{
  "target_id": "narrative:86590",
  "proof_status": "needs_context",
  "summary": "The passage announces a saving for the equal-parameter, equal-value subcase and attributes it to supplementary lemmas rather than to \\cS itself. The packet supplies neither the relevant definitions nor those lemmas, so the asserted transition cannot be verified. This is a scope limitation, not an identified proof defect. No external result is imported; latent model knowledge is not treated as evidence of historical availability.",
  "used_dependencies": [],
  "missing_context": [
    "Definitions and permitted ranges of a,a',f,f',m,n,m',n', including any structural restrictions on the forms.",
    "Definition of \\cS, the expression being estimated, its normalization and scale parameters, and the preceding case analysis explaining why this subcase remains.",
    "Statements and applicable proofs of the supplementary lemmas, together with the calculation converting their conclusions into the required saving."
  ],
  "findings": [
    {
      "kind": "context",
      "source_quote": "Finally we need some  savings in the case $a=a'$ and $f(m,-n)=f'(m',-n')$.",
      "explanation": "The parameters a,a', forms f,f', variables m,n,m',n', and their ranges are not defined in the supplied excerpts. Consequently, the hypotheses identifying this subcase cannot be checked, nor can one determine what quantity must exhibit a saving or how large that saving must be."
    },
    {
      "kind": "unverified",
      "source_quote": "This will no longer come from $\\cS$ itself, but from the following supplementary lemmata.",
      "explanation": "Neither \\cS and its preceding estimates nor the supplementary lemmas are supplied. The remaining obligation is to show that the subsequent lemmas apply under both displayed equalities and yield the quantitative saving required by the surrounding argument. The passage supports only an announcement of that strategy, not its execution or the density-one headline claim."
    }
  ],
  "dependency_proposals": [
    {
      "label": "Definitions and preceding estimates for the equal-parameter, equal-value subcase",
      "role": "proof_input",
      "source_quote": "Finally we need some  savings in the case $a=a'$ and $f(m,-n)=f'(m',-n')$.",
      "explanation": "These missing inputs are necessary to formulate and check the announced local estimate. No corresponding dependency is supplied in the frozen map."
    },
    {
      "label": "Unidentified supplementary lemmas following the target",
      "role": "conditional",
      "source_quote": "the following supplementary lemmata.",
      "explanation": "The prose explicitly proposes these lemmas as the source of the saving. Their identities, hypotheses, conclusions, and applicability remain unverified; this proposal does not establish them or alter the frozen map."
    }
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