Triple

Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).

Phrase: Langlands–Tunnell theorem | Statement: [Artin’s conjecture on L-functions, specialCaseProvedBy, Langlands–Tunnell theorem]

Target entity: Langlands–Tunnell theorem
Context triple: [Artin’s conjecture on L-functions, specialCaseProvedBy, Langlands–Tunnell theorem]

• A. Fontaine–Mazur conjecture
  The Fontaine–Mazur conjecture is a central open problem in number theory that predicts which p-adic Galois representations of number fields arise from geometry or from automorphic forms.
• B. Artin reciprocity law
  The Artin reciprocity law is a fundamental theorem in class field theory that generalizes quadratic reciprocity by describing abelian extensions of number fields in terms of characters of their idele class groups.
• C. Ribet's theorem
  Ribet's theorem is a result in number theory that linked certain modular forms to Galois representations and played a crucial role in the proof of Fermat's Last Theorem.
• D. Artin’s conjecture on L-functions
  Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.
• E. Chebotarev density theorem
  The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Langlands–Tunnell theorem
Target entity description: The Langlands–Tunnell theorem is a major result in number theory that proves the modularity of certain two-dimensional Galois representations, playing a key role in progress toward the Taniyama–Shimura–Weil conjecture and Fermat’s Last Theorem.

• A. Fontaine–Mazur conjecture
  The Fontaine–Mazur conjecture is a central open problem in number theory that predicts which p-adic Galois representations of number fields arise from geometry or from automorphic forms.
• B. Artin reciprocity law
  The Artin reciprocity law is a fundamental theorem in class field theory that generalizes quadratic reciprocity by describing abelian extensions of number fields in terms of characters of their idele class groups.
• C. Ribet's theorem
  Ribet's theorem is a result in number theory that linked certain modular forms to Galois representations and played a crucial role in the proof of Fermat's Last Theorem.
• D. Artin’s conjecture on L-functions
  Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.
• E. Chebotarev density theorem
  The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
• F. None of above. chosen

Target predicate: specialCaseProvedBy
Context triple: [Artin’s conjecture on L-functions, specialCaseProvedBy, Langlands–Tunnell theorem]

• A. specialCaseOf
  Indicates that one entity represents a more specific, exceptional, or restricted instance of the general situation, rule, or relationship expressed by another entity.
• B. provedInFullBy
  Indicates that something (such as a claim, theorem, or statement) has been completely and rigorously demonstrated or established by a particular agent or source.
• C. partiallyProvenFor chosen
  Indicates that something has been shown to hold or be true for part of a domain or set of cases, but not yet for all cases.
• D. typicalProofUses
  Indicates that a proof characteristically or commonly employs a particular method, technique, or component.
• E. independentlyProvedBy
  Indicates that a statement or result is established by a proof that does not rely on or derive from another specified proof or source.
• F. None of above.