Triple

T10991618
Position Surface form Disambiguated ID Type / Status
Subject Teichmüller theory E259765 entity
Predicate usesConcept P531 FINISHED
Object Beltrami differentials
Beltrami differentials are complex-valued tensor fields on Riemann surfaces that measure infinitesimal quasiconformal deformations of the complex structure.
E259765 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: Beltrami differentials | Statement: [Teichmüller theory, usesConcept, Beltrami differentials]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Beltrami differentials
Context triple: [Teichmüller theory, usesConcept, Beltrami differentials]
  • A. Teichmüller curve
    A Teichmüller curve is a complex geodesic in the moduli space of Riemann surfaces that arises from flat surface structures and has rich connections to dynamics, geometry, and number theory.
  • B. Teichmüller theory
    Teichmüller theory is a branch of complex analysis and geometry that studies the deformation spaces of Riemann surfaces and their moduli, often via quasiconformal mappings.
  • C. Conformal Invariants
    Conformal Invariants is a foundational mathematical work by Lars Ahlfors that systematically develops the theory of quantities preserved under conformal mappings in complex analysis and geometric function theory.
  • D. Riemann surfaces
    Riemann surfaces are one-dimensional complex manifolds that provide the natural geometric setting for studying complex analytic functions and their multi-valued behavior.
  • E. Lectures on Quasiconformal Mappings
    Lectures on Quasiconformal Mappings is a classic mathematical monograph by Lars Ahlfors that systematically develops the theory of quasiconformal mappings in the complex plane and higher dimensions.
  • 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: Beltrami differentials
Triple: [Teichmüller theory, usesConcept, Beltrami differentials]
Generated description
Beltrami differentials are complex-valued tensor fields on Riemann surfaces that measure infinitesimal quasiconformal deformations of the complex structure.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Beltrami differentials
Target entity description: Beltrami differentials are complex-valued tensor fields on Riemann surfaces that measure infinitesimal quasiconformal deformations of the complex structure.
  • A. Teichmüller curve
    A Teichmüller curve is a complex geodesic in the moduli space of Riemann surfaces that arises from flat surface structures and has rich connections to dynamics, geometry, and number theory.
  • B. Teichmüller theory chosen
    Teichmüller theory is a branch of complex analysis and geometry that studies the deformation spaces of Riemann surfaces and their moduli, often via quasiconformal mappings.
  • C. Conformal Invariants
    Conformal Invariants is a foundational mathematical work by Lars Ahlfors that systematically develops the theory of quantities preserved under conformal mappings in complex analysis and geometric function theory.
  • D. Riemann surfaces
    Riemann surfaces are one-dimensional complex manifolds that provide the natural geometric setting for studying complex analytic functions and their multi-valued behavior.
  • E. Lectures on Quasiconformal Mappings
    Lectures on Quasiconformal Mappings is a classic mathematical monograph by Lars Ahlfors that systematically develops the theory of quasiconformal mappings in the complex plane and higher dimensions.
  • F. None of above.

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_69d6aa8a6a548190a750f944ccdc8064 completed April 8, 2026, 7:20 p.m.
NER Named-entity recognition batch_69d795d1e918819090c71f5a077fa15a completed April 9, 2026, 12:04 p.m.
NED1 Entity disambiguation (via context triple) batch_69e34504ebec8190a78e4795765b0c24 completed April 18, 2026, 8:47 a.m.
NEDg Description generation batch_69e3556fd3548190a33f04604be947cf completed April 18, 2026, 9:57 a.m.
NED2 Entity disambiguation (via description) batch_69e3593b0f8481909ed7a90f8bb9839d completed April 18, 2026, 10:13 a.m.
Created at: April 8, 2026, 9:24 p.m.