Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
A tangent vector is not an ambient arrow by definition
Statement
False claim: a tangent vector at is, by definition, just a vector whose tail is drawn at .
Facts & Assumptions
Given: A point of a smooth manifold .
A tangent vector at is defined as a derivation on smooth germs at (Derivations at a point and the tangent space).
Refutation
The definition in [F1] uses only the local algebra of smooth germs and the Leibniz rule.
No ambient Euclidean arrow appears in that definition, so the picture of a drawn arrow is at best a later model, not the definition itself.
Therefore the claim is false.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
- Will J. Merry, Differential Geometry (standard reference, not scraped)
- Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)