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.

A constant map has a large critical locus and one critical value

Example

Let M be a nonempty smooth manifold, let N be positive-dimensional, and let F:M→N be constant with value q. Then every point of M is critical and the critical value set is {q}.

Facts & Assumptions

Given: A constant smooth map F:M→N with value q, where M is nonempty and dim⁡N>0.

[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
1.1

The differential of a constant map is zero at every point. Because TqN is nonzero, this differential is not surjective, so every point of M is critical.

givenalgebra
2.1

Hence the critical locus from [F1] is all of M, and because M is nonempty, its image is the singleton {q}.

F1step 1.1
3.1

Therefore a constant map has a large critical locus and one critical value.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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