Alphabeta MathHow 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.
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- 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.
Negative-gradient trajectories of a Morse function
Definition
Let be a Morse function and let be a Riemannian metric. A negative-gradient trajectory is a maximal integral curve of ; that is, it satisfies
Here is maximal among intervals on which this solution extends. It is a full trajectory when . The word “negative” fixes the descending convention used below.
Depends on
Used by
- A Morse trajectory from one critical point to another Definition
- A negative-gradient trajectory satisfies the energy identity Lemma
- Precompact trajectory tails have nonempty compact connected flow-invariant limit sets Lemma
- Proper Morse slabs prevent finite-time escape of connecting trajectories Proposition
Dependency tree · two levels
9 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)