{
  "target_id": "narrative:47545",
  "proof_status": "defective",
  "summary": "The displayed generating-function identity is incorrect in ordinary hypergeometric notation: integral examples with d=3 have zero lower parameters, making the stated series undefined. The companion-matrix calculation and eigenspace assertions are correct conditional on the displayed constant-term normalization. The integrality equivalence and complete family identification remain unverified because their supporting criterion, characterization, polynomial definitions, and sporadic tables are not supplied.",
  "used_dependencies": [
    "levthm",
    "theo"
  ],
  "missing_context": [
    "The introductory parameter conventions, nonresonance hypotheses, and the conversion from monodromy residues to hypergeometric series parameters.",
    "The almost-interlacing characterization needed to prove the d=3 criterion, including repeated-root cases.",
    "Definitions and parameter restrictions for P_m, Q_m, P_{m,k}, Q_{m,k}, together with Tables 2 and 3.",
    "The normalization or restriction giving P(0)=1 and Q(0)=-1 in the quadratic-form subsection.",
    "Exact external theorem statements and inspected versions. No external result has been imported as verified. Bibliographic years precede the 2013-08-11 cutoff, but do not verify the needed passages. Model knowledge may include later developments and is not used as dated evidence."
  ],
  "findings": [
    {
      "kind": "error",
      "source_quote": "$$\\sum_{m=0}^\\infty u_mz^m=_nF_{n-1}(\\alpha_1,\\dots,\\alpha_n; \\beta_1,\\dots,\\beta_{n-1}|Cz)$$",
      "explanation": "In standard hypergeometric notation, zero lower parameters give zero denominator Pochhammer symbols. An elementary check using the supplied family M_2(1,3), which satisfies its stated parameter restrictions, gives alpha=(1/4,1/2,3/4), beta=(0,0,0), R(t)=(t^4-1)/(t-1)^4, and u_m=(4m)!/(m!)^4. The left side has coefficient u_1=24, whereas the printed right side has lower parameters (0,0) and is undefined. An explicit convention converting monodromy residues into series parameters is required; omitting just one beta does not resolve the remaining zeros."
    },
    {
      "kind": "unverified",
      "source_quote": "It is easy to check using Landau's criterion \\cite{land} and our characterization of hyperbolic hypergeometric monodromies, that for the latter $u_m\\in \\mathbb Z$ iff $d=3$.",
      "explanation": "The key transition is asserted rather than demonstrated. The exact remaining obligation is to show, for every admissible hyperbolic pair in scope, that F(x)=sum_j floor(a_j x)-sum_k floor(b_k x) is nonnegative for every real x exactly when d=3, and then justify the invoked equivalence between this condition and integrality for every m. Equal polynomial degrees supply sum_j a_j=sum_k b_k, but the required almost-interlacing argument, including repeated parameters, is absent. Landau's theorem itself has not been inspected."
    },
    {
      "kind": "context",
      "source_quote": "Note that $d$ is the multiplicity of the $k$'s for which $\\beta_k=0$.",
      "explanation": "The factorization directly gives ord_{t=1}(P/Q)=K-L=-d, hence d=mult_Q(1)-mult_P(1), not automatically mult_Q(1). The stated equality requires, for d>0, the absence of a simultaneous root 1 of P and Q and representatives identifying beta=0 with that root. Nonresonance may be established in the omitted setup, but is not supplied here. Likewise, existence of the integer-power factorization requires cyclotomic P and Q, not merely Levelt's general setup."
    },
    {
      "kind": "unverified",
      "source_quote": "$M_2(j,n)$, $M_3(j,n)$, $N_3(j,k,n)$, and $N_4(j,k,n)$ are the infinite families with integral coefficients",
      "explanation": "The supplied theorem explicitly exhibits three zero beta entries for M_2, M_3, and N_3. Checking N_4 and excluding additional zeros in the other N-families requires the omitted definitions of P_{m,k}, Q_{m,k} and their admissible parameters. Completeness in the supplied theorem is restricted to n>9. The sporadic assertion additionally requires the absent Tables 2 and 3. Thus the full classification does not follow from the supplied statements alone."
