Alphabeta Math
False statementConstruction: AI-adaptedVerification: 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.

FALSE: every smooth vector field is complete

Statement

False claim: every smooth vector field on a smooth manifold is complete.

Facts & Assumptions

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

[L1]

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

Refutation

technique · direct
1.1

The integral curve through x0>0 is γ(t)=x01−x0t, because γ(0)=x0 and γ′(t)=γ(t)2.

given
2.1

The denominator in step 1.1 vanishes at t=1/x0, so the solution cannot be extended to all real times. Thus the flow is not global, and [L1] shows that X is not complete.

L1step 1.1
3.1

Hence a smooth vector field need not be complete.

step 2.1∎

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