{
  "target_id": "eq:cI42bnd",
  "proof_status": "needs_context",
  "summary": "Conditional on the two supplied lemmas, the displayed algebra is consistent, including X^{-1/8}T^{35/8}=N^{-29/1600}. The section supplies Q\u2265X, and a=a' makes the stated inequality of shifted-form values equivalent to the inequality required by eq:modBndaEqAp. However, the packet omits the growth restriction on U and the ensemble and modulus definitions needed for the intermediate estimates. Consequently, the claimed power saving is not yet locally established. All three supplied dependency edges are relevant; their premises have not been independently established. No external result is imported: the recorded cutoff is 2012-05-24, and model recall is not treated as dated evidence.",
  "used_dependencies": [
    "eq:TXNis",
    "eq:cIQ3def",
    "eq:modBndaEqAp"
  ],
  "missing_context": [
    "The definition and N-growth restriction of U, sufficient to make N^{-29/1600}U^{13}T^{\u03b5} a fixed negative power of N.",
    "The omitted modulus definition eq:tilQis, establishing q\u2080|q and q/q\u2080\u2264u\u00b2.",
    "The construction of F and its cardinality estimate, including multiplicities.",
    "The uniform coefficient bounds for forms in F and the identification of their parameter a with the stated nonzero quantity of order T.",
    "The definition of the subpiece I_Q^(=,\u2260), confirming that its restrictions are precisely a=a' and unequal form values as used in the first proof line."
  ],
  "findings": [
    {
      "kind": "unverified",
      "source_quote": "so we have again a power savings, as claimed.",
      "explanation": "The final factor is N^{-29/1600}U^{13}T^{\u03b5}. Thus a positive saving requires U^{13}T^{\u03b5}\u226aN^{29/1600\u2212\u03b7} for some \u03b7>0. For example, if U\u226aN^\u03ba, one needs 13\u03ba+\u03b5/100<29/1600. No definition or growth restriction for U is supplied. This is the exact outstanding obligation at the final transition, not an arithmetic error in the displayed N-exponent."
    },
    {
      "kind": "context",
      "source_quote": "\\sum_{\\tilde q_{1}\\mid a^{2}\\atop \\tilde q_{1}\\ll Q}",
      "explanation": "The modular reduction needs q\u2080|q and q/q\u2080\u2264u\u00b2 to replace (q/q\u2080)^5 by u^{10}. It also needs a\u22600 and |a|\u226aT. Then (a\u00b2,q\u2080) can be majorized by a divisor sum over divisors of a\u00b2 that divide q; summing q\u2248Q contributes O(Q) per divisor, and the elementary divisor bound gives T^{\u03b5}. The definition of q\u2080 is absent. The introductory requirement a_\u03b3\u2248T is stated, but its link to every indexed form parameter a is not supplied."
    },
    {
      "kind": "context",
      "source_quote": "\\sum_{\\ff,\\ff'\\in\\fF}",
      "explanation": "The next displayed line replaces this unrestricted pair sum by a factor T^{2\u03b4}. This needs an ensemble cardinality estimate such as #F\u226aT^\u03b4, or an estimate with arbitrarily small losses absorbed into T^{\u03b5}, counting any multiplicities. Neither the definition of F nor that estimate is included. Dropping a'=a after applying an a'-independent nonnegative majorant is otherwise legitimate."
    },
    {
      "kind": "context",
      "source_quote": "$\\ff(m,n)\\ll T (UQ/X)^{2}$ and $Q<XT$.",
      "explanation": "The value bound requires coefficient control for every form in F, including its constant term. Such control is asserted here but cannot be checked without the form construction. Conditional on that control and the preceding inputs, put L=UQ/X: summing u^{6} gives U^{7}, the four frequency variables give L^4, and the value-difference factor gives T^{1/2}L. The result is U^{13}X^{-2}T^{2\u03b4\u22121/2+\u03b5}Q^{31/8}. Using Q<XT yields exactly the displayed X\u00b2T^{2\u03b4\u22121}\u00b7X^{-1/8}T^{35/8}U^{13}T^{\u03b5}."
    }
  ],
  "dependency_proposals": [
    {
      "label": "Definition and growth restriction for U (anchor not supplied)",
      "role": "proof_input",
      "source_quote": "U^{13}",
      "explanation": "A quantitative upper bound on U is essential to the final power-saving conclusion and should be an explicit proof dependency."
    },
    {
      "label": "eq:tilQis",
      "role": "proof_input",
      "source_quote": "recall the notation \\eqref{eq:tilQis}",
      "explanation": "The omitted modulus reduction is needed for the u^{10} bound and the divisor majorization."
    },
    {
      "label": "Ensemble definition and cardinality estimate (anchor not supplied)",
      "role": "proof_input",
      "source_quote": "\\sum_{\\ff,\\ff'\\in\\fF}",
      "explanation": "The transition to T^{2\u03b4} requires an explicit bound on the number of indexed forms."
    },
    {
      "label": "Uniform form-coefficient and parameter bounds (anchor not supplied)",
      "role": "proof_input",
      "source_quote": "$\\ff(m,n)\\ll T (UQ/X)^{2}$",
      "explanation": "The proof needs uniform coefficient bounds and nonzero |a|\u226aT, not merely the scale relation TX\u00b2=N."
    }
  ],
  "_meta": {
    "pipeline": "claim_review",
    "model": "gpt-6.1-sol",
    "reasoning_effort": "high",
    "provider": "openai",
    "batch_job_id": "batch_6ad4023e64a48190a34e245075554b92",
    "custom_id": "claim-fff063b2f32949ebafd4a384",
    "source_sha256": "d8932a14df8232e5f747b73fa61656f4ce767ecd411d7870738e32260f9a740d",
    "usage": {
      "input_tokens": 10589,
      "input_tokens_details": {
        "cache_write_tokens": 10586,
        "cached_tokens": 0
      },
      "output_tokens": 3381,
      "output_tokens_details": {
        "reasoning_tokens": 2070
      },
      "total_tokens": 13970
    },
    "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
  }
}