Triple

T25432960

Position	Surface form	Disambiguated ID	Type / Status
Subject	Khintchine theorem	`E637305`	entity
Predicate	generalizationOf	`P2372`	FINISHED
Object	Borel–Cantelli lemma applications in Diophantine approximation Borel–Cantelli lemma applications in Diophantine approximation concern probabilistic methods used to determine how well and how often real numbers can be approximated by rationals, forming the foundational framework that results like Khintchine’s theorem refine and extend.	`E1681766`	NE FINISHED

How this triple was built (2 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: Borel–Cantelli lemma applications in Diophantine approximation | Statement: [Khintchine theorem, generalizationOf, Borel–Cantelli lemma applications in Diophantine approximation]

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: Borel–Cantelli lemma applications in Diophantine approximation
Triple: [Khintchine theorem, generalizationOf, Borel–Cantelli lemma applications in Diophantine approximation]

Generated description

Borel–Cantelli lemma applications in Diophantine approximation concern probabilistic methods used to determine how well and how often real numbers can be approximated by rationals, forming the foundational framework that results like Khintchine’s theorem refine and extend.

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_69e75db58a1c8190891b9ff7c2f8414e`	completed	April 21, 2026, 11:21 a.m.
NER	Named-entity recognition	`batch_69f5f6dc7d088190b1e4c191172ea256`	completed	May 2, 2026, 1:06 p.m.
NED1	Entity disambiguation (via context triple)	`batch_6a10ad594914819095995ca301fe5e2b`	completed	May 22, 2026, 7:24 p.m.
NEDg	Description generation	`batch_6a10adba8e0c8190af36d9471078272a`	completed	May 22, 2026, 7:25 p.m.
NED2	Entity disambiguation (via description)	`batch_6a10ae3652a88190afab1481d34ed1e5`	completed	May 22, 2026, 7:27 p.m.

Created at: April 21, 2026, 1:58 p.m.