Triple

T5425420
Position Surface form Disambiguated ID Type / Status
Subject Schauder fixed-point theorem E121350 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: [Schauder 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: [Schauder 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. Tarski’s fixed point theorem
    Tarski’s fixed point theorem is a fundamental result in order theory and lattice theory that guarantees the existence of fixed points for monotone functions on complete lattices, with wide applications in logic, computer science, and economics.
  • 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_69bd463b58d88190b258261573de9e91 completed March 20, 2026, 1:06 p.m.
NER Named-entity recognition batch_69bd881598448190a9bb456dee36004b completed March 20, 2026, 5:47 p.m.
NED1 Entity disambiguation (via context triple) batch_69bf3abfc7e88190b8f0a31b61c33973 completed March 22, 2026, 12:41 a.m.
Created at: March 20, 2026, 2:06 p.m.