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CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passverified 2026-09-23 (gpt-6-sol)
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  • 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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Two pairs of vector fields can agree at a point and still have different bracket values there

Statement refuted

False claim: the value of [X,Y]p is determined solely by the point values Xp and Yp.

Facts & Assumptions

Given: On R, the pairs (X,Y)=(d/dx,x d/dx) and (X′,Y′)=(d/dx,0) at the point p=0.

[L1]

The Lie bracket is the commutator of vector-field actions on smooth functions (The Lie bracket of smooth vector fields).

Counterexample

technique · direct
1.1given

At 0, both pairs have the same point values: X0=X0′=d/dx∣0 and Y0=Y0′=0.

1.2L1givenalgebra

For every smooth h:R→R, [L1] gives [X,Y]h=(xh′)′−xh′′=h′ and [X′,Y′]h=0. Thus [X,Y]=d/dx and [X′,Y′]=0, whence [X,Y]0=d/dx∣0≠0=[X′,Y′]0.

2.1step 1.1step 1.2∎

Hence equal point values do not determine the Lie bracket value at a point, giving the required counterexample.

Depends on

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