Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-generatedjudge pass (gpt-5.6-terra)audited 2026-09-04
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.

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An isolated critical point can be degenerate

Statement refuted

An isolated critical point of a smooth function must be nondegenerate.

Facts & Assumptions

Given: The smooth function f:R→R, f(x)=x4.

[F1]

Critical points are the points where the differential vanishes, and the critical-point Hessian is the second derivative in the standard coordinate (Critical points and critical values of a smooth function, The intrinsic Hessian of a smooth function at a critical point).

Counterexample

technique · direct computation
1.1F1givenalgebra

One has f′(x)=4x3, so f′(x)=0 only at x=0. Thus 0 is an isolated critical point.

2.1F1step 1.1algebra

Also f′′(x)=12x2, so the Hessian at the critical point is Hess⁡0(f)=0. Therefore the critical point is degenerate.

3.1step 1.1step 2.1∎

Hence an isolated critical point need not be nondegenerate.

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