{
  "target_id": "eq:cIQ11bnd",
  "proof_status": "needs_context",
  "summary": "The divisor-sum argument supports the claimed intermediate estimate conditionally on the supplied lemmas and missing family/range assumptions. There is a harmless strict-inequality error in the lcm comparison. The final N-power saving additionally requires the omitted relation among T, X, and N. No external results are imported; later-trained model knowledge is not treated as evidence of availability by the recorded cutoff, 2012-05-24.",
  "used_dependencies": [
    "eq:His",
    "eq:cIQ1def",
    "eq:fuIs",
    "eq:spec3"
  ],
  "missing_context": [
    "Definitions of a, the family of forms, and its multiplicity-preserving relationship to the gamma-family in eq:spec3; verification that the family parameters meet that lemma's hypotheses.",
    "Bounds ensuring a,a' are nonzero integers of polynomial size in T and every divisor modulus h used in eq:spec3 is less than N.",
    "The definition of the gcd-restricted contribution and the pairwise/localized estimate underlying eq:cIQ1def.",
    "The statement of eq:TXNis, including a fixed positive-power relationship between T and N sufficient to convert the T-saving into N^{-\u03b7}."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "where we used $[n,m]>(nm)^{1/2}$.",
      "explanation": "The strict inequality is false when n=m. Elementary correction (new local repair): for positive integers, [n,m]=nm/(n,m) and (n,m)\u2264sqrt(nm), so [n,m]\u2265sqrt(nm). This non-strict inequality suffices: writing L=[d,d'], the q-sum is O(1/L), and sqrt(dd')/L\u22641. Thus the typo does not invalidate the estimate."
    },
    {
      "kind": "context",
      "source_quote": "Assume $Q<X$. For $\\square\\in\\{=,\\neq\\}$, we have",
      "explanation": "The section supplies Q0\u2264Q<X, satisfying the displayed lemma's Q<X hypothesis. However, the lemma as stated bounds the entire off-diagonal contribution, not explicitly its gcd-restricted subpieces. The first line of this proof needs its pairwise/localized version. The preceding expansion suggests such a version may be available, but the lemma's proof and precise definition of the restricted contribution are omitted."
    },
    {
      "kind": "context",
      "source_quote": "Writing $(a,a')=h>H$,",
      "explanation": "The divisor manipulations require nonzero integer coefficients a,a' of polynomially bounded size in T. Under these conditions, replacing exact gcd conditions by divisibility only enlarges the nonnegative majorant; the two divisor counts contribute T^\u03b5. The later sum over h|a contributes another divisor factor, which can be absorbed by rescaling \u03b5. The required coefficient bounds are not supplied."
    },
    {
      "kind": "unverified",
      "source_quote": "Apply \\eqref{eq:spec3} to the innermost sum, getting",
      "explanation": "Conditionally on eq:spec3, the application gives at most T^\u03b4 h^{-\u03b70}, hence T^\u03b4 H^{-\u03b70}, provided every relevant modulus satisfies 1\u2264h<N. It also requires the forms indexed here to correspond to the gamma-family in eq:spec3 with the same counting multiplicities and coefficient a'=\u27e8e1,gamma v0\u27e9. With that identification, eq:spec3 at q=1 also supplies the outer family bound O(T^\u03b4). These prerequisites and the lemma's specified family parameters are not fully documented in the packet."
    },
    {
      "kind": "context",
      "source_quote": "By \\eqref{eq:His} and \\eqref{eq:fuIs}, this is a power savings, as claimed.",
      "explanation": "The parameter algebra checks: U^6 H^{-\u03b70}=U^{-4}=Q0^{-\u03b70\u00b2/25}=T^{-c}, where c=(\u03b4\u2212\u03981)\u03b70\u00b2/500>0 using the supplied definition of Q0. Thus the proved bound is X\u00b2 T^{2\u03b4\u22121+\u03b5\u2212c}. Its ratio to N T^{2(\u03b4\u22121)} is (X\u00b2T/N)T^{\u03b5\u2212c}. To obtain N^{-\u03b7}, one still needs the omitted scale relation. For example, if the source states N=TX\u00b2 and T=N^\u03c4 for fixed \u03c4>0, any \u03b5<c yields \u03b7=\u03c4(c\u2212\u03b5)>0. Those scale assumptions are not established by this packet."
    },
    {
      "kind": "map_correction",
      "source_quote": "\\be\\label{eq:Q0is}\nQ_{0}= T ^{(\\gd-\\gT_{1})/20},",
      "explanation": "The listed dependencies omit the definition making Q0 a fixed positive power of T. They also omit the T,X,N normalization needed for the proposition's stated scale. These are additional proof inputs, not consequences of eq:His and eq:fuIs alone."
    }
  ],
  "dependency_proposals": [
    {
      "label": "eq:Q0is",
      "role": "proof_input",
      "source_quote": "Q_{0}= T ^{(\\gd-\\gT_{1})/20},",
      "explanation": "Needed to express U^{-4} as a fixed positive power saving in T; the accompanying 1<\u03981<\u03b4 ensures positivity."
    },
    {
      "label": "eq:TXNis",
      "role": "conditional",
      "source_quote": "Then \\eqref{eq:fuIs} and \\eqref{eq:TXNis}\nimply the claimed power savings.",
      "explanation": "This nearby reference identifies a likely missing normalization input, but its statement is absent. Its exact content must establish the final comparison with N T^{2(\u03b4\u22121)}."
    },
    {
      "label": "eq:fFdef",
      "role": "conditional",
      "source_quote": "the product structure of the family $\\fF$\nin \\eqref{eq:fFdef}",
      "explanation": "Request this definition to check coefficient bounds, multiplicities, and compatibility with the family and parameter hypotheses of eq:spec3."
    }
  ],
  "_meta": {
    "pipeline": "claim_review",
    "model": "gpt-6.1-sol",
    "reasoning_effort": "high",
    "provider": "openai",
    "batch_job_id": "batch_6ad4049ce9288190b11b5b6f4c6caac5",
    "custom_id": "claim-ef4446de67fa0e682936ebdf",
    "source_sha256": "5cb4158b61b06be75e9edb2614eb39824c9e88f2bf474b5f61a335a1e0066b38",
    "usage": {
      "input_tokens": 12314,
      "input_tokens_details": {
        "cache_write_tokens": 12311,
        "cached_tokens": 0
      },
      "output_tokens": 2976,
      "output_tokens_details": {
        "reasoning_tokens": 1552
      },
      "total_tokens": 15290
    },
    "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
  }
}