Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-01
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.

Critical points and values of a height function on a sphere

Example

Let h:S2→R be the height function h(x,y,z)=z. Its critical points are the north and south poles, and its critical values are 1 and −1.

Facts & Assumptions

Given: The sphere S2={x2+y2+z2=1} and the height function h(x,y,z)=z.

[F1]

The critical locus and critical value set record the critical points and their images (The critical locus and critical value set).

Verification

technique · direct
1.1givenalgebra

The tangent space at (x,y,z)∈S2 consists of vectors orthogonal to (x,y,z). The differential of h is the projection v↦vz, so it vanishes on that tangent space exactly when every tangent vector has zero z-component, which happens only at (0,0,±1).

2.1F1step 1.1

Therefore the critical locus from [F1] is {(0,0,1),(0,0,−1)}, and the critical value set is {1,−1}.

3.1step 2.1∎

Thus the sphere height function has exactly the two expected critical points and values.

Depends on

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