Triple

T5256295

Position	Surface form	Disambiguated ID	Type / Status
Subject	Carathéodory’s theorem in convex geometry	`E118706`	entity
Predicate	relatedTo	`P37`	FINISHED
Object	Krein–Milman theorem The Krein–Milman theorem is a fundamental result in functional analysis and convex geometry stating that a compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points.	`E506849`	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: Krein–Milman theorem | Statement: [Carathéodory’s theorem in convex geometry, relatedTo, Krein–Milman theorem]

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Krein–Milman theorem
Context triple: [Carathéodory’s theorem in convex geometry, relatedTo, Krein–Milman theorem]

A. Banach–Alaoglu theorem
The Banach–Alaoglu theorem is a fundamental result in functional analysis stating that the closed unit ball in the dual of a normed space is compact in the weak-* topology.
B. Hahn–Banach theorem
The Hahn–Banach theorem is a fundamental result in functional analysis that guarantees the extension of bounded linear functionals from a subspace to the whole space without increasing their norm.
C. 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.
D. 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.
E. Banach–Steinhaus theorem
The Banach–Steinhaus theorem is a fundamental result in functional analysis that characterizes when a family of continuous linear operators is uniformly bounded, with major implications for the behavior of sequences of operators on Banach spaces.
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: Krein–Milman theorem
Triple: [Carathéodory’s theorem in convex geometry, relatedTo, Krein–Milman theorem]

Generated description

The Krein–Milman theorem is a fundamental result in functional analysis and convex geometry stating that a compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Krein–Milman theorem
Target entity description: The Krein–Milman theorem is a fundamental result in functional analysis and convex geometry stating that a compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points.

A. Banach–Alaoglu theorem
The Banach–Alaoglu theorem is a fundamental result in functional analysis stating that the closed unit ball in the dual of a normed space is compact in the weak-* topology.
B. Hahn–Banach theorem
The Hahn–Banach theorem is a fundamental result in functional analysis that guarantees the extension of bounded linear functionals from a subspace to the whole space without increasing their norm.
C. 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.
D. 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.
E. Banach–Steinhaus theorem
The Banach–Steinhaus theorem is a fundamental result in functional analysis that characterizes when a family of continuous linear operators is uniformly bounded, with major implications for the behavior of sequences of operators on Banach spaces.
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_69bd446978108190bb5f9c5c23d93f88`	completed	March 20, 2026, 12:58 p.m.
NER	Named-entity recognition	`batch_69bd7ba4ecd88190800b5e4eea3abed5`	completed	March 20, 2026, 4:53 p.m.
NED1	Entity disambiguation (via context triple)	`batch_69befe7a1f448190acfcdfe37c962028`	completed	March 21, 2026, 8:24 p.m.
NEDg	Description generation	`batch_69beff55faec8190a75a1b5f339a2c20`	completed	March 21, 2026, 8:28 p.m.
NED2	Entity disambiguation (via description)	`batch_69bf001f0d9c8190a67909a06ea41898`	completed	March 21, 2026, 8:31 p.m.

Created at: March 20, 2026, 1:50 p.m.