ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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 small gradient can hide error when the minimizer is ill conditioned

Example

For $f (u, v) = \frac{1}{2} (10^{- 8} u^{2} + v^{2})$ , the exact gradient norm can be tiny at a point a unit distance from the minimizer.

Facts & Assumptions

Given: $f (u, v) = \frac{1}{2} (10^{- 8} u^{2} + v^{2})$ on $R^{2}$ and candidate $\tilde{x} = (1, 0)$ .

[F1]

An inexact gradient residual certifies distance under strong convexity: with strong-convexity constant $μ$ , exact gradient residual $η$ , and discrepancy $δ = 0$ , the distance to the minimizer is at most $η / μ$ .

[F2]

The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case: the gradient consists of the standard-coordinate first partial derivatives.

Verification

1.1F2givenalgebra

The Hessian is $diag (10^{- 8}, 1)$ , so the unique minimizer is $x^{*} = (0, 0)$ and the strong-convexity constant is $μ = 10^{- 8}$ . At $\tilde{x} = (1, 0)$ , $\nabla f (\tilde{x}) = (10^{- 8}, 0)$ , so $η = ∥ \nabla f (\tilde{x}) ∥_{2} = 10^{- 8}$ and the true distance is $∥ \tilde{x} - x^{*} ∥_{2} = 1$ .

2.1F1step 1.1algebra∎

Applying [F1] gives the sharp bound $η / μ = 10^{- 8} / 10^{- 8} = 1$ . The residual is small in absolute size, but the weak curvature in the first coordinate makes this distance bound equal to one rather than a small number.

Depends on

Used by

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Dependency tree · two levels

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