Triple

T5256274
Position Surface form Disambiguated ID Type / Status
Subject Carathéodory’s extension theorem E118705 entity
Predicate alternativeName P39 FINISHED
Object Carathéodory’s measure extension theorem 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 measure extension theorem | Statement: [Carathéodory’s extension theorem, alternativeName, Carathéodory’s measure extension theorem]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07 
Target entity: Carathéodory’s measure extension theorem
Context triple: [Carathéodory’s extension theorem, alternativeName, Carathéodory’s measure extension theorem]
  • 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. Alexandrov–Hausdorff theorem
    The Alexandrov–Hausdorff theorem is a result in descriptive set theory that characterizes analytic sets as continuous images of Baire space, playing a key role in the study of definable sets in Polish spaces.
  • D. Lebesgue measure
    Lebesgue measure is the standard way of assigning a consistent notion of "length," "area," or "volume" to subsets of Euclidean space, forming the foundation of modern measure theory and integration.
  • E. Tychonoff theorem for products of compact spaces
    The Tychonoff theorem for products of compact spaces is a fundamental result in topology stating that any product of compact topological spaces is compact, a statement that is equivalent in strength to the axiom of choice.
  • 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_69befe7a1f448190acfcdfe37c962028 completed March 21, 2026, 8:24 p.m.
Created at: March 20, 2026, 1:50 p.m.