Triple

T37547085

Position	Surface form	Disambiguated ID	Type / Status
Subject	I. J. Schoenberg	`E933492`	entity
Predicate	notableFor	`P22`	FINISHED
Object	Schoenberg’s theorem on positive definite functions on spheres Schoenberg’s theorem on positive definite functions on spheres characterizes all positive definite kernels on spherical spaces by representing them as mixtures of Gegenbauer (ultraspherical) polynomials, providing a foundational tool in harmonic analysis and spatial statistics.	`E2232807`	NE FINISHED

How this triple was built (2 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: Schoenberg’s theorem on positive definite functions on spheres | Statement: [I. J. Schoenberg, notableFor, Schoenberg’s theorem on positive definite functions on spheres]

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: Schoenberg’s theorem on positive definite functions on spheres
Triple: [I. J. Schoenberg, notableFor, Schoenberg’s theorem on positive definite functions on spheres]

Generated description

Schoenberg’s theorem on positive definite functions on spheres characterizes all positive definite kernels on spherical spaces by representing them as mixtures of Gegenbauer (ultraspherical) polynomials, providing a foundational tool in harmonic analysis and spatial statistics.

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_69f76eca55bc8190acf25741793d5dac`	completed	May 3, 2026, 3:50 p.m.
NER	Named-entity recognition	`batch_69fba424e66c81908d42d7bf46e6938a`	completed	May 6, 2026, 8:27 p.m.
NED1	Entity disambiguation (via context triple)	`batch_6a409f0bf9c48190804daee8af64fcf5`	completed	June 28, 2026, 4:11 a.m.
NEDg	Description generation	`batch_6a40a0bb718081909f6f7d021b52070c`	completed	June 28, 2026, 4:19 a.m.
NED2	Entity disambiguation (via description)	`batch_6a40a180222c8190aa3f63942e798f2f`	completed	June 28, 2026, 4:22 a.m.

Created at: May 3, 2026, 4:17 p.m.