Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 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 vector field x^2 d/dx has finite-time escape

Example

The vector field X=x2 d/dx on R has integral curves

γx0(t)=x01−x0t.

For x0>0 this solution blows up at t=1/x0, so X is not complete.

Facts & Assumptions

Given: The vector field X=x2 d/dx on R.

[L1]

Completeness is equivalent to having a global flow (A vector field is complete if and only if its flow is global).

Verification

technique · direct
1.1

The curve γx0(t)=x0/(1−x0t) satisfies γx0(0)=x0 and γx0′(t)=γx0(t)2, so it is the integral curve through x0.

given
2.1

For x0>0, the denominator vanishes at t=1/x0, so this integral curve is not defined for all real times. Hence [L1] shows that X is not complete.

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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