ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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 multimodal nonconvex objective

Example

On $R$ , consider $f (x) = (x^{2} - 1)^{2}$ . It has two distinct global minimizers and a strict local maximum between them. The algorithms Gradient descent and Damped Newton method specify local update rules; this example proves no global convergence or initialization-independent claim for either procedure.

Facts & Assumptions

Given: The real objective $f (x) = (x^{2} - 1)^{2}$ .

[F1]

Gradient descent defines an update from the current negative gradient and its selected step.

[F2]

Damped Newton method defines a Hessian-based local update with line search and does not assert global minimization.

Verification

1.1givenalgebra

Factor $f (x) = (x - 1)^{2} (x + 1)^{2} \geq 0$ for every real $x$ , with equality at $x = 1$ and $x = - 1$ . Thus these are distinct global minimizers and the minimum value is zero.

1.2givenalgebra

For $∣ x ∣ < 1$ , $f (x) = (1 - x^{2})^{2} \leq 1$ , with strict inequality for $x \neq 0$ ; since $f (0) = 1$ , $x = 0$ is a strict local maximum.

2.1step 1.1step 1.2F1F2∎

Steps 1.1–1.2 exhibit a smooth objective with two global minima and an intervening local maximum. The procedures in [F1]–[F2] are defined by local updates and make no global-minimization or initialization-independent claim.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Mario A. T. Figueiredo, Lecture Notes on the EM Algorithm (2004) (standard reference, not scraped)