Triple

T4437405
Position Surface form Disambiguated ID Type / Status
Subject Rogers–Ramanujan-type identities E95684 entity
Predicate relatedTo P37 FINISHED
Object Schur identities
Schur identities are a family of partition identities in number theory that generalize and complement the Rogers–Ramanujan identities, often expressed through q-series and combinatorial interpretations.
E95684 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: Schur identities | Statement: [Rogers–Ramanujan-type identities, relatedTo, Schur identities]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Schur identities
Context triple: [Rogers–Ramanujan-type identities, relatedTo, Schur identities]
  • A. Rogers–Ramanujan-type identities
    Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
  • B. Clebsch–Aronhold invariants
    The Clebsch–Aronhold invariants are classical algebraic invariants associated with binary forms, particularly quartic forms, that play a key role in invariant theory and the classification of algebraic curves.
  • C. Cauchy determinant
    The Cauchy determinant is a classical determinant formula in linear algebra that gives a closed-form expression for matrices with entries of the form 1/(x_i + y_j), named after the French mathematician Augustin-Louis Cauchy.
  • D. Jacobi’s four-square theorem
    Jacobi’s four-square theorem is a fundamental result in number theory that gives a precise formula for the number of ways an integer can be expressed as a sum of four squares.
  • E. Vandermonde's identity
    Vandermonde's identity is a fundamental combinatorial formula that expresses a binomial coefficient with a sum index as a sum of products of binomial coefficients, often visualized via counting arguments or generating functions.
  • 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: Schur identities
Triple: [Rogers–Ramanujan-type identities, relatedTo, Schur identities]
Generated description
Schur identities are a family of partition identities in number theory that generalize and complement the Rogers–Ramanujan identities, often expressed through q-series and combinatorial interpretations.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Schur identities
Target entity description: Schur identities are a family of partition identities in number theory that generalize and complement the Rogers–Ramanujan identities, often expressed through q-series and combinatorial interpretations.
  • A. Rogers–Ramanujan-type identities chosen
    Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
  • B. Clebsch–Aronhold invariants
    The Clebsch–Aronhold invariants are classical algebraic invariants associated with binary forms, particularly quartic forms, that play a key role in invariant theory and the classification of algebraic curves.
  • C. Cauchy determinant
    The Cauchy determinant is a classical determinant formula in linear algebra that gives a closed-form expression for matrices with entries of the form 1/(x_i + y_j), named after the French mathematician Augustin-Louis Cauchy.
  • D. Jacobi’s four-square theorem
    Jacobi’s four-square theorem is a fundamental result in number theory that gives a precise formula for the number of ways an integer can be expressed as a sum of four squares.
  • E. Vandermonde's identity
    Vandermonde's identity is a fundamental combinatorial formula that expresses a binomial coefficient with a sum index as a sum of products of binomial coefficients, often visualized via counting arguments or generating functions.
  • 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_69b3453ea2b48190a26f154b3b8fece5 completed March 12, 2026, 10:59 p.m.
NER Named-entity recognition batch_69b3558b1d4481909060ede5e0ded4bc completed March 13, 2026, 12:08 a.m.
NED1 Entity disambiguation (via context triple) batch_69b6137960908190814cbdf0b4e56542 completed March 15, 2026, 2:03 a.m.
NEDg Description generation batch_69b61439a86c8190849c5af718ddc647 completed March 15, 2026, 2:06 a.m.
NED2 Entity disambiguation (via description) batch_69b614d6106c81908a601f540622f934 completed March 15, 2026, 2:09 a.m.
Created at: March 12, 2026, 11:31 p.m.