{
  "target_id": "thm:BF",
  "proof_status": "needs_context",
  "summary": "The excerpt presents an attributed external theorem, not a local proof. Conditional on the cited theorem having the stated content and all its ambient hypotheses being satisfied, the existence and coset conclusions are supported by that premise. The packet does not supply the representation category, the full hypotheses, or the relevant external version. The subsequent minimal-stabilizer characterization requires separate verification; the same-map definition supplies no proof of the target. No mathematical defect is established by the missing proof anchor.",
  "used_dependencies": [
    "external:BaderFurman"
  ],
  "missing_context": [
    "Definitions of T-algebraic representation, coset T-algebraic representation, and morphism, including equality and equivariance conventions modulo null sets.",
    "The complete standing setup for G, Gamma, T, k, the algebraic target, and any fixed representation; these are needed to check every hypothesis of the cited theorem.",
    "The exact statement and applicable definitions of Bader\u2013Furman Theorem 4.3 from a documented version available on or before 2019-06-20, together with dated version evidence.",
    "The definition of the stabilizer J and a source or argument establishing the claimed equivalence between minimal stabilizer and initiality.",
    "If the second reference edge was intended to support a later assertion rather than the displayed definition, the corresponding assertion and proof scope."
  ],
  "findings": [
    {
      "kind": "context",
      "source_quote": "The collection of $T$-algebraic representations of $G$ forms a category.",
      "explanation": "Neither the objects nor their morphisms are defined in the supplied excerpts. The standing assumptions on G, Gamma, T, k, and any algebraic target or fixed representation are also missing. Consequently, category formation and the applicability of the external theorem cannot be checked. The introductory lattice statement and section-level measure assumption do not supply these definitions."
    },
    {
      "kind": "unverified",
      "source_quote": "\\begin{thm}[Thm.\\ 4.3 \\cite{BaderFurman}]\\label{thm:BF}",
      "explanation": "The external text has not been inspected. The displayed hypothesis is weak mixing of the T-action on G/Gamma, but the complete hypotheses of the cited theorem remain unknown. Verification must compare its exact conclusion and ambient assumptions with the manuscript's category. The missing local proof anchor is a mapping limitation, not evidence that this cited theorem lacks a proof."
    },
    {
      "kind": "unverified",
      "source_quote": "\\newblock Preprint: arXiv:1810.01608.",
      "explanation": "The bibliography identifies a preprint but not the version containing the needed theorem. Availability record: available_from=unresolved; date_evidence=the supplied arXiv identifier only; date_basis=bibliographic identifier, not a checked version history; version=unspecified. The required theorem and its full hypotheses must be verified in a version available by 2019-06-20. The supplied manuscript edition, dated 26 April 2020, does not establish that availability. Model knowledge may include later developments; none is used here to close this obligation."
    },
    {
      "kind": "unverified",
      "source_quote": "An initial object is characterized by the fact that $\\bfJ$ is the minimal subgroup, up to conjugacy, that can arise as a stabilizer in any coset $T$-algebraic representation in the category.",
      "explanation": "This characterization does not follow merely from the quoted assertion that an initial coset object exists. The missing definitions or external result must explain the stabilizer ordering and establish both implications. In particular, minimality of a stabilizer alone is not the categorical universal property: existence and uniqueness of morphisms to every object must also be accounted for. This is an unverified additional assertion, not a demonstrated false statement."
    },
    {
      "kind": "map_correction",
      "source_quote": "Though not stated explicitly, the following is also implicit in \\cite{BaderFurman}.",
      "explanation": "The second supplied source-reference edge points into following prose introducing the same-map terminology. In this packet that prose is a definition, not an argument for existence of an initial object. It should not be counted as an independent proof input for thm:BF. Any subsequent result intended by 'the following' is not supplied."
    }
  ],
  "dependency_proposals": [
    {
      "label": "external:BaderFurman",
      "role": "conditional",
      "source_quote": "If the $T$-action on $G / \\Gam$ is weakly mixing, then this category has an initial object and this initial object is a coset $T$-algebraic representation.",
      "explanation": "This is the intended external proof input, conditional on exact theorem matching, all ambient hypotheses, and cutoff-admissible version evidence. It is not independently established in this review."
    },
    {
      "label": "Following same-map definition",
      "role": "excluded",
      "source_quote": "Given two subgroups $S$ and $T$ of $G$, we say that their initial objects",
      "explanation": "This passage defines terminology assuming initial objects are already available. It does not prove the target's existence or coset conclusion."
    }
  ],
  "_public_provenance": {
    "origin": "returned_native_final_assessment_not_campaign_verdict",
    "model": "gpt-6.1-sol",
    "effort": "high",
    "provider": "openai",
    "manuscript_sha256": "6776902687fa24498e9bc0084edb02b1630009b90b7b06a7e2fba0e6d697397c",
    "review_input_sha256": "0791c7051cd025d4032e12d0dc609c676baaa5e4bc0a0355f4d2b83923a16afc",
    "original_assessment_sha256": "76842b8e95c59bb9157e688627c71369d3451e79d7b38877d594f7d66ee68eac",
    "note": "Original final assessment fields retained; embedded roadmap and private provider/run metadata omitted. No raw reasoning stream is exported."
  }
}