ExampleConstruction: Literature-sourcedVerification: 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.

Meridian trajectories for height on the sphere

Example

On the round $S^{2} \subset R^{3}$ , let $f (x, y, z) = z$ . The intrinsic gradient is $grad f = e_{z} - z (x, y, z)$ , so the negative gradient is $z (x, y, z) - e_{z}$ . Its nonconstant orbits are the meridians from the north pole to the south pole.

Verification

Given: The round unit sphere and the height function $f (x, y, z) = z$ .

1.1

The tangential projection of $e_{z}$ is $e_{z} - z (x, y, z)$ , so the displayed vector field is the negative gradient. Along it, $\dot{z} = z^{2} - 1 < 0$ whenever $∣ z ∣ < 1$ .

givenalgebra

2.1

The longitude is constant along the flow, while $z$ decreases from $1$ to $- 1$ . Therefore every nonpolar longitude gives a meridian from north to south, a one-parameter family before quotienting by time translation.

step 1.1∎

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Ralph L. Cohen, Bundles, Manifolds, and Homotopy, §13.1 (standard reference, not scraped)