Triple

T10991824
Position Surface form Disambiguated ID Type / Status
Subject Schwarz lemma E259769 entity
Predicate generalization P2372 FINISHED
Object 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.
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–Pick theorem | Statement: [Schwarz lemma, generalization, Schwarz–Pick theorem]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Schwarz–Pick theorem
Context triple: [Schwarz lemma, generalization, Schwarz–Pick theorem]
  • 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. 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.
  • 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. Riemann mapping theorem
    The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
  • E. 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.
  • 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–Pick theorem
Triple: [Schwarz lemma, generalization, Schwarz–Pick theorem]
Generated description
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.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Schwarz–Pick theorem
Target entity description: 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.
  • 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. 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.
  • 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. Riemann mapping theorem
    The Riemann mapping theorem is a fundamental result in complex analysis stating that any non-empty simply connected open subset of the complex plane (other than the whole plane) can be conformally mapped onto the open unit disk.
  • E. 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.
  • 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_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_69e374639f1481908ea372b81a834f6f completed April 18, 2026, 12:09 p.m.
NEDg Description generation batch_69e37ab6ca788190ac41f9494ad9a47f completed April 18, 2026, 12:36 p.m.
NED2 Entity disambiguation (via description) batch_69e37c9439fc8190a69cfb1a13da4c19 completed April 18, 2026, 12:44 p.m.
Created at: April 8, 2026, 9:24 p.m.