{
  "target_id": "narrative:105412",
  "proof_status": "defective",
  "summary": "The narrative describes a conditional route from classification and graph certificates to the tables, but the supplied excerpts do not verify completeness or the reported thinness computations. Direct checks identify malformed parameter tuples and an incompatible invariant-factor entry. These are local tabulation defects, not a verdict on the classification or thinness theorems. No external theorem is imported to close an obligation; external texts remain uninspected, and model knowledge is not treated as dated evidence.",
  "used_dependencies": [
    "familyclassification",
    "interlace",
    "table3",
    "table4"
  ],
  "missing_context": [
    "Complete almost-interlacing reduction, classification theorem, family definitions and parameter restrictions, and the finite sporadic input list.",
    "Enumeration conventions: equivalence under simultaneous half-shift, primitivity test, disjointness of alpha and beta spectra, cyclotomic integrality, and signature verification.",
    "Definitions of even/odd monodromy, H_r, R_f, the chosen lattice and its normalization, and the Factors convention.",
    "Complete tables, Gram matrices and Smith-normal-form computations; graph certificates and proofs of starred index assertions.",
    "Exact Nikulin theorem and hypotheses, dimension-three arithmeticity proofs, and a dated Park version available by the cutoff."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "${\\scriptstyle \\alpha=}(0,\\frac{1}{12},\\frac{5}{12},\\frac{1}{2},\\frac{ 7}{12},\\frac{11}{12})$",
      "explanation": "In the dimension-five sporadic row with left-hand beta=(0,0,0,1/4,3/4), the shifted alpha has six entries while its beta has five. Direct addition of 1/2 modulo 1 to the left-hand alpha gives five entries and does not retain 1/2. Thus the displayed pair is not the claimed shift and does not define the stated dimension-five group."
    },
    {
      "kind": "error",
      "source_quote": "\\frac{7}{12}\\frac{11}{12}",
      "explanation": "This occurs in both alpha tuples of another dimension-five sporadic row without a separating comma. Literally it is a product rather than two tuple entries, so the displayed tuples do not have the required five components. The numerical table is malformed independently of any external theorem."
    },
    {
      "kind": "error",
      "source_quote": "2,  3,  3 & Sporadic &Yes",
      "explanation": "The narrative calls these entries invariant factors. Under that convention, successive positive factors must divide one another; 2 does not divide 3. Omitting unit factors cannot resolve this. The supplied data do not determine the correct replacement."
    },
    {
      "kind": "unverified",
      "source_quote": "We note that the sporadic groups are all in dimension $n\\leq 9$ and are easily determined using the method from Section~\\ref{interlace}",
      "explanation": "The interlace excerpt supplies an interlacing definition, not the almost-interlacing reduction and its converse. The familyclassification excerpt announces seven families but omits their complete theorem and proof. Consequently neither the dimension bound nor exhaustive enumeration, primitive filtering, and family assignments can be checked here. This is a packet limitation, not evidence that the classification is false."
    },
    {
      "kind": "unverified",
      "source_quote": "thinness here is always confirmed using the minimal distance graph method described above, with the longest path encountered being of length $3$",
      "explanation": "No certificate theorem, Gram matrices, root vectors, or paths are supplied. Each Yes needs the theorem's lattice hypotheses, graph connectivity witnesses, and a justified passage from H_r to H. Infinite index of H_r in H explains why its thinness alone would not suffice, but the starred index assertions themselves are not verified. The dimension-three No entries likewise lack their arithmeticity arguments."
    },
    {
      "kind": "context",
      "source_quote": "Nikulin's results from \\cite{nik} do not apply to the odd case and hence we cannot immediately deduce thinness",
      "explanation": "This appropriately refrains from claiming thinness merely from containment in R_f. However, even/odd, H_r, R_f, and the relevant Nikulin hypotheses are not defined in the supplied scope. The precise applicability distinction therefore remains unchecked; containment alone supplies no displayed infinite-index conclusion."
    },
    {
      "kind": "map_correction",
      "source_quote": "In his recent thesis, H.~Park \\cite{park} shows that this $H(\\alpha,\\beta)$ is geometrically finite and determines a presentation for it.",
      "explanation": "The Park edge supports a supplementary footnote about one example, not the asserted minimal-distance certificates or exhaustive classification. Its bibliography gives only August 2013, insufficient to establish availability by 2013-08-11. Record: available_from=unknown; date_evidence=August 2013 bibliography; date_basis=month-only bibliographic date; version=unspecified thesis. Neither the exact result nor its hypotheses has been inspected."
    }
  ],
  "dependency_proposals": [
    {
      "label": "familyclassification",
      "role": "conditional",
      "source_quote": "seven infinite families of primitive integer hypergeometric groups",
      "explanation": "Needed for family assignments and the nonsporadic classification; the supplied announcement does not establish exhaustiveness."
    },
    {
      "label": "interlace",
      "role": "conditional",
      "source_quote": "This classification will be used in the next section to classify all hyperbolic hypergeometric monodromies in dimension $n>9$.",
      "explanation": "The actual reduction and classification, rather than this announcement, are required proof inputs."
    },
    {
      "label": "external:bh",
      "role": "conditional",
      "source_quote": "Invent. Math. {\\bf 95}, pp. 325-354 (1989)",
      "explanation": "Not imported. Candidate record: available_from=1989, year precision; date_evidence=cited bibliography; date_basis=publication year; version=cited journal article. Exact finite-classification theorem, hypotheses, and sporadic list remain uninspected."
    },
    {
      "label": "external:nik",
      "role": "conditional",
      "source_quote": "Soviet Math. Dokl. {\\bf 20}, pp. 1156-1158 (1979)",
      "explanation": "Not imported. Candidate record: available_from=1979, year precision; date_evidence=cited bibliography; date_basis=publication year; version=cited journal article. Exact reflection-quotient theorem and lattice hypotheses are missing."
    },
    {
      "label": "external:park",
      "role": "background",
      "source_quote": "Ph.D. Thesis, Princeton University (August 2013).",
      "explanation": "Supplementary geometric-finiteness/presentation claim; cutoff availability is unresolved and cannot support the original proof obligation."
    },
    {
      "label": "table3",
      "role": "proof_input",
      "source_quote": "Even monodromy groups: dimensions 3, 5, 7, 9.",
      "explanation": "Relevant data evidence, not an independent proof of its completeness or Thin column."
    },
    {
      "label": "table4",
      "role": "background",
      "source_quote": "Odd monodromy groups: dimensions 5, 7, 9.$^7$",
      "explanation": "Only fragments are supplied; the entire odd containment table cannot be audited."
    },
    {
      "label": "mindistgraph",
      "role": "proof_input",
      "source_quote": "minimal distance graph method described above",
      "explanation": "The certificate theorem and its per-entry witnesses are omitted prerequisites to the thinness transition."
    }
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