ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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 vector field with global solutions

Example

Let $ρ : R^{n} \to [0, 1]$ be a smooth bump function supported in the closed unit ball, and fix $v \in R^{n}$ . Then

$V (x) : = ρ (x) v$

is a compactly supported smooth vector field, so all of its maximal solutions are global.

Facts & Assumptions

Given: A smooth bump function $ρ$ supported in $\overline{B} (0, 1)$ and a vector $v \in R^{n}$ .

[L1]

Every compactly supported smooth Euclidean vector field is complete (A compactly supported smooth Euclidean vector field is complete).

Verification

technique · direct

1.1

The support of $V (x) = ρ (x) v$ is contained in the compact support of $ρ$ , [given] and $V$ is smooth because it is a scalar multiple of the constant vector $v$ by a smooth scalar function.

given

2.1

Therefore [L1] applies and makes every maximal trajectory of $V$ global. [L1, step 1.1] Outside the support of $ρ$ , the field vanishes and the solution is locally constant, which is consistent with that completeness conclusion.

L1step 1.1∎

Depends on

A compactly supported smooth Euclidean vector field is complete

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

Chin-Lung Wang, Banach Calculus, §4.4 (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds, Appendix §10.2 (standard reference, not scraped)