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.

Nonconstant negative-gradient trajectories strictly decrease the function

Statement

If $γ : I \to M$ is a nonconstant negative-gradient trajectory, then $(f \circ γ)^{'} (t) < 0$ for every $t \in I$ .

Facts & Assumptions

Given: A nonconstant negative-gradient trajectory $γ : I \to M$ .

[F1]

$(f \circ γ)^{'} = - ∥ {grad}_{g} f ∥_{g}^{2}$ along $γ$ (A negative-gradient trajectory satisfies the energy identity).

[F2]

The gradient vanishes exactly at a critical point (The Riemannian gradient vanishes exactly at the critical points).

[F3]

An integral curve through a prescribed point is unique on its maximal interval (Through each point there is a unique maximal integral curve).

Proof

technique · direct

1.1

If $γ (t_{0})$ were critical, [F2] would make the vector field $- {grad}_{g} f$ vanish there, so the constant curve at $γ (t_{0})$ is an integral curve through that point.

F2given

2.1

By [F3], that constant integral curve and $γ$ agree on $I$ , contradicting the hypothesis that $γ$ is nonconstant. Hence $γ (t)$ is never critical.

F3step 1.1

3.1

By [F2] the gradient is nonzero at every $γ (t)$ , and [F1] now gives $(f \circ γ)^{'} (t) < 0$ for every $t \in I$ .

F1F2step 2.1∎

Depends on

Used by

Dependency tree · two levels

9 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, Lemma 13.1 (standard reference, not scraped)