Triple

T179304
Position Surface form Disambiguated ID Type / Status
Subject Kakutani fixed-point theorem E3648 entity
Predicate relatedTo P37 FINISHED
Object Brouwer fixed-point theorem E22815 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: Brouwer fixed-point theorem | Statement: [Kakutani fixed-point theorem, relatedTo, Brouwer fixed-point theorem]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07 
Target entity: Brouwer fixed-point theorem
Context triple: [Kakutani fixed-point theorem, relatedTo, Brouwer fixed-point theorem]
  • A. Brouwer fixed-point theorem chosen
    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.
  • B. 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.
  • C. 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.
  • D. 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 \(\mathbb{R}^{2n}\)).
  • E. 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.
  • 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_69a25900709c8190a65e778936be5dd5 completed Feb. 28, 2026, 2:54 a.m.
NED1 Entity disambiguation (via context triple) batch_69a2fa7c7a288190bc547d5e7010732b completed Feb. 28, 2026, 2:23 p.m.
Created at: Feb. 28, 2026, 2:39 a.m.