Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-02
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.

Peano's surface has a strict minimum on every line through the origin but no local extremum

Statement refuted

Refuted: a strict minimum of a function on every line through a point is a local minimum.

Facts & Assumptions

Given: p(x,y)=(y−3x2)(y−x2).

Counterexample

Proof

technique · direct
1.1givenalgebra

On a nonvertical line y=mx, p(x,mx)=x2(m−3x)(m−x), which is positive for sufficiently small nonzero x when m≠0; on y=0 it is 3x4>0, and on the vertical line x=0 it is y2>0.

1.2givenalgebra

Along the parabola y=2x2, p(x,2x2)=−x4<0 for x≠0, whereas p(0,0)=0.

2.1step 1.1step 1.2∎

Hence (0,0) is a strict linewise minimum but is not a local minimum in the sense of Local and strict local extrema for scalar fields on Euclidean open sets.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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