LemmaStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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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.

Broken Morse trajectories have strictly decreasing critical values and indices

Statement

For a finite string of nonconstant Morse--Smale trajectory components $p_{0} \to p_{1} \to \dots \to p_{r}$ , both $f (p_{i})$ and $λ (p_{i})$ strictly decrease with $i$ . Consequently $r \leq λ (p_{0})$ .

Facts & Assumptions

Given: A finite string of nonconstant connecting trajectories in a Morse--Smale pair.

[F1]

Along each nonconstant component, $d f (X) < 0$ (Downward gradient-like vector fields for a Morse function).

[F2]

A nonpositive index drop admits no Morse--Smale trajectory (No Morse--Smale trajectories for nonpositive index drop).

Proof

technique · direct

1.1F1given

Integrating the strict decrease in [F1] along each component gives $f (p_{i}) > f (p_{i + 1})$ .

1.2F2given

Since each component exists, the contrapositive of [F2] gives $λ (p_{i}) > λ (p_{i + 1})$ .

2.1step 1.2algebra∎

Each strict index drop is at least one and indices are nonnegative, so there can be at most $λ (p_{0})$ components.

Depends on

Used by

Broken Morse trajectories Definition
Breaking length is bounded by the index drop Lemma

Dependency tree · two levels

11 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)