LemmaStatement: Literature-sourcedProof: 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 global mesh ratio bounds the local degree-zero ratio

Statement

Let $x_{0} < \dots < x_{n}$ , $n \geq 1$ , put $h = \max_{0 \leq j < n} (x_{j + 1} - x_{j})$ and $h_{\min} = \min_{0 \leq j < n} (x_{j + 1} - x_{j})$ , and let $ρ = h / h_{\min}$ . The local ratio $β$ in Local mesh ratio for degree-zero Floater–Hormann interpolation satisfies $0 \leq β \leq ρ$ .

Facts & Assumptions

Given: Ordered distinct nodes, their largest and smallest adjacent gaps $h, h_{\min}$ , and the local ratio $β$ .

[F1]

Local mesh ratio for degree-zero Floater–Hormann interpolation: for $n \geq 3$ , $β$ is the maximum over interior indices of the smaller of two adjacent-gap ratios; for $n \leq 2$ , $β = 0$ .

Proof

1.1givenF1algebra

Since every adjacent gap is at most $h$ and at least $h_{\min} > 0$ , each ratio appearing in [F1] is positive and at most $h / h_{\min} = ρ$ .

2.1givenF1step 1.1algebra∎

By step 1.1, their minimum is in $[0, ρ]$ , and the maximum of these minima also lies in $[0, ρ]$ . When $n \leq 2$ , $β = 0 \leq ρ$ .

Depends on

Local mesh ratio for degree-zero Floater–Hormann interpolation

Used by

Dependency tree · two levels

2 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

Michael S. Floater and Kai Hormann, Barycentric rational interpolation with no poles and high rates of approximation, Theorem 3 (standard reference, not scraped)