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: Gaussian curvature is invariant under local isometries of surfaces | Statement: [Theorema Egregium, proves, Gaussian curvature is invariant under local isometries of surfaces]

Target entity: Gaussian curvature is invariant under local isometries of surfaces
Context triple: [Theorema Egregium, proves, Gaussian curvature is invariant under local isometries of surfaces]

• A. Theorema Egregium
  Theorema Egregium is Gauss’s celebrated theorem in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.
• B. Gaussian curvature chosen
  Gaussian curvature is a fundamental concept in differential geometry that measures how a surface bends at a point by combining its principal curvatures into a single intrinsic quantity.
• C. Gauss–Bonnet theorem (early form)
  The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.
• D. Nash embedding theorem
  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.
• E. Lie sphere geometry
  Lie sphere geometry is a branch of differential geometry that studies the properties and transformations of spheres (and related objects like planes and points) using the methods of Lie groups and projective geometry.
• F. None of above.
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target predicate: proves
Context triple: [Theorema Egregium, proves, Gaussian curvature is invariant under local isometries of surfaces]

• A. proved chosen
  Indicates that one entity has demonstrated the truth or validity of another entity (such as a statement, theorem, or claim) through logical or evidential means.
• B. proposes
  Indicates that one entity formally suggests or puts forward an idea, plan, or course of action to another entity for consideration or approval.
• C. providesEvidenceFor
  Indicates that one entity serves as support, justification, or proof for the validity or truth of another entity.
• D. approves
  Indicates that one entity formally accepts, authorizes, or agrees to a proposal, action, or decision made by another entity.
• E. promotes
  Indicates that one entity actively supports, advances, or encourages the growth, adoption, or success of another entity or outcome.
• F. None of above.