ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

The degree-one minimax approximation to x squared

Example

On $[- 1, 1]$ , the unique best degree-one uniform approximation to $f (x) = x^{2}$ is the constant polynomial $p (x) = 1 / 2$ , with error $1 / 2$ .

Verification

Given: $I = [- 1, 1]$ , $f (x) = x^{2}$ , $p (x) = 1 / 2 \in P_{1} (I)$ .

[F1] Best uniform polynomial error on a compact interval: $E_{1} (f; I)$ is the infimum of $∥ f - q ∥_{\infty, I}$ over real polynomials $q$ of degree at most $1$ .

[F2] Exact polynomial equioscillation certifies best approximation: a degree- $n$ candidate with exact uniform error $E > 0$ and $n + 2$ ordered alternating extrema is a best approximant.

[F3] The best uniform polynomial approximant is unique: the best degree- $n$ approximant exists and is unique for every continuous real target on a nondegenerate compact interval.

1.1givenalgebra

For $x \in [- 1, 1]$ , $0 \leq x^{2} \leq 1$ , so $- 1 / 2 \leq x^{2} - 1 / 2 \leq 1 / 2$ . Equality occurs at $x = 0$ and at $x = \pm 1$ , hence $∥ f - p ∥_{\infty, I} = 1 / 2$ .

2.1F1F2F3step 1.1algebra∎

The ordered points $- 1 < 0 < 1$ give residual values $1 / 2, - 1 / 2, 1 / 2$ . By [F2], $p$ is a best degree-one approximant; [F1] identifies its error with $E_{1} (f; I)$ , and [F3] makes it unique.

Depends on

Used by

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Dependency tree · two levels

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Sources

Lloyd N. Trefethen, Approximation Theory and Approximation Practice, Chapter 10 (standard reference, not scraped)