DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Parametrized Morse trajectory space

Definition

Fix a complete downward gradient-like vector field $X$ for $f$ . For distinct critical points $p, q$ , let ${\tilde{M}}_{X} (p, q)$ be the set of full $X$ -orbits $γ : R \to M$ satisfying $\dot{γ} = X (γ), \lim_{t \to - \infty} γ (t) = p, q q u a d \lim_{t \to + \infty} γ (t) = q .$ When $X$ is understood we write $\tilde{M} (p, q)$ . Existence and uniqueness for the complete flow make evaluation at zero a bijection with the point-marked intersection

${ev}_{0} : \tilde{M} (p, q) ⟶ W^{u} (p) \cap W^{s} (q), γ ⟼ γ (0) .$

We use this bijection whenever the intersection is given its smooth structure. For the special choice $X = - {grad}_{g} f$ it agrees with the usual negative-gradient trajectory convention.

Depends on

Used by

Dependency tree · two levels

8 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

Alexander F. Ritter, Part III Morse Homology, Lecture 9 (standard reference, not scraped)