Triple

T7304623
Position Surface form Disambiguated ID Type / Status
Subject J. W. S. Cassels E167942 entity
Predicate notableWork P4 FINISHED
Object Rational Quadratic Forms
Rational Quadratic Forms is a classic monograph in number theory that systematically develops the arithmetic theory of quadratic forms over the rational numbers.
E654583 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: Rational Quadratic Forms | Statement: [J. W. S. Cassels, notableWork, Rational Quadratic Forms]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Rational Quadratic Forms
Context triple: [J. W. S. Cassels, notableWork, Rational Quadratic Forms]
  • A. Hermitian forms (work on quadratic forms)
    Hermitian forms (work on quadratic forms) are a class of complex-valued quadratic forms that are linear in one variable and conjugate-linear in the other, generalizing real symmetric quadratic forms and playing a central role in linear algebra and functional analysis.
  • B. Über die Bildung des Formensystems der ternären biquadratischen Form
    "Über die Bildung des Formensystems der ternären biquadratischen Form" is the 1907 doctoral dissertation of mathematician Emmy Noether, in which she investigates the invariant theory of certain higher-degree algebraic forms.
  • C. Higher composition laws I–IV
    Higher composition laws I–IV is a landmark four-part series of papers by Manjul Bhargava that generalizes Gauss’s composition of binary quadratic forms and develops new structures and methods in algebraic number theory.
  • D. Gaussian rationals ℚ(i)
    Gaussian rationals ℚ(i) are the field of complex numbers whose real and imaginary parts are rational, formed by adjoining the imaginary unit i to the rational numbers.
  • E. Hodge–Riemann bilinear relations
    The Hodge–Riemann bilinear relations are fundamental positivity and orthogonality conditions on the intersection form in Hodge theory that underpin results such as the hard Lefschetz theorem and the Hodge index theorem.
  • 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: Rational Quadratic Forms
Triple: [J. W. S. Cassels, notableWork, Rational Quadratic Forms]
Generated description
Rational Quadratic Forms is a classic monograph in number theory that systematically develops the arithmetic theory of quadratic forms over the rational numbers.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Rational Quadratic Forms
Target entity description: Rational Quadratic Forms is a classic monograph in number theory that systematically develops the arithmetic theory of quadratic forms over the rational numbers.
  • A. Hermitian forms (work on quadratic forms)
    Hermitian forms (work on quadratic forms) are a class of complex-valued quadratic forms that are linear in one variable and conjugate-linear in the other, generalizing real symmetric quadratic forms and playing a central role in linear algebra and functional analysis.
  • B. Über die Bildung des Formensystems der ternären biquadratischen Form
    "Über die Bildung des Formensystems der ternären biquadratischen Form" is the 1907 doctoral dissertation of mathematician Emmy Noether, in which she investigates the invariant theory of certain higher-degree algebraic forms.
  • C. Higher composition laws I–IV
    Higher composition laws I–IV is a landmark four-part series of papers by Manjul Bhargava that generalizes Gauss’s composition of binary quadratic forms and develops new structures and methods in algebraic number theory.
  • D. Gaussian rationals ℚ(i)
    Gaussian rationals ℚ(i) are the field of complex numbers whose real and imaginary parts are rational, formed by adjoining the imaginary unit i to the rational numbers.
  • E. Hodge–Riemann bilinear relations
    The Hodge–Riemann bilinear relations are fundamental positivity and orthogonality conditions on the intersection form in Hodge theory that underpin results such as the hard Lefschetz theorem and the Hodge index theorem.
  • 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.