Alphabeta Math
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: the point values X_p and Y_p determine the bracket value [X,Y]_p

Statement

False claim: if two pairs of vector fields agree pointwise at p, then they have the same Lie bracket value at p.

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 [X,Y]1=X1∂xY1−Y1∂xX1 on R (Coordinate formula for the Lie bracket).

Refutation

technique · direct
1.1

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

given
1.2

Applying [L1] to (X,Y) gives [X,Y]=d/dx, while applying it to (X′,Y′) gives [X′,Y′]=0. Thus [X,Y]0=d/dx∣0≠0=[X′,Y′]0.

L1given
2.1

Therefore the point values Xp and Yp do not determine the bracket value at p.

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