Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck 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 need not be isolated

Statement

False claim: every critical point of a smooth map is isolated.

Facts & Assumptions

Given: The constant smooth map F:R→R, F(x)=0.

[F1]

The critical locus is the set of nonregular points (The critical locus and critical value set).

Refutation

technique · direct
1.1

The differential of F is zero at every point and is not surjective onto T0R, so every point of R is critical.

givenalgebra
2.1

The critical locus is therefore all of R, which has no isolated points. This is the critical locus from [F1].

F1step 1.1
3.1

Therefore critical points need not be isolated.

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