Alphabeta Math
ExampleConstruction: 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.

The planar rotation field has the circle rotation flow

Example

On R2, let

X=−y∂∂x+x∂∂y.

Then the flow is rotation:

Φt(x,y)=(xcos⁡t−ysin⁡t, xsin⁡t+ycos⁡t).

Facts & Assumptions

Given: The rotation vector field X=−y ∂/∂x+x ∂/∂y.

[L1]

A vector field is complete if and only if its maximal flow is global (A vector field is complete if and only if its flow is global).

Verification

technique · direct
1.1

Differentiating the displayed formula gives ddtΦt(x,y)=(−xsin⁡t−ycos⁡t, xcos⁡t−ysin⁡t)=X(Φt(x,y)). Also Φ0(x,y)=(x,y), so Φt(x,y) is the integral curve through (x,y).

given
2.1

The formula is defined for all t∈R, so the flow is global. Therefore [L1] implies that the rotation field is complete.

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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