RemarkRemark: Literature-sourcedProof: Not applicablePipeline-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.

Completeness of a gradient flow is an extra hypothesis on a noncompact manifold

On a noncompact manifold, a gradient or gradient-like field need not be complete. A compact slab gives only the conditional nonescape conclusion of Proper Morse slabs prevent finite-time escape of connecting trajectories; it does not make all trajectories global. The explicit escape calculation is already visible on $R$ : for $f (x) = - x^{3} / 3$ , the negative-gradient equation is $\dot{x} = x^{2}$ , and the solution from $x_{0} > 0$ is $x (t) = x_{0} / (1 - x_{0} t)$ , which escapes to $+ \infty$ as $t ↑ 1 / x_{0}$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

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