Triple

T5256275
Position Surface form Disambiguated ID Type / Status
Subject Carathéodory’s extension theorem E118705 entity
Predicate alternativeName P39 FINISHED
Object Carathéodory’s theorem on extension of measures E118705 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: Carathéodory’s theorem on extension of measures | Statement: [Carathéodory’s extension theorem, alternativeName, Carathéodory’s theorem on extension of measures]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07 
Target entity: Carathéodory’s theorem on extension of measures
Context triple: [Carathéodory’s extension theorem, alternativeName, Carathéodory’s theorem on extension of measures]
  • A. Carathéodory’s extension theorem chosen
    Carathéodory’s extension theorem is a fundamental result in measure theory that guarantees a unique extension of a pre-measure defined on an algebra of sets to a complete measure on the generated σ-algebra.
  • B. Carathéodory’s theorem in convex geometry
    Carathéodory’s theorem in convex geometry is a fundamental result stating that any point in the convex hull of a set in ℝⁿ can be expressed as a convex combination of at most n+1 points from that set.
  • C. Carathéodory measurability criterion
    The Carathéodory measurability criterion is a fundamental condition in measure theory that characterizes measurable sets via an outer measure by requiring additivity over intersections and complements.
  • D. Minkowski’s theorem on convex sets
    Minkowski’s theorem on convex sets is a fundamental result in convex geometry that characterizes lattice points in convex bodies, underpinning much of the theory of convex polytopes and the geometry of numbers.
  • E. Helly’s theorem
    Helly’s theorem is a fundamental result in convex geometry that gives conditions under which a family of convex sets in Euclidean space has a nonempty common intersection.
  • 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_69bd446978108190bb5f9c5c23d93f88 completed March 20, 2026, 12:58 p.m.
NER Named-entity recognition batch_69bd7ba4ecd88190800b5e4eea3abed5 completed March 20, 2026, 4:53 p.m.
NED1 Entity disambiguation (via context triple) batch_69bf06bfa5e48190bc9313a39d95531e completed March 21, 2026, 8:59 p.m.
Created at: March 20, 2026, 1:50 p.m.