ExampleConstruction: Literature-sourcedVerification: 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.

A Morse--Smale flow on the circle

Example

On $S^{1} = R / 2 π Z$ , take $f (θ) = \cos θ$ with its standard metric. Its maximum $p = 0$ has index $1$ and its minimum $q = π$ has index $0$ . There are two unparametrized trajectories from $p$ to $q$ , one through each open semicircle.

Facts & Assumptions

Given: The negative-gradient equation $\dot{θ} = \sin θ$ .

[F1]

For a Morse--Smale pair, an index-drop-one unparametrized space is discrete (Index-one trajectory spaces are zero-dimensional).

Verification

technique · direct

1.1

The complement of ${0, π}$ has two connected arcs. On each, $\sin θ$ has fixed sign and every orbit has backward limit $0$ and forward limit $π$ , so each arc is one time-translation orbit.

given

2.1

Here $W^{u} (p) = S^{1} ∖ {q}$ and $W^{s} (q) = S^{1} ∖ {p}$ . Along their two open-arc intersection both tangent spaces equal $T S^{1}$ , so their sum is $T S^{1}$ . The reversed distinct pair has empty intersection, and at each equal critical-point pair one of the stable or unstable tangent spaces is $T S^{1}$ . Thus all stable--unstable intersections are transverse and the specified standard-metric pair is Morse--Smale by Morse--Smale pairs.

step 1.1given

3.1

Hence $M (p, q)$ has exactly two points. This agrees with [F1] and directly displays the quotient by translation.

F1step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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)