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.