CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

How 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.

Every smooth vector field on a compact manifold is complete

Statement

Every smooth vector field on a compact manifold is complete.

Facts & Assumptions

Given: A compact smooth manifold $M$ and a smooth vector field $X$ on $M$ .

[L1]

A compactly supported smooth vector field is complete (Compactly supported smooth vector fields are complete).

[L2]

The support of a section is a closed subset of the base manifold (Smooth sections, local sections, and support).

[L3]

Closed subsets of compact spaces are compact (A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact).

Proof

technique · direct

1.1

By [L2], the support of $X$ is a closed subset of $M$ . Since $M$ is compact, [L3] shows that $supp (X)$ is compact.

L2L3given

2.1

Thus $X$ is compactly supported, so [L1] implies that $X$ is complete.

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)