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.

Zero AAA sample residual can coexist with arbitrary off-grid error

Example

Let the sample set be $Z = (0, 1)$ and, for any $M > 0$ , let $f_{M} (t) = M t (1 - t)$ on $[0, 1]$ . Exact AAA with support budget $B = 1$ , sweep budget $S = 0$ , and tolerance zero starts at the constant mean of the two samples and returns immediately with zero sampled residual, but its uniform error is $M / 4$ . For a continuous $q$ on $[0, 1]$ , write $∥ q ∥_{\infty, [0, 1]} : = \sup_{t \in [0, 1]} ∣ q (t) ∣$ .

Facts & Assumptions

Given: $M > 0$ , the two sample points $0, 1$ , data $f_{M} (t) = M t (1 - t)$ , support budget $B = 1$ , sweep budget $S = 0$ , tolerance $0$ , and the exact AAA procedure.

[F1]

Finite-sample AAA rational approximation with exact SVD: the initial candidate is the sample mean, and the procedure stops when the maximum sample residual is at most the supplied tolerance.

Verification

1.1givenF1algebra

The sample values are $f_{M} (0) = f_{M} (1) = 0$ , so their mean is zero and the initial candidate $r_{0}$ is the zero function. Its residual at both sample points is zero, hence the tolerance-zero stopping rule in [F1] returns $r_{0}$ before selecting any support node.

2.1givenstep 1.1algebra∎

At the unsampled point $t = 1 / 2$ , $f_{M} (1 / 2) = M / 4$ and $r_{0} (1 / 2) = 0$ . Since $t (1 - t) = 1 / 4 - (t - 1 / 2)^{2} \leq 1 / 4$ on $[0, 1]$ , the uniform error is exactly $M / 4$ , while the sampled residual is zero. Since $M$ is arbitrary, no finite uniform off-grid error bound follows from this sample residual alone.

Depends on

Finite-sample AAA rational approximation with exact SVD

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Yuji Nakatsukasa, Olivier Sète, and Lloyd N. Trefethen, The AAA algorithm for rational approximation, Sections 3–4 (standard reference, not scraped)