ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge 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 black-box quadratic simulator fit using direct search

Example

Treat the simulator response $f (θ) = (θ - 2)^{2}$ as a black-box evaluation, with no derivative supplied. Run Finite-budget deterministic direct search with positive-spanning polling from $θ_{0} = 0$ using ordered polls $D = (+ 1, - 1)$ , $Δ_{0} = 1$ , cap $Δ_{\max} = 2$ , forcing $ρ (Δ) = Δ^{2} / 10$ , increase factor $2$ , decrease factor $1 / 2$ , and poll budget $N = 3$ . The successive centers are $0, 1, 1, 2$ ; the third poll reaches the exact simulator fit $θ = 2$ , and the finite run returns status budget-exhausted.

Facts & Assumptions

Given: Positive-spanning direct-search polling and forcing rule Finite-budget deterministic direct search with positive-spanning polling.

[F1]

In one dimension $D = (+ 1, - 1)$ is positive spanning; the acceptance test compares only function values and the stated forcing threshold (Finite-budget deterministic direct search with positive-spanning polling).

Verification

1.1givenF1algebra

At center $0$ , $f (0) = 4$ and the first poll point $1$ has $f (1) = 1 < 4 - 1 / 10$ , so the first poll succeeds at $1$ and the mesh grows to $2$ .

2.1givenF1step 1.1algebra∎

At center $1$ with mesh $2$ , the ordered trials $3, - 1$ have values $1, 9$ , neither below $f (1) - 4 / 10 = 3 / 5$ ; the second poll is unsuccessful and halves the mesh to $1$ . At center $1$ with mesh $1$ , the first trial $2$ has value $0 < 1 - 1 / 10$ , so the third poll succeeds at $2$ . The budget is now exhausted; the observed exact minimizer is returned without a derivative-based certificate.

Depends on

Finite-budget deterministic direct search with positive-spanning polling

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.

Sources

A. R. Conn, K. Scheinberg, and L. N. Vicente, Introduction to Derivative-Free Optimization, §§2.1–2.2 and ch. 8 (standard reference, not scraped)
Tamara G. Kolda, Robert Michael Lewis, and Virginia Torczon, Optimization by Direct Search, §§3.1–3.6 (standard reference, not scraped)