Triple

T1057158
Position Surface form Disambiguated ID Type / Status
Subject local existence and uniqueness theorem E22820 entity
Predicate isProvedBy P21917 FINISHED
Object Banach fixed-point theorem E126344 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: Banach fixed-point theorem | Statement: [local existence and uniqueness theorem, isProvedBy, Banach fixed-point theorem]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07 
Target entity: Banach fixed-point theorem
Context triple: [local existence and uniqueness theorem, isProvedBy, Banach fixed-point theorem]
  • A. Banach fixed-point theorem chosen
    The Banach fixed-point theorem is a fundamental result in metric space theory that guarantees the existence and uniqueness of a fixed point for any contraction mapping and provides a method for finding it via iterative approximation.
  • B. 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.
  • 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. 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.
  • 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_69a493dada0481909c43649f9843ea91 completed March 1, 2026, 7:30 p.m.
NER Named-entity recognition batch_69a4bb7320f88190a8428946541df157 completed March 1, 2026, 10:19 p.m.
NED1 Entity disambiguation (via context triple) batch_69ac53864a20819081fc59e7102a6e00 completed March 7, 2026, 4:34 p.m.
Created at: March 1, 2026, 7:42 p.m.