Triple

T10313547
Position Surface form Disambiguated ID Type / Status
Subject Herbert Busemann E241955 entity
Predicate hasConceptNamedAfter P3325 FINISHED
Object Busemann–Feller theorem
The Busemann–Feller theorem is a result in geometric measure theory that characterizes when a metric space is geodesic by relating distance properties to the existence of shortest paths between points.
E855798 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: Busemann–Feller theorem | Statement: [Herbert Busemann, hasConceptNamedAfter, Busemann–Feller theorem]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Busemann–Feller theorem
Context triple: [Herbert Busemann, hasConceptNamedAfter, Busemann–Feller theorem]
  • A. Hopf–Rinow theorem
    The Hopf–Rinow theorem is a fundamental result in Riemannian geometry that characterizes when a Riemannian manifold is geodesically complete, relating metric completeness, compactness of closed and bounded sets, and the existence of minimizing geodesics between points.
  • B. Bernstein theorem
    Bernstein theorem is a fundamental result in set theory stating that if each of two sets can be injected into the other, then there exists a bijection between them, so the sets have the same cardinality.
  • C. Banach–Saks theorem
    The Banach–Saks theorem is a result in functional analysis stating that every bounded sequence in a reflexive Banach space has a subsequence whose Cesàro means converge in norm.
  • D. Banach–Mazur theorem
    The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
  • E. Carathéodory existence theorem
    The Carathéodory existence theorem is a result in the theory of ordinary differential equations that guarantees the existence (and sometimes uniqueness) of solutions under weaker regularity conditions on the right-hand side than those required by classical theorems like Picard–Lindelöf.
  • 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: Busemann–Feller theorem
Triple: [Herbert Busemann, hasConceptNamedAfter, Busemann–Feller theorem]
Generated description
The Busemann–Feller theorem is a result in geometric measure theory that characterizes when a metric space is geodesic by relating distance properties to the existence of shortest paths between points.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Busemann–Feller theorem
Target entity description: The Busemann–Feller theorem is a result in geometric measure theory that characterizes when a metric space is geodesic by relating distance properties to the existence of shortest paths between points.
  • A. Hopf–Rinow theorem
    The Hopf–Rinow theorem is a fundamental result in Riemannian geometry that characterizes when a Riemannian manifold is geodesically complete, relating metric completeness, compactness of closed and bounded sets, and the existence of minimizing geodesics between points.
  • B. Bernstein theorem
    Bernstein theorem is a fundamental result in set theory stating that if each of two sets can be injected into the other, then there exists a bijection between them, so the sets have the same cardinality.
  • C. Banach–Saks theorem
    The Banach–Saks theorem is a result in functional analysis stating that every bounded sequence in a reflexive Banach space has a subsequence whose Cesàro means converge in norm.
  • D. Banach–Mazur theorem
    The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
  • E. Carathéodory existence theorem
    The Carathéodory existence theorem is a result in the theory of ordinary differential equations that guarantees the existence (and sometimes uniqueness) of solutions under weaker regularity conditions on the right-hand side than those required by classical theorems like Picard–Lindelöf.
  • 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_69d381ac38808190a8ca7457c85b625b completed April 6, 2026, 9:49 a.m.
NER Named-entity recognition batch_69d4d35a292c8190bc8c467e522bba92 completed April 7, 2026, 9:50 a.m.
NED1 Entity disambiguation (via context triple) batch_69d71d86c7e481908a0d5e65f66ab2c0 completed April 9, 2026, 3:31 a.m.
NEDg Description generation batch_69d73186831481909555e2205d8783a7 completed April 9, 2026, 4:56 a.m.
NED2 Entity disambiguation (via description) batch_69d732bfc76c819089287477b54a7b77 completed April 9, 2026, 5:01 a.m.
Created at: April 6, 2026, 11:48 a.m.