Triple

T13660503
Position Surface form Disambiguated ID Type / Status
Subject Picard theorem E326981 entity
Predicate usedInProofOf P27215 FINISHED
Object Casorati–Weierstrass theorem (converse directions and related results)
Casorati–Weierstrass theorem (converse directions and related results) refers to a collection of results in complex analysis that explore when the chaotic boundary behavior of holomorphic functions near a point implies the presence of an essential singularity, often using tools such as Picard’s theorem.
E326981 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: Casorati–Weierstrass theorem (converse directions and related results) | Statement: [Picard theorem, usedInProofOf, Casorati–Weierstrass theorem (converse directions and related results)]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Casorati–Weierstrass theorem (converse directions and related results)
Context triple: [Picard theorem, usedInProofOf, Casorati–Weierstrass theorem (converse directions and related results)]
  • A. 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.
  • B. Malgrange–Ehrenpreis theorem
    The Malgrange–Ehrenpreis theorem is a fundamental result in the theory of partial differential equations stating that every linear partial differential operator with constant coefficients admits a fundamental solution.
  • C. 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.
  • D. 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."
  • E. Malgrange preparation theorem
    The Malgrange preparation theorem is a fundamental result in analysis and singularity theory that generalizes the Weierstrass preparation theorem to smooth functions, providing a local factorization of such functions near singular points.
  • 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: Casorati–Weierstrass theorem (converse directions and related results)
Triple: [Picard theorem, usedInProofOf, Casorati–Weierstrass theorem (converse directions and related results)]
Generated description
Casorati–Weierstrass theorem (converse directions and related results) refers to a collection of results in complex analysis that explore when the chaotic boundary behavior of holomorphic functions near a point implies the presence of an essential singularity, often using tools such as Picard’s theorem.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Casorati–Weierstrass theorem (converse directions and related results)
Target entity description: Casorati–Weierstrass theorem (converse directions and related results) refers to a collection of results in complex analysis that explore when the chaotic boundary behavior of holomorphic functions near a point implies the presence of an essential singularity, often using tools such as Picard’s theorem.
  • A. Picard theorem chosen
    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.
  • B. Malgrange–Ehrenpreis theorem
    The Malgrange–Ehrenpreis theorem is a fundamental result in the theory of partial differential equations stating that every linear partial differential operator with constant coefficients admits a fundamental solution.
  • C. 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.
  • D. 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."
  • E. Malgrange preparation theorem
    The Malgrange preparation theorem is a fundamental result in analysis and singularity theory that generalizes the Weierstrass preparation theorem to smooth functions, providing a local factorization of such functions near singular points.
  • 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_69d8076d8270819092afc2f0e9c359a8 completed April 9, 2026, 8:09 p.m.
NER Named-entity recognition batch_69dbc620df208190afaccf3ddd10aa60 completed April 12, 2026, 4:19 p.m.
NED1 Entity disambiguation (via context triple) batch_69f78b08d27c8190badc612c26423c0e completed May 3, 2026, 5:51 p.m.
NEDg Description generation batch_69f78fd0d29481908bd44bda28e3b2c1 completed May 3, 2026, 6:11 p.m.
NED2 Entity disambiguation (via description) batch_69f7908d92e08190918525c59cb37b55 completed May 3, 2026, 6:14 p.m.
Created at: April 9, 2026, 9:52 p.m.