Triple

T4461377
Position Surface form Disambiguated ID Type / Status
Subject Wigner–Eckart theorem E98261 entity
Predicate relatedTo P37 FINISHED
Object Wigner 6-j symbols
Wigner 6-j symbols are mathematical coefficients in quantum angular momentum theory that encode the recoupling of three angular momenta and play a key role in simplifying calculations involving the Wigner–Eckart theorem.
E443146 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: Wigner 6-j symbols | Statement: [Wigner–Eckart theorem, relatedTo, Wigner 6-j symbols]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Wigner 6-j symbols
Context triple: [Wigner–Eckart theorem, relatedTo, Wigner 6-j symbols]
  • A. Clebsch–Gordan coefficients
    Clebsch–Gordan coefficients are numerical factors in quantum mechanics and representation theory that describe how to combine two angular momenta (or group representations) into a single resultant one.
  • B. Wigner–Eckart theorem
    The Wigner–Eckart theorem is a fundamental result in quantum mechanics that factorizes matrix elements of tensor operators into a reduced matrix element and a purely geometric part given by Clebsch–Gordan coefficients, greatly simplifying angular momentum calculations.
  • C. Racah algebra
    Racah algebra is a mathematical structure in representation theory and quantum mechanics that encodes the symmetries and coupling properties of angular momenta, particularly through Racah coefficients and related special functions.
  • D. Brillouin–Wigner perturbation theory
    Brillouin–Wigner perturbation theory is a formulation of quantum mechanical perturbation theory that uses an energy-dependent effective Hamiltonian to obtain improved approximations to eigenvalues and eigenstates.
  • E. Infeld–van der Waerden formalism
    The Infeld–van der Waerden formalism is a mathematical framework in general relativity that reformulates the theory using spinor calculus to describe gravitational and electromagnetic fields.
  • 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: Wigner 6-j symbols
Triple: [Wigner–Eckart theorem, relatedTo, Wigner 6-j symbols]
Generated description
Wigner 6-j symbols are mathematical coefficients in quantum angular momentum theory that encode the recoupling of three angular momenta and play a key role in simplifying calculations involving the Wigner–Eckart theorem.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Wigner 6-j symbols
Target entity description: Wigner 6-j symbols are mathematical coefficients in quantum angular momentum theory that encode the recoupling of three angular momenta and play a key role in simplifying calculations involving the Wigner–Eckart theorem.
  • A. Clebsch–Gordan coefficients
    Clebsch–Gordan coefficients are numerical factors in quantum mechanics and representation theory that describe how to combine two angular momenta (or group representations) into a single resultant one.
  • B. Wigner–Eckart theorem
    The Wigner–Eckart theorem is a fundamental result in quantum mechanics that factorizes matrix elements of tensor operators into a reduced matrix element and a purely geometric part given by Clebsch–Gordan coefficients, greatly simplifying angular momentum calculations.
  • C. Racah algebra chosen
    Racah algebra is a mathematical structure in representation theory and quantum mechanics that encodes the symmetries and coupling properties of angular momenta, particularly through Racah coefficients and related special functions.
  • D. Brillouin–Wigner perturbation theory
    Brillouin–Wigner perturbation theory is a formulation of quantum mechanical perturbation theory that uses an energy-dependent effective Hamiltonian to obtain improved approximations to eigenvalues and eigenstates.
  • E. Infeld–van der Waerden formalism
    The Infeld–van der Waerden formalism is a mathematical framework in general relativity that reformulates the theory using spinor calculus to describe gravitational and electromagnetic fields.
  • 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_69b3454a7c608190944f5455c8031d73 completed March 12, 2026, 10:59 p.m.
NER Named-entity recognition batch_69b35674f718819089388c3924dd1414 completed March 13, 2026, 12:12 a.m.
NED1 Entity disambiguation (via context triple) batch_69b6376c8e74819087aa2ff3e4429674 completed March 15, 2026, 4:37 a.m.
NEDg Description generation batch_69b63b3e39608190890b6da972f28eeb completed March 15, 2026, 4:53 a.m.
NED2 Entity disambiguation (via description) batch_69b63baa6e9081909d9f87d528b4f2bb completed March 15, 2026, 4:55 a.m.
Created at: March 12, 2026, 11:34 p.m.