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.