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 computed-gradient residual needs its discrepancy bound

Example

On a one-dimensional quadratic, omitting the computed-gradient discrepancy would give a false distance certificate.

Facts & Assumptions

Given: $f (x) = \frac{1}{2} x^{2}$ , minimizer $x^{*} = 0$ , candidate $\tilde{x} = 1 / 10$ , and computed gradient $\hat{g} = 1 / 50$ .

[F1]

An inexact gradient residual certifies distance under strong convexity: if $∥ \hat{g} ∥_{2} \leq η$ and $∥ \hat{g} - \nabla f (\tilde{x}) ∥_{2} \leq δ$ , then $∥ \tilde{x} - x^{*} ∥_{2} \leq (η + δ) / μ$ .

[F2]

The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case: in one dimension the gradient is the ordinary first derivative.

Verification

1.1F2givenalgebra

On $U = R$ , $f$ is $C^{2}$ and $f^{''} (x) = 1$ , so $μ = 1$ . Thus $\nabla f (\tilde{x}) = 1 / 10$ , and the computed residual is $η = ∣ \hat{g} ∣ = 1 / 50$ . The independently verified discrepancy is $δ = ∣ 1 / 50 - 1 / 10 ∣ = 2 / 25$ . The actual distance is $∣ \tilde{x} - x^{*} ∣ = 1 / 10$ .

2.1F1step 1.1algebra∎

The certificate [F1] gives $(η + δ) / μ = (1 / 50 + 2 / 25) = 1 / 10$ , exactly the actual distance. If the discrepancy were omitted, the proposed bound $1 / 50$ would be false.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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.