{
  "target_id": "narrative:135233",
  "proof_status": "needs_context",
  "summary": "The target is a continuation of a table, not a narrative argument or proof. Direct inspection supports basic consistency of its parameter data: disregarding punctuation artifacts, the tuples have nine entries, lie in [0,1), have disjoint alpha and beta multisets, and give products of cyclotomic polynomials. Each paired entry is obtained by adding 1/2 modulo 1 to both parameter multisets. These checks do not establish primitivity, hyperbolic signature, the numerical columns, family membership, or the apparent thinness assertions marked Yes. The omitted table setup and relevant internal criteria are needed. No substantive mathematical error is established by this packet.",
  "used_dependencies": [],
  "missing_context": [
    "The omitted section setup, table caption, column headings, and definitions of Yes, ?, and the asterisk.",
    "Definitions and admissible parameter ranges for the displayed families, together with the classification result used to assign family membership or Sporadic status.",
    "The definition and verification of primitivity, and the signature criterion applied to each displayed group.",
    "The numerical invariant's definition, lattice construction, normalization, and reproducible calculations.",
    "The exact thinness certificate and row-specific witnesses establishing its hypotheses for each Yes entry.",
    "The result relating the paired common half-shifts at the group and lattice level, if conclusions are transferred between the two columns.",
    "For any external proof input ultimately invoked: its exact theorem and hypotheses, version, available_from, date_evidence, and date_basis establishing availability by 2013-08-11. No external result is imported here. Model knowledge may include later developments, but none is used to close these obligations."
  ],
  "findings": [
    {
      "kind": "map_correction",
      "source_quote": "$\\mathcal N_1(1,9,9)$&Yes",
      "explanation": "The mapped target consists of table rows, despite its narrative designation. Its column headings, caption, legend, and introductory explanation are outside the supplied slice. It should be assessed as tabulated claims rather than as a proof of the introductory headline results. The empty dependency map does not establish that these claims are independent of earlier results."
    },
    {
      "kind": "unverified",
      "source_quote": "$\\mathcal M_2(3,9)$&Yes",
      "explanation": "The meaning of Yes is not supplied explicitly; the introduction suggests thinness. If so, this entry needs an identified certificate and verification of every hypothesis for the stated group and lattice. No such verification is present here. The table can record a conclusion, but does not itself justify the transition from parameter data to thinness."
    },
    {
      "kind": "context",
      "source_quote": "$\\mathcal M_3(1,9)$&?$^\\ast$",
      "explanation": "The distinction between ?, ? with an asterisk, and Yes requires the omitted legend. In particular, question marks must not be interpreted as assertions of arithmeticity or as counterexamples to a thinness criterion."
    },
    {
      "kind": "unverified",
      "source_quote": "2, 2, 112",
      "explanation": "The numerical column is undefined in the supplied text. Its values cannot be checked without knowing the invariant, the integral lattice and normalization, and the computation used. Likewise, the family labels and Sporadic classifications require their definitions and parameter restrictions. These rows alone do not establish completeness of the n=9 classification."
    },
    {
      "kind": "context",
      "source_quote": "We assume throughout that $H(\\alpha,\\beta)$ is primitive -- see Section~\\ref{beuksum}.",
      "explanation": "Primitivity is an explicit standing hypothesis, but neither its definition nor a verification for these rows is supplied. The target also needs the criterion establishing signature (8,1). Rationality, the range restriction, cross-disjointness, cyclotomic factorization, and the common half-shift pairing can be checked directly from the data; they do not by themselves establish either missing property."
    },
    {
      "kind": "context",
      "source_quote": "and the $\\alpha$'s and $\\beta$'s are distinct",
      "explanation": "The table repeatedly includes multiplicities such as three zeros or three copies of 1/2. Thus this introductory phrase must be distinguished from pairwise distinctness within each tuple. If it means only that no alpha equals a beta, the displayed data satisfy it. If it means within-tuple distinctness, these rows fall outside that introductory restriction, and the extension to repeated parameters needs the omitted setup. This ambiguity is not sufficient to establish a defect."
    }
  ],
  "dependency_proposals": [
    {
      "label": "table_setup_and_legend",
      "role": "proof_input",
      "source_quote": "$\\mathcal N_4(1,7,9)$&?$^\\ast$",
      "explanation": "The omitted setup is necessary to determine exactly what the table asserts and how its status markers should be interpreted."
    },
    {
      "label": "theo",
      "role": "conditional",
      "source_quote": "Our first result is the determination of the $\\alpha,\\beta$'s for which $G(\\alpha,\\beta)$ is hyperbolic, see Theorem~\\ref{theo}.",
      "explanation": "This is a proposed internal input for classification and hyperbolicity, not an established premise in this packet. Its statement and the definitions of its families are required."
    },
    {
      "label": "beuksum",
      "role": "conditional",
      "source_quote": "We assume throughout that $H(\\alpha,\\beta)$ is primitive -- see Section~\\ref{beuksum}.",
      "explanation": "The referenced section is needed to interpret and verify the standing primitivity hypothesis and any associated signature test."
    },
    {
      "label": "quadform",
      "role": "conditional",
      "source_quote": "In Section~\\ref{quadform} we determine a commensurable quadratic sublattice explicitly which facilitates many further calculations.",
      "explanation": "This may supply the lattice underlying the numerical column and thinness tests. Its relevance cannot be confirmed without the table headings and section text."
    },
    {
      "label": "thinness_certificate",
      "role": "conditional",
      "source_quote": "Our main result is a robust certificate for showing that certain $H(\\alpha,\\beta)$'s are thin.",
      "explanation": "If Yes denotes certified thinness, the exact internal certificate and its row-specific application are necessary proof inputs. This introductory announcement is not itself that certificate."
    }
  ],
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      "basis": "journal_received",
      "evidence": "Publisher PDF p.1617 states Received August 11, 2013; publisher.html independently records Submitted 11 August 2013. Reviewed native source is arXiv v2 dated 2013-08-12; that does not advance the external proof-input cutoff. ArXiv v1 is 2013-05-03. Later sources may provide retrospective context, not proof inputs for the original paper.",
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