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.

Negative-gradient lines for a quadratic Morse function

Example

On $R^{r} \times R^{s}$ with its Euclidean metric, let $f (u, v) = - ∣ u ∣^{2} + ∣ v ∣^{2}$ . Its negative-gradient equation is

$\dot{u} = 2 u, \dot{v} = - 2 v .$

Verification

Given: The quadratic function $f (u, v) = - ∣ u ∣^{2} + ∣ v ∣^{2}$ on Euclidean space.

1.1givenalgebra

Solving the separated equations gives $u (t) = e^{2 t} u_{0}$ and $v (t) = e^{- 2 t} v_{0}$ . Thus the forward-stable coordinate disk is ${u = 0}$ and the backward-unstable disk is ${v = 0}$ .

2.1step 1.1algebra∎

Directly, $\frac{d}{d t} f (u (t), v (t)) = - 4 ∣ u (t) ∣^{2} - 4 ∣ v (t) ∣^{2}$ , which is strictly negative away from the origin.

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)