Triple

T21610212
Position Surface form Disambiguated ID Type / Status
Subject Jakob Steiner E533284 entity
Predicate notableWork P4 FINISHED
Object Steiner’s theorem in projective geometry
Steiner’s theorem in projective geometry is a classical result that characterizes the locus and incidence properties of points or conics associated with a complete quadrilateral (or related projective configurations), illustrating fundamental principles of projective transformations and duality.
E1493313 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: Steiner’s theorem in projective geometry | Statement: [Jakob Steiner, notableWork, Steiner’s theorem in projective geometry]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Steiner’s theorem in projective geometry
Context triple: [Jakob Steiner, notableWork, Steiner’s theorem in projective geometry]
  • A. Veblen axioms for projective geometry
    The Veblen axioms for projective geometry are a foundational set of incidence-based axioms introduced by Oswald Veblen to rigorously formalize the structure of projective spaces.
  • B. Traité des propriétés projectives des figures
    Traité des propriétés projectives des figures is a foundational 19th-century mathematical treatise that systematically develops projective geometry and helped establish it as an independent discipline.
  • C. On the Principles of Geometry
    "On the Principles of Geometry" is Nikolai Lobachevsky’s foundational work that introduced non-Euclidean (hyperbolic) geometry, challenging the universality of Euclid’s parallel postulate.
  • D. Cremona group of the projective plane
    The Cremona group of the projective plane is the group of all birational self-maps of the complex projective plane, serving as a fundamental object in algebraic geometry and the study of plane transformations.
  • E. Sylvester–Gallai theorem
    The Sylvester–Gallai theorem is a result in incidence geometry stating that for any finite set of points in the Euclidean plane not all on a single line, there exists a line that passes through exactly two of the points.
  • 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: Steiner’s theorem in projective geometry
Triple: [Jakob Steiner, notableWork, Steiner’s theorem in projective geometry]
Generated description
Steiner’s theorem in projective geometry is a classical result that characterizes the locus and incidence properties of points or conics associated with a complete quadrilateral (or related projective configurations), illustrating fundamental principles of projective transformations and duality.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Steiner’s theorem in projective geometry
Target entity description: Steiner’s theorem in projective geometry is a classical result that characterizes the locus and incidence properties of points or conics associated with a complete quadrilateral (or related projective configurations), illustrating fundamental principles of projective transformations and duality.
  • A. Veblen axioms for projective geometry
    The Veblen axioms for projective geometry are a foundational set of incidence-based axioms introduced by Oswald Veblen to rigorously formalize the structure of projective spaces.
  • B. Traité des propriétés projectives des figures
    Traité des propriétés projectives des figures is a foundational 19th-century mathematical treatise that systematically develops projective geometry and helped establish it as an independent discipline.
  • C. On the Principles of Geometry
    "On the Principles of Geometry" is Nikolai Lobachevsky’s foundational work that introduced non-Euclidean (hyperbolic) geometry, challenging the universality of Euclid’s parallel postulate.
  • D. Cremona group of the projective plane
    The Cremona group of the projective plane is the group of all birational self-maps of the complex projective plane, serving as a fundamental object in algebraic geometry and the study of plane transformations.
  • E. Sylvester–Gallai theorem
    The Sylvester–Gallai theorem is a result in incidence geometry stating that for any finite set of points in the Euclidean plane not all on a single line, there exists a line that passes through exactly two of the points.
  • 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_69e0c46411108190bba0d4176dffc9f3 completed April 16, 2026, 11:13 a.m.
NER Named-entity recognition batch_69ef17e7d1388190922a90cb91ec9fc4 completed April 27, 2026, 8:01 a.m.
NED1 Entity disambiguation (via context triple) batch_6a09fd3d0d9881909d04a901cf65c211 completed May 17, 2026, 5:39 p.m.
NEDg Description generation batch_6a0a0c1bb6188190966c2db0f61048f7 completed May 17, 2026, 6:42 p.m.
NED2 Entity disambiguation (via description) batch_6a0a0cb1b9d08190aa0edccd5a49b289 completed May 17, 2026, 6:45 p.m.
Created at: April 16, 2026, 6:33 p.m.