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: Nash C^k isometric embedding theorem | Statement: [Janet–Cartan theorem, strengthenedBy, Nash C^k isometric embedding theorem]

Target entity: Nash C^k isometric embedding theorem
Context triple: [Janet–Cartan theorem, strengthenedBy, Nash C^k isometric embedding theorem]

• A. Nash embedding theorem chosen
  The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
• B. Whitney embedding theorem
  The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
• C. Janet–Cartan theorem
  The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
• D. Riemannian manifolds
  Riemannian manifolds are smooth manifolds equipped with an inner product on each tangent space that allows one to measure lengths, angles, and curvature in a curved geometric setting.
• E. Kakutani fixed-point theorem
  The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
• F. None of above.
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target predicate: strengthenedBy
Context triple: [Janet–Cartan theorem, strengthenedBy, Nash C^k isometric embedding theorem]

• A. strengthens chosen
  Indicates that one entity increases the power, effectiveness, or resilience of another.
• B. isStrongerThan
  Indicates that one entity possesses greater physical power, force, or effectiveness than another entity.
• C. estimatedStrength
  Indicates that a value represents an approximate or inferred level, magnitude, or intensity of something rather than a precisely measured strength.
• D. forcedBy
  Indicates that an entity is compelled or coerced into an action or state by another entity.
• E. enteredIntoForce
  Indicates that a law, treaty, agreement, or regulation has become legally effective and binding from a specific point in time.
• F. None of above.