False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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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.

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FALSE: a vanishing Lie bracket forces the vector fields to be pointwise linearly dependent

Statement

False claim: if $[X, Y] = 0$ , then $X_{p}$ and $Y_{p}$ are linearly dependent for every $p$ .

Facts & Assumptions

Given: On $R^{2}$ , the coordinate vector fields $X = \partial / \partial x$ and $Y = \partial / \partial y$ .

[L1]

Coordinate vector fields commute (Coordinate vector fields commute).

Refutation

technique · direct

1.1

By [L1], one has $[X, Y] = 0$ .

L1given

1.2

At every point of $R^{2}$ , the vectors $\partial / \partial x$ and $\partial / \partial y$ are linearly independent.

given

2.1

Thus vanishing Lie bracket does not force pointwise linear dependence.

step 1.1step 1.2∎

Depends on

Coordinate vector fields commute

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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)