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.