ExampleConstruction: AI-generatedVerification: 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.

A compactly supported cutoff of an incomplete vector field is complete

Example

Choose a smooth bump function $χ : R \to [0, 1]$ with $χ = 1$ on $[- 1, 1]$ and $supp χ \subseteq [- 2, 2]$ . Then

$Y = χ (x) x^{2} \frac{\partial}{\partial x}$

agrees with the incomplete field $x^{2} d / d x$ near the origin but is complete.

Facts & Assumptions

Given: A bump function $χ$ equal to $1$ on $[- 1, 1]$ and supported in $[- 2, 2]$ .

[L1]

Smooth bump functions with prescribed compact support exist (A manifold bump for a compact set inside an open set).

[L2]

Compactly supported smooth vector fields are complete (Compactly supported smooth vector fields are complete).

Verification

technique · direct

1.1

By [L1], such a bump function $χ$ exists. The field $Y = χ (x) x^{2} d / d x$ is smooth, agrees with $x^{2} d / d x$ on $[- 1, 1]$ , and vanishes outside the compact interval $[- 2, 2]$ .

L1given

2.1

Since $Y$ has compact support, [L2] implies that $Y$ is complete. Thus a compactly supported cutoff can preserve the local model of an incomplete field while restoring global existence.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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)
John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)