PropositionStatement: Literature-sourcedProof: Literature-sourcedPipeline-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.

Index-one trajectory spaces are zero-dimensional

Statement

If $(f, X)$ is Morse--Smale and $λ (p) - λ (q) = 1$ , then $M (p, q)$ is a discrete smooth manifold. This statement does not assert finiteness.

Facts & Assumptions

Given: A Morse--Smale pair and critical points with index drop one.

[F1]

The unparametrized space has dimension $λ (p) - λ (q) - 1$ (The unparametrized trajectory space is a smooth manifold).

Proof

technique · direct

1.1

Substitution in [F1] gives $\dim M (p, q) = 1 - 1 = 0$ .

F1givenalgebra

2.1

A zero-dimensional smooth manifold is discrete, proving the claim without any compactness assertion.

step 1.1∎

Depends on

The unparametrized trajectory space is a smooth manifold

Used by

A Morse--Smale flow on the circle Example

Dependency tree · two levels

6 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)