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.
Orbit circles of rotation as a foliation away from the origin
Example
On , the rotation field is nowhere zero. Its flow is rotation about the origin, so its leaves are the circles centered at the origin. The origin is excluded because there the field vanishes and the rank drops.
Facts & Assumptions
Given: The rotation vector field on the punctured plane.
Its flow preserves the radius function .
Verification
The field is nowhere zero on , so it defines a [given] regular one-dimensional distribution there.
Along the flow, [given] so the radius is constant on every orbit. The orbits are therefore contained in circles about the origin, and conversely each such circle is an orbit.
Hence the punctured plane is foliated by the rotation circles. At the [given] origin the field vanishes, so the regular rank-one hypothesis fails there.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
26 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
- Will J. Merry, Differential Geometry (standard reference, not scraped)