Triple

T7593279
Position Surface form Disambiguated ID Type / Status
Subject Condon–Shortley phase E179791 entity
Predicate relatedTo P37 FINISHED
Object Wigner 3j symbols
Wigner 3j symbols are mathematical coefficients used in quantum mechanics and angular momentum theory to describe the coupling and recoupling of three angular momenta with well-defined symmetry properties.
E674699 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 3j symbols | Statement: [Condon–Shortley phase, relatedTo, Wigner 3j symbols]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Wigner 3j symbols
Context triple: [Condon–Shortley phase, relatedTo, Wigner 3j 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. Jacobi polynomials
    Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
  • E. 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.
  • 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 3j symbols
Triple: [Condon–Shortley phase, relatedTo, Wigner 3j symbols]
Generated description
Wigner 3j symbols are mathematical coefficients used in quantum mechanics and angular momentum theory to describe the coupling and recoupling of three angular momenta with well-defined symmetry properties.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Wigner 3j symbols
Target entity description: Wigner 3j symbols are mathematical coefficients used in quantum mechanics and angular momentum theory to describe the coupling and recoupling of three angular momenta with well-defined symmetry properties.
  • 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. Jacobi polynomials
    Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
  • E. 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.
  • 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_69c69f3487ec8190bf7acdf2dd91e6d6 completed March 27, 2026, 3:16 p.m.
NER Named-entity recognition batch_69c6f9bab3a08190a2c36b2c72a1de25 completed March 27, 2026, 9:42 p.m.
NED1 Entity disambiguation (via context triple) batch_69c86197fe0881908307a411cabdca7f completed March 28, 2026, 11:17 p.m.
NEDg Description generation batch_69c86223bfec8190b47f840e39c9a51a completed March 28, 2026, 11:20 p.m.
NED2 Entity disambiguation (via description) batch_69c862bb95e881909a60608a5279238d completed March 28, 2026, 11:22 p.m.
Created at: March 27, 2026, 3:53 p.m.