Triple

T2364526
Position Surface form Disambiguated ID Type / Status
Subject Riemann mapping theorem E47349 entity
Predicate relatedTo P37 FINISHED
Object 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.
E259769 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 lemma | Statement: [Riemann mapping theorem, relatedTo, Schwarz lemma]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Schwarz lemma
Context triple: [Riemann mapping theorem, relatedTo, Schwarz lemma]
  • A. 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.
  • B. Cauchy–Hadamard theorem
    The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
  • C. Riemann–Lebesgue lemma
    The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
  • D. Cauchy integral formula
    The Cauchy integral formula is a fundamental result in complex analysis that expresses the value of a holomorphic function inside a disk in terms of a contour integral of the function around the disk’s boundary.
  • E. Weierstrass preparation theorem
    The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
  • 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 lemma
Triple: [Riemann mapping theorem, relatedTo, Schwarz lemma]
Generated description
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.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Schwarz lemma
Target entity description: 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.
  • A. 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.
  • B. Cauchy–Hadamard theorem
    The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
  • C. Riemann–Lebesgue lemma
    The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
  • D. Cauchy integral formula
    The Cauchy integral formula is a fundamental result in complex analysis that expresses the value of a holomorphic function inside a disk in terms of a contour integral of the function around the disk’s boundary.
  • E. Weierstrass preparation theorem
    The Weierstrass preparation theorem is a fundamental result in complex analysis and analytic geometry that locally expresses analytic functions near a zero as a product of a polynomial and a unit, enabling a power-series analogue of factorization.
  • 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_69a88a1a4a6081908645b0f2914521ab completed March 4, 2026, 7:38 p.m.
NER Named-entity recognition batch_69abc7486cb48190acef1891cc87bdb1 completed March 7, 2026, 6:35 a.m.
NED1 Entity disambiguation (via context triple) batch_69aea896e0388190aabff2d70787dc43 completed March 9, 2026, 11:01 a.m.
NEDg Description generation batch_69aea91ce164819091aa24b287f9fb8e completed March 9, 2026, 11:03 a.m.
NED2 Entity disambiguation (via description) batch_69aea999b864819084134c670e7c5d9c completed March 9, 2026, 11:06 a.m.
Created at: March 4, 2026, 7:55 p.m.