Triple

T19315455
Position Surface form Disambiguated ID Type / Status
Subject Modigliani–Miller theorem E483085 entity
Predicate alsoKnownAs P39 FINISHED
Object MM theorem
The MM theorem is a foundational result in corporate finance stating that, under certain idealized conditions, a firm's value is unaffected by its capital structure or dividend policy.
E1369826 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: MM theorem | Statement: [Modigliani–Miller theorem, alsoKnownAs, MM theorem]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: MM theorem
Context triple: [Modigliani–Miller theorem, alsoKnownAs, MM theorem]
  • A. Page theorem
    The Page theorem is a result in quantum information theory and black hole physics that predicts how the entanglement entropy of a subsystem typically evolves, underpinning the characteristic "Page curve" behavior in discussions of the black hole information paradox.
  • B. Haag’s theorem
    Haag’s theorem is a result in axiomatic quantum field theory showing that the interaction picture cannot be consistently defined for interacting fields in the same Hilbert space as free fields, undermining the standard formulation of quantum field theory.
  • C. de Bruijn–van Aardenne–Ehrenfest theorem
    The de Bruijn–van Aardenne–Ehrenfest theorem is a fundamental result in combinatorics that characterizes the number of Eulerian circuits in directed graphs, particularly de Bruijn graphs, and underpins constructions in coding theory and discrete mathematics.
  • D. Łoś's theorem
    Łoś's theorem is a fundamental result in model theory that characterizes the truth of first-order formulas in ultraproducts by showing they hold exactly when they are true in "almost all" component structures.
  • E. Kesten’s theorem
    Kesten’s theorem is a fundamental result in probability theory that characterizes when a random walk on a group is transient or recurrent, with deep implications for random walks on groups and percolation theory.
  • 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: MM theorem
Triple: [Modigliani–Miller theorem, alsoKnownAs, MM theorem]
Generated description
The MM theorem is a foundational result in corporate finance stating that, under certain idealized conditions, a firm's value is unaffected by its capital structure or dividend policy.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: MM theorem
Target entity description: The MM theorem is a foundational result in corporate finance stating that, under certain idealized conditions, a firm's value is unaffected by its capital structure or dividend policy.
  • A. Page theorem
    The Page theorem is a result in quantum information theory and black hole physics that predicts how the entanglement entropy of a subsystem typically evolves, underpinning the characteristic "Page curve" behavior in discussions of the black hole information paradox.
  • B. Haag’s theorem
    Haag’s theorem is a result in axiomatic quantum field theory showing that the interaction picture cannot be consistently defined for interacting fields in the same Hilbert space as free fields, undermining the standard formulation of quantum field theory.
  • C. de Bruijn–van Aardenne–Ehrenfest theorem
    The de Bruijn–van Aardenne–Ehrenfest theorem is a fundamental result in combinatorics that characterizes the number of Eulerian circuits in directed graphs, particularly de Bruijn graphs, and underpins constructions in coding theory and discrete mathematics.
  • D. Łoś's theorem
    Łoś's theorem is a fundamental result in model theory that characterizes the truth of first-order formulas in ultraproducts by showing they hold exactly when they are true in "almost all" component structures.
  • E. Kesten’s theorem
    Kesten’s theorem is a fundamental result in probability theory that characterizes when a random walk on a group is transient or recurrent, with deep implications for random walks on groups and percolation theory.
  • 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_69d8e8d04d5c8190baa816986f2b1d1e completed April 10, 2026, 12:10 p.m.
NER Named-entity recognition batch_69e60d833034819092a8414d5e0fc26e completed April 20, 2026, 11:26 a.m.
NED1 Entity disambiguation (via context triple) batch_6a071461923081909deb4c114cab2560 completed May 15, 2026, 12:41 p.m.
NEDg Description generation batch_6a0714eb6a0c8190bec7ae8ef785a4ab completed May 15, 2026, 12:43 p.m.
NED2 Entity disambiguation (via description) batch_6a0718ee8adc8190a74685f9783ba5d5 completed May 15, 2026, 1 p.m.
Created at: April 10, 2026, 1:32 p.m.