ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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 two-node degree-zero Remez certificate

Example

For $f (x) = ∣ x ∣$ on $[- 1, 1]$ , degree $n = 0$ , and initial nodes $x_{0} = 0, x_{1} = 1$ , the Remez linear system returns $p (x) = 1 / 2$ and signed level $λ = - 1 / 2$ . An exact global-extrema oracle returns error $1 / 2$ , so the current reference is an optimality certificate.

Verification

Given: $I = [- 1, 1]$ , $f (x) = ∣ x ∣$ , $n = 0$ , reference $X = (0, 1)$ , and an exact extrema oracle as in Finite-budget polynomial Remez exchange with an exact extrema oracle that returns maximum $(1 / 2, 0)$ on the first candidate $p = 1 / 2$ .

[F1] The Remez alternation linear system has a unique solution: at two distinct nodes the degree-zero alternation equations have a unique solution $(p, λ)$ .

[F2] Finite-budget polynomial Remez exchange with an exact extrema oracle: with equations $p (x_{i}) + (- 1)^{i} λ = f (x_{i})$ , a maximum $(E, y)$ response with $E = ∣ λ ∣ > 0$ returns certified $(p, E, X)$ using the current reference.

[F3] Exact polynomial equioscillation certifies best approximation: exact global equioscillation at $n + 2$ ordered nodes certifies best approximation.

1.1givenF1algebra

The equations at $x_{0} = 0, x_{1} = 1$ are $p + λ = 0$ and $p - λ = 1$ . Adding them gives $2 p = 1$ , so $p = 1 / 2$ and then $λ = - 1 / 2$ , in agreement with the unique solve [F1].

2.1F2F3step 1.1givenalgebra∎

For every $x \in [- 1, 1]$ , $∣ x ∣ \in [0, 1]$ , so $- ∣ 1 / 2 ∣ \leq ∣ x ∣ - 1 / 2 \leq 1 / 2$ and the absolute error is at most $1 / 2$ . It equals $1 / 2$ at $x = 0$ and at $x = 1$ (also at $x = - 1$ ), so the stipulated response maximum $(1 / 2, 0)$ is valid and has $E = 1 / 2 = ∣ λ ∣$ . The residuals at $(0, 1)$ are $- 1 / 2, 1 / 2$ , hence with this maximum response the procedure returns certified $(p, 1 / 2, X)$ by [F2], and [F3] makes $p$ best.

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.

Sources

Ricardo Pachón and Lloyd N. Trefethen, Barycentric-Remez Algorithms for Best Polynomial Approximation in the Chebfun System, §2 (standard reference, not scraped)