CorollaryStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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 downward gradient flow has no nonconstant periodic or recurrent orbit

Statement

A nonconstant orbit of a negative-gradient flow is neither periodic nor recurrent. Here recurrent means that for some point $x$ on the orbit there are $t_{k} \to \infty$ with $Φ_{t_{k}} (x) \to x$ .

Facts & Assumptions

Given: A nonconstant negative-gradient orbit $γ (t) = Φ_{t} (x)$ .

[F1]

The function $f \circ γ$ is strictly decreasing (Nonconstant negative-gradient trajectories strictly decrease the function).

Proof

technique · direct

1.1

If the orbit had period $T > 0$ , then $f (γ (T)) = f (γ (0))$ , contradicting [F1].

F1given

1.2

If $Φ_{t_{k}} (x) \to x$ with $t_{k} \to \infty$ , fix $s > 0$ . For all sufficiently large $k$ , $t_{k} \geq s$ , so [F1] gives $f (Φ_{t_{k}} (x)) \leq f (Φ_{s} (x)) < f (x)$ .

F1given

2.1

Continuity of $f$ makes the left side of step 1.2 tend to $f (x)$ , a contradiction. Thus the orbit is not recurrent either.

step 1.2∎

Depends on

Nonconstant negative-gradient trajectories strictly decrease the function

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)