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
1.1

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.

givenalgebra
1.2

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

givenalgebra
2.1

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.

step 1.1step 1.2∎

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