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.

The radial vector field has the dilation flow

Example

On $R^{n}$ , let

$X = \sum_{i = 1}^{n} x^{i} \frac{\partial}{\partial x^{i}} .$

Then the integral curve through $x_{0}$ is $γ_{x_{0}} (t) = e^{t} x_{0}$ , so the flow is the dilation $Φ_{t} (x) = e^{t} x$ .

Facts & Assumptions

Given: The radial vector field $X = \sum_{i} x^{i} \partial / \partial x^{i}$ .

[L1]

A vector field is complete if and only if its maximal flow is global (A vector field is complete if and only if its flow is global).

Verification

technique · direct

1.1

The curve $γ_{x_{0}} (t) = e^{t} x_{0}$ satisfies $γ_{x_{0}} (0) = x_{0}$ and $γ_{x_{0}}^{'} (t) = e^{t} x_{0} = X (γ_{x_{0}} (t))$ , so it is the integral curve through $x_{0}$ .

given

2.1

Since $e^{t} x_{0}$ is defined for all $t \in R$ , the flow is global. By [L1], the radial vector field is complete, and its flow maps are the dilations $Φ_{t} (x) = e^{t} x$ .

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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)