PropositionStatement: 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.

The generating vector field is invariant under its own flow

Statement

Let $Φ : D \to M$ be the maximal flow of a smooth vector field $X$ . Then for each $t$ ,

$(Φ_{t})_{*} X = X$

on the common domain of definition.

Facts & Assumptions

Given: The maximal flow $Φ$ of $X$ , a time $t$ , and a point $p \in D_{t}$ .

[L1]

Each time- $t$ flow map is a diffeomorphism between open domains (Time-t flow maps are diffeomorphisms between open domains).

[L2]

The flow satisfies the local group law and its time slices are integral curves of $X$ (The fundamental theorem on flows).

Proof

technique · direct

1.1

By [L1], $Φ_{t}$ is a diffeomorphism near $p$ , so $(Φ_{t})_{*} X$ is defined there. Consider the curve $η (s) : = Φ_{t} (Φ_{s} (p)) = Φ_{t + s} (p)$ , where the second equality is the local group law from [L2].

L1L2given

2.1

Differentiating $η$ at $s = 0$ yields $η^{'} (0) = d (Φ_{t})_{p} (X_{p}) .$ But $η$ is also the integral curve of $X$ through $Φ_{t} (p)$ , so $η^{'} (0) = X_{Φ_{t} (p)}$ by [L2]. Hence $d (Φ_{t})_{p} (X_{p}) = X_{Φ_{t} (p)}$ .

L2step 1.1

3.1

Since $p$ was arbitrary, $(Φ_{t})_{*} X = X$ on the common domain.

step 2.1∎

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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)