Triple

T7304622
Position Surface form Disambiguated ID Type / Status
Subject J. W. S. Cassels E167942 entity
Predicate notableWork P4 FINISHED
Object An Introduction to Diophantine Approximation
"An Introduction to Diophantine Approximation" is a classic mathematical monograph that systematically develops the theory of approximating real numbers by rationals, aimed at advanced undergraduates and researchers in number theory.
E654582 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: An Introduction to Diophantine Approximation | Statement: [J. W. S. Cassels, notableWork, An Introduction to Diophantine Approximation]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: An Introduction to Diophantine Approximation
Context triple: [J. W. S. Cassels, notableWork, An Introduction to Diophantine Approximation]
  • A. Diophantine approximation
    Diophantine approximation is a branch of number theory that studies how closely real numbers can be approximated by rational numbers, often with quantitative bounds on the quality of approximation.
  • B. Baker theorem on linear forms in logarithms
    The Baker theorem on linear forms in logarithms is a fundamental result in transcendental number theory that provides explicit lower bounds for nonzero linear combinations of logarithms of algebraic numbers, with powerful applications to Diophantine equations and Diophantine approximation.
  • C. An Introduction to the Theory of Numbers
    An Introduction to the Theory of Numbers is a classic textbook in number theory, co-authored by G. H. Hardy, that systematically develops fundamental concepts such as divisibility, prime numbers, Diophantine equations, and quadratic forms.
  • D. Dirichlet approximation theorem
    The Dirichlet approximation theorem is a fundamental result in Diophantine approximation that guarantees, for any real number and positive integer, the existence of a nearby rational number with bounded denominator and small approximation error.
  • E. Three Pearls of Number Theory
    Three Pearls of Number Theory is a classic mathematical text that presents three elegant and accessible problems in number theory, illustrating deep ideas through simple, beautifully explained examples.
  • 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: An Introduction to Diophantine Approximation
Triple: [J. W. S. Cassels, notableWork, An Introduction to Diophantine Approximation]
Generated description
"An Introduction to Diophantine Approximation" is a classic mathematical monograph that systematically develops the theory of approximating real numbers by rationals, aimed at advanced undergraduates and researchers in number theory.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: An Introduction to Diophantine Approximation
Target entity description: "An Introduction to Diophantine Approximation" is a classic mathematical monograph that systematically develops the theory of approximating real numbers by rationals, aimed at advanced undergraduates and researchers in number theory.
  • A. Diophantine approximation
    Diophantine approximation is a branch of number theory that studies how closely real numbers can be approximated by rational numbers, often with quantitative bounds on the quality of approximation.
  • B. Baker theorem on linear forms in logarithms
    The Baker theorem on linear forms in logarithms is a fundamental result in transcendental number theory that provides explicit lower bounds for nonzero linear combinations of logarithms of algebraic numbers, with powerful applications to Diophantine equations and Diophantine approximation.
  • C. An Introduction to the Theory of Numbers
    An Introduction to the Theory of Numbers is a classic textbook in number theory, co-authored by G. H. Hardy, that systematically develops fundamental concepts such as divisibility, prime numbers, Diophantine equations, and quadratic forms.
  • D. Dirichlet approximation theorem
    The Dirichlet approximation theorem is a fundamental result in Diophantine approximation that guarantees, for any real number and positive integer, the existence of a nearby rational number with bounded denominator and small approximation error.
  • E. Three Pearls of Number Theory
    Three Pearls of Number Theory is a classic mathematical text that presents three elegant and accessible problems in number theory, illustrating deep ideas through simple, beautifully explained examples.
  • 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_69c6888c820881909fc68f689fe1c251 completed March 27, 2026, 1:39 p.m.
NER Named-entity recognition batch_69c6ebb352ec8190846eff044e08805e completed March 27, 2026, 8:42 p.m.
NED1 Entity disambiguation (via context triple) batch_69c7e55bfccc8190a46067c60c3c1a3f completed March 28, 2026, 2:27 p.m.
NEDg Description generation batch_69c7e5fbe8a8819083a892f4e54013eb completed March 28, 2026, 2:30 p.m.
NED2 Entity disambiguation (via description) batch_69c7e69ca1ac8190a398da894c6cc04e completed March 28, 2026, 2:33 p.m.
Created at: March 27, 2026, 3:01 p.m.