Triple

T9297066
Position Surface form Disambiguated ID Type / Status
Subject Chebotarev density theorem E223663 entity
Predicate strengthens P3261 FINISHED
Object Frobenius density theorem
The Frobenius density theorem is a result in algebraic number theory that describes how often prime ideals in a number field have a given Frobenius conjugacy class, providing a precursor to the more general Chebotarev density theorem.
E223663 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: Frobenius density theorem | Statement: [Chebotarev density theorem, strengthens, Frobenius density theorem]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Frobenius density theorem
Context triple: [Chebotarev density theorem, strengthens, Frobenius density theorem]
  • A. Chebotarev density theorem
    The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
  • B. Artin’s conjecture on L-functions
    Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.
  • C. Sato–Tate distribution (for families of elliptic curves)
    The Sato–Tate distribution (for families of elliptic curves) is a probabilistic law describing how the normalized Frobenius traces (or equivalently, the angles in the Hasse bound) of elliptic curves are distributed, typically following a specific sine-squared measure on the interval [0, π].
  • D. Siegel’s theorem on zeros of L-functions
    Siegel’s theorem on zeros of L-functions is a result in analytic number theory that gives strong bounds on how close nontrivial zeros of Dirichlet L-functions can approach 1, with deep implications for the distribution of primes in arithmetic progressions.
  • E. Hasse–Arf theorem
    The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
  • 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: Frobenius density theorem
Triple: [Chebotarev density theorem, strengthens, Frobenius density theorem]
Generated description
The Frobenius density theorem is a result in algebraic number theory that describes how often prime ideals in a number field have a given Frobenius conjugacy class, providing a precursor to the more general Chebotarev density theorem.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Frobenius density theorem
Target entity description: The Frobenius density theorem is a result in algebraic number theory that describes how often prime ideals in a number field have a given Frobenius conjugacy class, providing a precursor to the more general Chebotarev density theorem.
  • A. Chebotarev density theorem chosen
    The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
  • B. Artin’s conjecture on L-functions
    Artin’s conjecture on L-functions is a major unproven hypothesis in number theory asserting that nontrivial Artin L-functions associated to Galois representations are entire, with deep implications for the distribution of primes and the structure of number fields.
  • C. Sato–Tate distribution (for families of elliptic curves)
    The Sato–Tate distribution (for families of elliptic curves) is a probabilistic law describing how the normalized Frobenius traces (or equivalently, the angles in the Hasse bound) of elliptic curves are distributed, typically following a specific sine-squared measure on the interval [0, π].
  • D. Siegel’s theorem on zeros of L-functions
    Siegel’s theorem on zeros of L-functions is a result in analytic number theory that gives strong bounds on how close nontrivial zeros of Dirichlet L-functions can approach 1, with deep implications for the distribution of primes in arithmetic progressions.
  • E. Hasse–Arf theorem
    The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
  • 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_69ca8423edb08190bc0c91287a484768 completed March 30, 2026, 2:09 p.m.
NER Named-entity recognition batch_69cd089c9c588190967404c9eb938dfb completed April 1, 2026, 11:59 a.m.
NED1 Entity disambiguation (via context triple) batch_69d0b251c4148190a94fafdc23a601d6 completed April 4, 2026, 6:40 a.m.
NEDg Description generation batch_69d0b65ea4548190b445563ac695b008 completed April 4, 2026, 6:57 a.m.
NED2 Entity disambiguation (via description) batch_69d0b6f504a48190878db828312e8a97 completed April 4, 2026, 7 a.m.
Created at: March 30, 2026, 7:36 p.m.