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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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 $f : R \to R$ , $f (x) = x^{4}$ .

[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

1.1

One has $f^{'} (x) = 4 x^{3}$ , so $f^{'} (x) = 0$ only at $x = 0$ . Thus $0$ is an isolated critical point.

F1givenalgebra

2.1

Also $f^{''} (x) = 12 x^{2}$ , so the Hessian at the critical point is ${Hess}_{0} (f) = 0$ . Therefore the critical point is degenerate.

F1step 1.1algebra

3.1

Hence an isolated critical point need not be nondegenerate.

step 1.1step 2.1∎

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

Michele Audin and Mihai Damian, Morse Theory and Floer Homology (standard reference, not scraped)