Triple

T21047248
Position Surface form Disambiguated ID Type / Status
Subject Lindelöf theorem in complex analysis E518479 entity
Predicate relatedTo P37 FINISHED
Object Fatou theorem
The Fatou theorem is a fundamental result in complex analysis that describes the almost-everywhere existence of radial boundary limits for bounded analytic (or harmonic) functions on the unit disk.
E1463339 NE FINISHED

How this triple was built (4 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: Fatou theorem | Statement: [Lindelöf theorem in complex analysis, relatedTo, Fatou theorem]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Fatou theorem
Context triple: [Lindelöf theorem in complex analysis, relatedTo, Fatou theorem]
  • A. Montel theorem
    Montel's theorem is a fundamental result in complex analysis stating that a family of holomorphic functions that is uniformly bounded on every compact subset of a domain is a normal family, meaning every sequence in it has a subsequence that converges uniformly on compact sets.
  • B. Borel–Carathéodory theorem
    The Borel–Carathéodory theorem is a result in complex analysis that provides bounds on the modulus of a holomorphic function inside a disk in terms of the maximum of its real part on a larger concentric disk.
  • C. Corona theorem
    The Corona theorem is a fundamental result in complex analysis that characterizes when bounded analytic functions on the unit disk can be solved in a certain type of division problem, showing that the maximal ideal space of the disk algebra has no "corona."
  • D. Mergelyan's theorem
    Mergelyan's theorem is a fundamental result in complex analysis stating that every function continuous on a compact subset of the complex plane with connected complement and holomorphic in its interior can be uniformly approximated by polynomials.
  • E. Picard theorem
    Picard theorem is a fundamental result in complex analysis stating that entire non-constant functions take on all possible complex values, with at most one exception.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Fatou theorem
Triple: [Lindelöf theorem in complex analysis, relatedTo, Fatou theorem]
Generated description
The Fatou theorem is a fundamental result in complex analysis that describes the almost-everywhere existence of radial boundary limits for bounded analytic (or harmonic) functions on the unit disk.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Fatou theorem
Target entity description: The Fatou theorem is a fundamental result in complex analysis that describes the almost-everywhere existence of radial boundary limits for bounded analytic (or harmonic) functions on the unit disk.
  • A. Montel theorem
    Montel's theorem is a fundamental result in complex analysis stating that a family of holomorphic functions that is uniformly bounded on every compact subset of a domain is a normal family, meaning every sequence in it has a subsequence that converges uniformly on compact sets.
  • B. Borel–Carathéodory theorem
    The Borel–Carathéodory theorem is a result in complex analysis that provides bounds on the modulus of a holomorphic function inside a disk in terms of the maximum of its real part on a larger concentric disk.
  • C. Corona theorem
    The Corona theorem is a fundamental result in complex analysis that characterizes when bounded analytic functions on the unit disk can be solved in a certain type of division problem, showing that the maximal ideal space of the disk algebra has no "corona."
  • D. Mergelyan's theorem
    Mergelyan's theorem is a fundamental result in complex analysis stating that every function continuous on a compact subset of the complex plane with connected complement and holomorphic in its interior can be uniformly approximated by polynomials.
  • E. Picard theorem
    Picard theorem is a fundamental result in complex analysis stating that entire non-constant functions take on all possible complex values, with at most one exception.
  • F. None of above. chosen

Provenance (5 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_69e0b50438e08190917e2538bb8bc034 completed April 16, 2026, 10:08 a.m.
NER Named-entity recognition batch_69e6fcf4d26481908b639996500a8319 completed April 21, 2026, 4:28 a.m.
NED1 Entity disambiguation (via context triple) batch_6a09475512b081908ceb38a118b0f026 completed May 17, 2026, 4:43 a.m.
NEDg Description generation batch_6a09489e80288190b401d11ccde39dc1 completed May 17, 2026, 4:48 a.m.
NED2 Entity disambiguation (via description) batch_6a094995d5bc8190b25d328dbf06b8b1 completed May 17, 2026, 4:52 a.m.
Created at: April 16, 2026, 2:34 p.m.