Triple

T4927201

Position	Surface form	Disambiguated ID	Type / Status
Subject	Weierstrass approximation theorem	`E110605`	entity
Predicate	generalizationOf	`P2372`	FINISHED
Object	Stone–Weierstrass theorem The Stone–Weierstrass theorem is a fundamental result in functional analysis that characterizes when a subalgebra of continuous functions on a compact space is dense, thereby generalizing classical polynomial approximation results.	`E480871`	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: Stone–Weierstrass theorem | Statement: [Weierstrass approximation theorem, generalizationOf, Stone–Weierstrass theorem]

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

Target entity: Stone–Weierstrass theorem
Context triple: [Weierstrass approximation theorem, generalizationOf, Stone–Weierstrass theorem]

A. Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
B. Banach–Stone theorem
The Banach–Stone theorem is a fundamental result in functional analysis that characterizes compact Hausdorff spaces via isometric isomorphisms between their spaces of continuous real- or complex-valued functions.
C. 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.
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. 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.
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: Stone–Weierstrass theorem
Triple: [Weierstrass approximation theorem, generalizationOf, Stone–Weierstrass theorem]

Generated description

The Stone–Weierstrass theorem is a fundamental result in functional analysis that characterizes when a subalgebra of continuous functions on a compact space is dense, thereby generalizing classical polynomial approximation results.

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

Target entity: Stone–Weierstrass theorem
Target entity description: The Stone–Weierstrass theorem is a fundamental result in functional analysis that characterizes when a subalgebra of continuous functions on a compact space is dense, thereby generalizing classical polynomial approximation results.

A. Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
B. Banach–Stone theorem
The Banach–Stone theorem is a fundamental result in functional analysis that characterizes compact Hausdorff spaces via isometric isomorphisms between their spaces of continuous real- or complex-valued functions.
C. 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.
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. 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.
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_69bd4415190c8190817bee7ec9f9f944`	completed	March 20, 2026, 12:56 p.m.
NER	Named-entity recognition	`batch_69bd70354bd081909291a43439f42ed3`	completed	March 20, 2026, 4:05 p.m.
NED1	Entity disambiguation (via context triple)	`batch_69be77ac88148190a51fa2e9085d6897`	completed	March 21, 2026, 10:49 a.m.
NEDg	Description generation	`batch_69be7847a378819081687ec783a8b862`	completed	March 21, 2026, 10:51 a.m.
NED2	Entity disambiguation (via description)	`batch_69be7915a26c81909b21a128daebf5b3`	completed	March 21, 2026, 10:55 a.m.

Created at: March 20, 2026, 1:30 p.m.