Triple

T179452
Position Surface form Disambiguated ID Type / Status
Subject Janet–Cartan theorem E3651 entity
Predicate strengthenedBy P3261 FINISHED
Object Nash C^1 isometric 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 C^1 isometric embedding theorem | Statement: [Janet–Cartan theorem, strengthenedBy, Nash C^1 isometric embedding theorem]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07 
Target entity: Nash C^1 isometric embedding theorem
Context triple: [Janet–Cartan theorem, strengthenedBy, Nash C^1 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. Brouwer fixed-point theorem
    The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
  • 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_69a25374990081909766d30c79a18e0e completed Feb. 28, 2026, 2:31 a.m.
NER Named-entity recognition batch_69a25bafd5808190a0a0cb2b21ce007f completed Feb. 28, 2026, 3:06 a.m.
NED1 Entity disambiguation (via context triple) batch_69a2fd0405608190bf4c1aa8a8fa9ad1 completed Feb. 28, 2026, 2:34 p.m.
Created at: Feb. 28, 2026, 2:39 a.m.