Triple

T209665
Position Surface form Disambiguated ID Type / Status
Subject binomial theorem E4686 entity
Predicate isSpecialCaseOf P2372 FINISHED
Object multinomial theorem
The multinomial theorem is a fundamental algebraic formula that generalizes the binomial theorem to express powers of sums with any number of terms using multinomial coefficients.
E26831 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: multinomial theorem | Statement: [binomial theorem, isSpecialCaseOf, multinomial theorem]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: multinomial theorem
Context triple: [binomial theorem, isSpecialCaseOf, multinomial theorem]
  • A. binomial theorem
    The binomial theorem is a fundamental algebraic formula that provides a systematic way to expand powers of binomial expressions, playing a key role in combinatorics and mathematical analysis.
  • B. central limit theorem
    The central limit theorem is a fundamental result in probability theory stating that the sum (or average) of many independent, identically distributed random variables tends to follow a normal distribution, regardless of the original variables’ distribution, under mild conditions.
  • C. Minkowski sum
    The Minkowski sum is a fundamental operation in geometry and convex analysis that combines two sets by adding every vector in one set to every vector in the other, widely used in areas such as optimization, robotics, and computational geometry.
  • D. Born–Huang expansion
    The Born–Huang expansion is a quantum mechanical method that systematically improves upon the Born–Oppenheimer approximation by including couplings between electronic and nuclear motions in molecular systems.
  • E. Bose–Einstein statistics
    Bose–Einstein statistics is a quantum statistical framework that describes the distribution and collective behavior of indistinguishable bosons, underpinning phenomena such as Bose–Einstein condensation.
  • 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: multinomial theorem
Triple: [binomial theorem, isSpecialCaseOf, multinomial theorem]
Generated description
The multinomial theorem is a fundamental algebraic formula that generalizes the binomial theorem to express powers of sums with any number of terms using multinomial coefficients.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: multinomial theorem
Target entity description: The multinomial theorem is a fundamental algebraic formula that generalizes the binomial theorem to express powers of sums with any number of terms using multinomial coefficients.
  • A. binomial theorem
    The binomial theorem is a fundamental algebraic formula that provides a systematic way to expand powers of binomial expressions, playing a key role in combinatorics and mathematical analysis.
  • B. central limit theorem
    The central limit theorem is a fundamental result in probability theory stating that the sum (or average) of many independent, identically distributed random variables tends to follow a normal distribution, regardless of the original variables’ distribution, under mild conditions.
  • C. Minkowski sum
    The Minkowski sum is a fundamental operation in geometry and convex analysis that combines two sets by adding every vector in one set to every vector in the other, widely used in areas such as optimization, robotics, and computational geometry.
  • D. Born–Huang expansion
    The Born–Huang expansion is a quantum mechanical method that systematically improves upon the Born–Oppenheimer approximation by including couplings between electronic and nuclear motions in molecular systems.
  • E. Bose–Einstein statistics
    Bose–Einstein statistics is a quantum statistical framework that describes the distribution and collective behavior of indistinguishable bosons, underpinning phenomena such as Bose–Einstein condensation.
  • 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_69a25737567c81908f9c505300239181 completed Feb. 28, 2026, 2:47 a.m.
NER Named-entity recognition batch_69a25c082fa08190b4d097bf68300222 completed Feb. 28, 2026, 3:07 a.m.
NED1 Entity disambiguation (via context triple) batch_69a332c9e99081909026bf5bfeb6c86c completed Feb. 28, 2026, 6:24 p.m.
NEDg Description generation batch_69a33326c71c81908c02320901915ce3 completed Feb. 28, 2026, 6:25 p.m.
NED2 Entity disambiguation (via description) batch_69a3338948808190b6fd60524c721fd5 completed Feb. 28, 2026, 6:27 p.m.
Created at: Feb. 28, 2026, 2:51 a.m.