LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

Radial mollification fixes local mean-value functions

Statement

Let $u \in C (Ω)$ have the spherical mean-value property, and let $(ρ_{ε})$ be a radial mollifier family as in A radial mollifier family in Rn. Then $(u * ρ_{ε}) (x) = u (x)$ whenever $B_{ε} (x) ⋐ Ω$ .

Proof

Given: $u$ has the local spherical mean-value property and $B_{ε} (x) ⋐ Ω$ .

1.1

Polar coordinates give $(u * ρ_{ε}) (x) = ω_{n - 1} \int_{0}^{ε} q_{ε} (t) t^{n - 1} M_{u} (x, t) d t$ [given].

2.1

Substitute $M_{u} (x, t) = u (x)$ and use $\int ρ_{ε} = 1$ to obtain $(u * ρ_{ε}) (x) = u (x)$ [given]. ∎

Depends on

Used by

Continuous ball-mean-value functions are harmonic Theorem
Weyl's lemma for the Laplacian Theorem

Dependency tree · two levels

7 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)