Triple

T1358586
Position Surface form Disambiguated ID Type / Status
Subject Joseph-Louis Lagrange E29045 entity
Predicate knownFor P22 FINISHED
Object Lagrange's four-square theorem
Lagrange's four-square theorem is a fundamental result in number theory stating that every natural number can be expressed as the sum of four integer squares.
E156185 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: Lagrange's four-square theorem | Statement: [Joseph-Louis Lagrange, knownFor, Lagrange's four-square theorem]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Lagrange's four-square theorem
Context triple: [Joseph-Louis Lagrange, knownFor, Lagrange's four-square theorem]
  • A. Fermat's theorem on sums of two squares
    Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
  • B. Fermat polygonal number theorem
    The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
  • C. Fermat's Last Theorem
    Fermat's Last Theorem is a famous statement in number theory asserting that there are no whole-number solutions to the equation xⁿ + yⁿ = zⁿ for integers n greater than 2, a problem that remained unsolved for over three centuries until it was proved by Andrew Wiles in the 1990s.
  • D. Hardy–Ramanujan asymptotic formula
    The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
  • E. Gauss’s lemma in number theory
    Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
  • 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: Lagrange's four-square theorem
Triple: [Joseph-Louis Lagrange, knownFor, Lagrange's four-square theorem]
Generated description
Lagrange's four-square theorem is a fundamental result in number theory stating that every natural number can be expressed as the sum of four integer squares.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Lagrange's four-square theorem
Target entity description: Lagrange's four-square theorem is a fundamental result in number theory stating that every natural number can be expressed as the sum of four integer squares.
  • A. Fermat's theorem on sums of two squares
    Fermat's theorem on sums of two squares is a result in number theory stating exactly which prime numbers (and, more generally, which integers) can be expressed as the sum of two perfect squares.
  • B. Fermat polygonal number theorem
    The Fermat polygonal number theorem is a result in number theory stating that every positive integer can be expressed as a sum of a fixed number of polygonal numbers of a given order.
  • C. Fermat's Last Theorem
    Fermat's Last Theorem is a famous statement in number theory asserting that there are no whole-number solutions to the equation xⁿ + yⁿ = zⁿ for integers n greater than 2, a problem that remained unsolved for over three centuries until it was proved by Andrew Wiles in the 1990s.
  • D. Hardy–Ramanujan asymptotic formula
    The Hardy–Ramanujan asymptotic formula is a landmark result in number theory that gives an approximate expression for the partition function p(n), describing how the number of integer partitions of n grows rapidly with n.
  • E. Gauss’s lemma in number theory
    Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
  • 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_69a498d77abc8190913bf57e5f51d2c4 completed March 1, 2026, 7:51 p.m.
NER Named-entity recognition batch_69a4c28f5b988190b0be4504eabb919d completed March 1, 2026, 10:49 p.m.
NED1 Entity disambiguation (via context triple) batch_69acce701a84819094815ab6e8383b76 completed March 8, 2026, 1:18 a.m.
NEDg Description generation batch_69accf15d910819098e9a4b24247881b completed March 8, 2026, 1:21 a.m.
NED2 Entity disambiguation (via description) batch_69accff8af488190a7580cf7c02ceed9 completed March 8, 2026, 1:25 a.m.
Created at: March 1, 2026, 7:56 p.m.