ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generated

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.

A local maximum has the wrong Morse index for the saddle certificate

Example

Let $V (x, y) = - x^{2} - y^{2}$ on $U = R^{2}$ . The origin is a critical point and a strict local maximum, but its Hessian has two negative eigenvalues. It therefore fails the index-one enclosure required by Validated ordered eigenvalue enclosures for an index-one Hessian and cannot be certified by Validated Hessian enclosures and a small gradient residual certify a nearby index-one saddle.

Facts & Assumptions

Given: The quadratic potential on $R^{2}$ and the definitions of index-one enclosure and the local saddle certificate.

[F1]

Validated ordered eigenvalue enclosures for an index-one Hessian: an index-one enclosure requires $u_{1} \leq - γ$ and $ℓ_{j} \geq γ$ for every $j \geq 2$ .

[F2]

Validated Hessian enclosures and a small gradient residual certify a nearby index-one saddle: the nearby-saddle claim requires such an index-one enclosure at the reported point.

Given: $V (x, y) = - x^{2} - y^{2}$ and the origin.

Proof

1.1givenalgebra

The gradient is $\nabla V = (- 2 x, - 2 y)$ , so it vanishes at $(0, 0)$ ; the Hessian is $H = - 2 I$ , with ordered eigenvalues $λ_{1} = λ_{2} = - 2$ .

2.1F1step 1.1

For any $γ > 0$ , an interval enclosing $λ_{2} = - 2$ cannot have lower endpoint $ℓ_{2} \geq γ > 0$ . Thus no exact validated enclosure of this Hessian satisfies [F1].

3.1F2step 2.1∎

The point is a maximum with Morse index two, not index one. Since the required enclosure premise in [F2] fails by step 2.1, the theorem makes no certification claim for it; this example illustrates why a small gradient or critical-point report alone does not establish the desired saddle type.

Depends on

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Dependency tree · two levels

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Sources

Graeme Henkelman, Blas P. Uberuaga, and Hannes Jónsson, A climbing image nudged elastic band method for finding saddle points and minimum energy paths (standard reference, not scraped)
Xuanyu Liu, Huajie Chen, and Christoph Ortner, Convergence of the Discrete Minimum Energy Path (standard reference, not scraped)