    },
    {
      "kind": "context",
      "source_quote": "$P(x):=x^n+a_1x^{n-1}+\\cdots +a_{n-1}x+1$",
      "explanation": "The quadratic-form subsection assumes P(0)=1 and Q(0)=-1 without supplying the normalization establishing this for its claimed scope. The listed M_1 family instead has a root 1 in P and, for odd n, the opposite constant-term signs. A preceding restriction or scalar-shift convention is needed. Conditional on the displayed signs, direct multiplication verifies C=I-v e_n^t, so C fixes e_1,...,e_{n-1}, sends v to -v, and has exactly the asserted eigenspace dimensions."
    },
    {
      "kind": "map_correction",
      "source_quote": "This will be a necessary ingredient in our minimal distance graph method described in Section~\\ref{mindistgraph}.",
      "explanation": "This is a forward application reference, not a proof input from narrative:56875 to the present calculation. That dependency should be classified as background/application context. Levelt supports the companion-matrix model, but does not by itself establish the factorial-integrality claims."
    }
  ],
  "dependency_proposals": [
    {
      "label": "external:land",
      "role": "conditional",
      "source_quote": "using Landau's criterion \\cite{land}",
      "explanation": "Required input for factorial-ratio integrality, not verified here. available_from: 1900 at year precision; date_evidence: supplied bibliography; date_basis: publication year; version: uninspected journal article. Request the exact theorem and hypotheses."
    },
    {
      "label": "external:bob",
      "role": "conditional",
      "source_quote": "(see \\cite{vill}, \\cite{bob})",
      "explanation": "Proposed support for the factorial-ratio/series correspondence; it cannot validate the printed zero-parameter identity without a parameter convention. available_from: 2009 at year precision; date_evidence: supplied bibliography; date_basis: thesis year; version: uninspected Michigan thesis."
    },
    {
      "label": "external:vill",
      "role": "conditional",
      "source_quote": "(see \\cite{vill}, \\cite{bob})",
      "explanation": "Proposed support for the correspondence and finite-monodromy comparison. available_from: 2003 at year precision; date_evidence: supplied bibliography; date_basis: publication year; version: uninspected Fields Institute article. Exact results and hypotheses remain unavailable."
    },
    {
      "label": "levthm",
      "role": "proof_input",
      "source_quote": "Then $H(\\alpha,\\beta)$ is the group generated by",
      "explanation": "Conditional input for the companion-matrix model, not for integrality or the constant-term normalization."
    },
    {
      "label": "theo",
      "role": "conditional",
      "source_quote": "If $n>9$, this list of families completely describes (up to scalar shift)",
      "explanation": "Supports family identification only conditionally, with its completeness bound and scalar-shift qualification retained."
    },
    {
      "label": "narrative:56875",
      "role": "background",
      "source_quote": "This will be a necessary ingredient in our minimal distance graph method described in Section~\\ref{mindistgraph}.",
      "explanation": "Forward application context rather than a premise proving the present assertions."
    }
  ],
  "_meta": {
    "pipeline": "claim_review",
    "model": "gpt-6.1-sol",
    "reasoning_effort": "high",
    "provider": "openai",
    "batch_job_id": "batch_6ad5925185988190adf2ddc0ca0c0808",
    "custom_id": "claim-b2996d01d1b76e64bd1d95f1",
    "source_sha256": "800913940f206d957e5521c61e15362a7e82f0b65e4330d2699a543fb1fffd36",
    "usage": {
      "input_tokens": 14935,
      "input_tokens_details": {
        "cache_write_tokens": 14932,
        "cached_tokens": 0
      },
      "output_tokens": 5392,
      "output_tokens_details": {
        "reasoning_tokens": 3558
      },
      "total_tokens": 20327
    },
    "max_output_tokens": 12288,
    "prompt_profile": "claim_review-v2",
    "prompt_profile_sha256": "600c451625bbe1d6d19828dfd9fb687fea4ac45fe9189b2bf52bc25ef9e1edcb",
    "review_policy": {
      "schema": 1,
      "cutoff_date": "2013-08-11",
      "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.",
      "status": "recorded",
      "rule": "external-results-available-by-cutoff"
    },
    "chronology_prompted": true
  }
}