LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-07

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.

Sphere and ball measures scale in Rn

Statement

For $n \geq 1$ and $r > 0$ , $∣ \partial B_{r} ∣ = ω_{n - 1} r^{n - 1}$ and $∣ B_{r} ∣ = ω_{n - 1} r^{n} / n$ ; both factors are finite and positive.

Proof

Given: $n \geq 1$ and $r > 0$ .

1.1

The parametrization $θ \mapsto r θ$ gives $∣ \partial B_{r} ∣ = ω_{n - 1} r^{n - 1}$ [given].

2.1

Applying Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma to $1_{B_{r}}$ gives $∣ B_{r} ∣ = ω_{n - 1} \int_{0}^{r} t^{n - 1} d t = ω_{n - 1} r^{n} / n$ [given, algebra]. ∎

Depends on

Used by

Dependency tree · two levels

12 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

PDE source treatment (standard reference, not scraped)