Triple

T10991825
Position Surface form Disambiguated ID Type / Status
Subject Schwarz lemma E259769 entity
Predicate generalization P2372 FINISHED
Object Schwarz–Ahlfors lemma
The Schwarz–Ahlfors lemma is a result in complex analysis that extends the classical Schwarz lemma to holomorphic maps between hyperbolic Riemann surfaces, providing curvature-based bounds on such mappings.
E899966 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: Schwarz–Ahlfors lemma | Statement: [Schwarz lemma, generalization, Schwarz–Ahlfors lemma]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Schwarz–Ahlfors lemma
Context triple: [Schwarz lemma, generalization, Schwarz–Ahlfors lemma]
  • A. Schwarz lemma
    Schwarz lemma is a fundamental result in complex analysis that constrains holomorphic self-maps of the unit disk, particularly bounding their magnitude and derivative at the origin.
  • B. Schwarz–Pick theorem
    The Schwarz–Pick theorem is a fundamental result in complex analysis that characterizes holomorphic self-maps of the unit disk by showing they are distance-decreasing with respect to the hyperbolic (Poincaré) metric.
  • C. Bieberbach conjecture
    The Bieberbach conjecture, now a theorem, is a landmark result in complex analysis that characterizes the size of Taylor coefficients of normalized univalent (injective) holomorphic functions on the unit disk.
  • D. Koebe quarter theorem
    The Koebe quarter theorem is a fundamental result in complex analysis stating that any univalent holomorphic function on the unit disk maps it onto a domain containing a disk of radius one quarter, providing a sharp bound on the size of the image.
  • E. 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.
  • 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: Schwarz–Ahlfors lemma
Triple: [Schwarz lemma, generalization, Schwarz–Ahlfors lemma]
Generated description
The Schwarz–Ahlfors lemma is a result in complex analysis that extends the classical Schwarz lemma to holomorphic maps between hyperbolic Riemann surfaces, providing curvature-based bounds on such mappings.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Schwarz–Ahlfors lemma
Target entity description: The Schwarz–Ahlfors lemma is a result in complex analysis that extends the classical Schwarz lemma to holomorphic maps between hyperbolic Riemann surfaces, providing curvature-based bounds on such mappings.
  • A. Schwarz lemma
    Schwarz lemma is a fundamental result in complex analysis that constrains holomorphic self-maps of the unit disk, particularly bounding their magnitude and derivative at the origin.
  • B. Schwarz–Pick theorem chosen
    The Schwarz–Pick theorem is a fundamental result in complex analysis that characterizes holomorphic self-maps of the unit disk by showing they are distance-decreasing with respect to the hyperbolic (Poincaré) metric.
  • C. Bieberbach conjecture
    The Bieberbach conjecture, now a theorem, is a landmark result in complex analysis that characterizes the size of Taylor coefficients of normalized univalent (injective) holomorphic functions on the unit disk.
  • D. Koebe quarter theorem
    The Koebe quarter theorem is a fundamental result in complex analysis stating that any univalent holomorphic function on the unit disk maps it onto a domain containing a disk of radius one quarter, providing a sharp bound on the size of the image.
  • E. 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.
  • F. None of above.

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_69d6aa8a6a548190a750f944ccdc8064 completed April 8, 2026, 7:20 p.m.
NER Named-entity recognition batch_69d795d1e918819090c71f5a077fa15a completed April 9, 2026, 12:04 p.m.
NED1 Entity disambiguation (via context triple) batch_69e3a9644ff08190a3005e4f6a8243fe completed April 18, 2026, 3:55 p.m.
NEDg Description generation batch_69e3ad00b5c08190a7bf3ecbeae76d88 completed April 18, 2026, 4:10 p.m.
NED2 Entity disambiguation (via description) batch_69e3b1efe4a88190884eb5186954cf39 completed April 18, 2026, 4:31 p.m.
Created at: April 8, 2026, 9:24 p.m.