Triple

T31634
Position Surface form Disambiguated ID Type / Status
Subject Nash embedding theorem E631 entity
Predicate hasVersion P455 FINISHED
Object Nash Ck embedding theorem E631 NE FINISHED

How this triple was built (2 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
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).
Input
Phrase: Nash Ck embedding theorem | Statement: [Nash embedding theorem, hasVersion, Nash Ck embedding theorem]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07 
Target entity: Nash Ck embedding theorem
Context triple: [Nash embedding theorem, hasVersion, Nash Ck 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. 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.
  • C. 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.
  • D. implicit function theorem
    The implicit function theorem is a fundamental result in calculus and differential geometry that guarantees, under suitable smoothness and nondegeneracy conditions, the local solvability of equations for some variables as differentiable functions of others.
  • E. 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.
  • F. None of above.
  • G. Unsure - the case is ambiguous/there is not enough information to decide.

Provenance (3 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69a2479dec388190967ba648663442c9 completed Feb. 28, 2026, 1:40 a.m.
NER Named-entity recognition batch_69a24ac8d59c8190aaf6607f2792ba3a completed Feb. 28, 2026, 1:54 a.m.
NED1 Entity disambiguation (via context triple) batch_69a255307eac81908d0055775ef3c965 completed Feb. 28, 2026, 2:38 a.m.
Created at: Feb. 28, 2026, 1:44 a.m.