Triple

T1056915
Position Surface form Disambiguated ID Type / Status
Subject Brouwer fixed-point theorem E22815 entity
Predicate relatedTo P37 FINISHED
Object Kakutani fixed-point theorem E3648 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: Kakutani fixed-point theorem | Statement: [Brouwer fixed-point theorem, relatedTo, Kakutani fixed-point theorem]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07 
Target entity: Kakutani fixed-point theorem
Context triple: [Brouwer fixed-point theorem, relatedTo, Kakutani fixed-point theorem]
  • A. Kakutani fixed-point theorem chosen
    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.
  • B. Glicksberg fixed-point theorem
    The Glicksberg fixed-point theorem is a result in functional analysis that extends Kakutani’s fixed-point theorem to certain infinite-dimensional or compact convex subsets of locally convex topological vector spaces.
  • C. Schauder fixed-point theorem
    The Schauder fixed-point theorem is a fundamental result in functional analysis that guarantees the existence of fixed points for continuous compact mappings on convex closed subsets of Banach spaces, generalizing the Brouwer fixed-point theorem to infinite-dimensional settings.
  • D. 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.
  • E. Sperner's lemma
    Sperner's lemma is a fundamental result in combinatorial topology that guarantees the existence of a fully labeled simplex in certain labeled triangulations, and is widely used to prove fixed-point and equilibrium theorems.
  • 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_69a493dada0481909c43649f9843ea91 completed March 1, 2026, 7:30 p.m.
NER Named-entity recognition batch_69a4b8da80dc8190b79beaf509910725 completed March 1, 2026, 10:08 p.m.
NED1 Entity disambiguation (via context triple) batch_69ac429f0580819082d5a2cc6129cb1e completed March 7, 2026, 3:22 p.m.
Created at: March 1, 2026, 7:42 p.m.