Triple

T16991699
Position Surface form Disambiguated ID Type / Status
Subject Non-Euclidean geometry E412207 entity
Predicate hasModel P2390 FINISHED
Object Beltrami–Klein model
The Beltrami–Klein model is a projective representation of hyperbolic geometry in which the entire hyperbolic plane is mapped inside a disk and geodesics appear as straight line segments.
E1245022 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: Beltrami–Klein model | Statement: [Non-Euclidean geometry, hasModel, Beltrami–Klein model]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Beltrami–Klein model
Context triple: [Non-Euclidean geometry, hasModel, Beltrami–Klein model]
  • A. Poincaré upper half-plane model
    The Poincaré upper half-plane model is a standard representation of the hyperbolic plane using the complex numbers with positive imaginary part, equipped with a specific metric that makes geodesics appear as semicircles and vertical lines.
  • B. Poincaré metric
    The Poincaré metric is the canonical complete Riemannian metric of constant negative curvature on simply connected Riemann surfaces like the unit disk or upper half-plane, fundamental in complex analysis and hyperbolic geometry.
  • C. Möbius geometry
    Möbius geometry is a branch of geometry that studies properties of figures invariant under Möbius (conformal) transformations of the extended complex plane or higher-dimensional spheres.
  • D. Riemann sphere
    The Riemann sphere is the complex plane plus a point at infinity, forming a one-dimensional complex manifold topologically equivalent to a sphere and used to study meromorphic functions and complex analysis.
  • E. Busemann
    Busemann is a surname most notably associated with Herbert Busemann, a German-American mathematician known for his work in geometry.
  • 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: Beltrami–Klein model
Triple: [Non-Euclidean geometry, hasModel, Beltrami–Klein model]
Generated description
The Beltrami–Klein model is a projective representation of hyperbolic geometry in which the entire hyperbolic plane is mapped inside a disk and geodesics appear as straight line segments.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Beltrami–Klein model
Target entity description: The Beltrami–Klein model is a projective representation of hyperbolic geometry in which the entire hyperbolic plane is mapped inside a disk and geodesics appear as straight line segments.
  • A. Poincaré upper half-plane model
    The Poincaré upper half-plane model is a standard representation of the hyperbolic plane using the complex numbers with positive imaginary part, equipped with a specific metric that makes geodesics appear as semicircles and vertical lines.
  • B. Poincaré metric
    The Poincaré metric is the canonical complete Riemannian metric of constant negative curvature on simply connected Riemann surfaces like the unit disk or upper half-plane, fundamental in complex analysis and hyperbolic geometry.
  • C. Möbius geometry
    Möbius geometry is a branch of geometry that studies properties of figures invariant under Möbius (conformal) transformations of the extended complex plane or higher-dimensional spheres.
  • D. Riemann sphere
    The Riemann sphere is the complex plane plus a point at infinity, forming a one-dimensional complex manifold topologically equivalent to a sphere and used to study meromorphic functions and complex analysis.
  • E. Busemann
    Busemann is a surname most notably associated with Herbert Busemann, a German-American mathematician known for his work in geometry.
  • 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_69d886cb581c8190ab05f4b429c9cd85 completed April 10, 2026, 5:12 a.m.
NER Named-entity recognition batch_69e3d280e3348190a27bd5dc7cf87c0e completed April 18, 2026, 6:50 p.m.
NED1 Entity disambiguation (via context triple) batch_6a00dc14d5688190945f7ae72f724922 completed May 10, 2026, 7:27 p.m.
NEDg Description generation batch_6a0114d5aeb0819086f1a5d279ac0d0f completed May 10, 2026, 11:29 p.m.
NED2 Entity disambiguation (via description) batch_6a0115c967b0819088e2335fd45d755b completed May 10, 2026, 11:33 p.m.
Created at: April 10, 2026, 5:32 a.m.