Triple

T20627234

Position	Surface form	Disambiguated ID	Type / Status
Subject	Minkowski’s theorem on convex sets	`E506850`	entity
Predicate	relatedTo	`P37`	FINISHED
Object	Minkowski’s second theorem Minkowski’s second theorem is a fundamental result in the geometry of numbers that relates the successive minima of a convex, symmetric body in Euclidean space to its volume and the determinant of a lattice.	`E1444953`	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: Minkowski’s second theorem | Statement: [Minkowski’s theorem on convex sets, relatedTo, Minkowski’s second theorem]

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

Target entity: Minkowski’s second theorem
Context triple: [Minkowski’s theorem on convex sets, relatedTo, Minkowski’s second theorem]

A. Minkowski’s theorem on convex sets
Minkowski’s theorem on convex sets is a fundamental result in convex geometry that characterizes lattice points in convex bodies, underpinning much of the theory of convex polytopes and the geometry of numbers.
B. Hermite–Minkowski theorem
The Hermite–Minkowski theorem is a fundamental result in algebraic number theory that gives a finiteness bound on the number of number fields of a given degree and discriminant.
C. Mazur’s theorem on convex sets
Mazur’s theorem on convex sets is a fundamental result in functional analysis that characterizes the structure and approximation properties of convex sets in Banach spaces, particularly via convex combinations of sequences.
D. Siegel’s lemma
Siegel’s lemma is a result in number theory that guarantees the existence of small-height integer solutions to systems of linear equations with integer coefficients.
E. Carathéodory’s theorem in convex geometry
Carathéodory’s theorem in convex geometry is a fundamental result stating that any point in the convex hull of a set in ℝⁿ can be expressed as a convex combination of at most n+1 points from that set.
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: Minkowski’s second theorem
Triple: [Minkowski’s theorem on convex sets, relatedTo, Minkowski’s second theorem]

Generated description

Minkowski’s second theorem is a fundamental result in the geometry of numbers that relates the successive minima of a convex, symmetric body in Euclidean space to its volume and the determinant of a lattice.

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

Target entity: Minkowski’s second theorem
Target entity description: Minkowski’s second theorem is a fundamental result in the geometry of numbers that relates the successive minima of a convex, symmetric body in Euclidean space to its volume and the determinant of a lattice.

A. Minkowski’s theorem on convex sets
Minkowski’s theorem on convex sets is a fundamental result in convex geometry that characterizes lattice points in convex bodies, underpinning much of the theory of convex polytopes and the geometry of numbers.
B. Hermite–Minkowski theorem
The Hermite–Minkowski theorem is a fundamental result in algebraic number theory that gives a finiteness bound on the number of number fields of a given degree and discriminant.
C. Mazur’s theorem on convex sets
Mazur’s theorem on convex sets is a fundamental result in functional analysis that characterizes the structure and approximation properties of convex sets in Banach spaces, particularly via convex combinations of sequences.
D. Siegel’s lemma
Siegel’s lemma is a result in number theory that guarantees the existence of small-height integer solutions to systems of linear equations with integer coefficients.
E. Carathéodory’s theorem in convex geometry
Carathéodory’s theorem in convex geometry is a fundamental result stating that any point in the convex hull of a set in ℝⁿ can be expressed as a convex combination of at most n+1 points from that set.
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_69e0b4bd4a0081908d4e97a590a33fb2`	completed	April 16, 2026, 10:06 a.m.
NER	Named-entity recognition	`batch_69e6abe576c081909231dc0d7304b9a9`	completed	April 20, 2026, 10:42 p.m.
NED1	Entity disambiguation (via context triple)	`batch_6a08d7d3315881908e06c34b0f1e758b`	completed	May 16, 2026, 8:47 p.m.
NEDg	Description generation	`batch_6a08d86fd0b08190a611203e6cd663b6`	completed	May 16, 2026, 8:49 p.m.
NED2	Entity disambiguation (via description)	`batch_6a08d8e7d944819094c61059a1f489f5`	completed	May 16, 2026, 8:51 p.m.

Created at: April 16, 2026, 11:42 a.m.