
      "role": "background",
      "source_quote": "\\cI_{Q}^{(=,=)}\n\\ll\nN\\, T^{2(\\gd-1)}\\, N^{-\\eta}",
      "explanation": "This supplies the headline bound that the missing estimate is intended to support. It is not evidence that the estimate follows from earlier lemmata, and its own proof remains outside the supplied coverage."
    }
  ],
  "_meta": {
    "pipeline": "claim_review",
    "model": "gpt-6.1-sol",
    "reasoning_effort": "high",
    "provider": "openai",
    "batch_job_id": "batch_6ad3ffda23ec8190bb2333acdad49cf6",
    "custom_id": "claim-87db2770661d87eee3c0971f",
    "source_sha256": "b9cefae7c32f848769eb2765fe5295b67031f15afb13ce6e552ef74838f82ee6",
    "usage": {
      "input_tokens": 7697,
      "input_tokens_details": {
        "cache_write_tokens": 7694,
        "cached_tokens": 0
      },
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      "output_tokens_details": {
        "reasoning_tokens": 800
      },
      "total_tokens": 9337
    },
    "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
  }
}