Alphabeta Math
CounterexampleConstruction: 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.

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 has the coordinate formula on R (Coordinate formula for the Lie bracket).

Counterexample

technique · direct
1.1

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

given
1.2

Using [L1], one computes [X,Y]=d/dx and [X′,Y′]=0. Therefore [X,Y]0=d/dx∣0≠0=[X′,Y′]0.

L1given
2.1

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

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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