CounterexampleConstruction: Literature-sourcedVerification: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

A gradient flow on a noncompact manifold can be incomplete

Statement refuted

Every negative-gradient vector field on a noncompact manifold is complete.

Counterexample

Given: The Euclidean line and $f (x) = - x^{3} / 3$ .

1.1givenalgebra

Since $grad f = - x^{2} \partial_{x}$ , the negative-gradient field is $x^{2} \partial_{x}$ and its equation is $\dot{x} = x^{2}$ .

2.1step 1.1algebra∎

For $x_{0} > 0$ , the solution is $x (t) = x_{0} / (1 - x_{0} t)$ , which tends to $+ \infty$ as $t ↑ 1 / x_{0}$ . Hence its maximal interval has a finite positive endpoint and the field is incomplete.

This witnesses incompleteness only: $f$ is not offered as a Morse-function example.

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Ralph L. Cohen, Bundles, Manifolds, and Homotopy, §13.1 (standard reference, not scraped)