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.
Induced boundary orientation
Definition
For an oriented manifold with boundary, orient by the outward-normal-first rule: an outward vector first, followed by a positive boundary determinant, is a positive determinant of .
Depends on
Used by
- A cylinder with oppositely oriented boundary components Example
- Boundary orientation of the unit sphere by the outward normal Example
- Positive-dimensional real projective space is orientable exactly in odd dimension Example
- The boundary of an oriented interval Example
- Boundary orientation is inward-normal-first False statement
- Boundary orientation is independent of the outward vector field Proposition
- Boundary orientation of a product with at most one boundary factor Proposition
Dependency tree · two levels
8 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
- Ioan Mărcuț, Manifolds (2017 lecture notes), §§14.5, 15.1 (standard reference, not scraped)
- Will Merry, Differential Geometry (2021), Lecture 24 (standard reference, not scraped)