CorollaryStatement: 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.

Ball mean-value property for harmonic functions

Statement

Under Spherical mean-value property for harmonic functions, $u (x) = A_{u} (x, r)$ for every $B_{r} (x) ⋐ Ω$ .

Proof

Given: $u$ is harmonic and $B_{r} (x) ⋐ Ω$ .

1.1

Polar coordinates express $\int_{B_{r} (x)} u = ω_{n - 1} \int_{0}^{r} t^{n - 1} M_{u} (x, t) d t$ [given].

1.2

The spherical identity makes this $ω_{n - 1} u (x) r^{n} / n = u (x) ∣ B_{r} ∣$ [given, algebra].

2.1

Divide by the positive ball volume from Sphere and ball measures scale in Rn [step 1.2]. ∎

Depends on

Used by

Locally uniform limits of harmonic functions are harmonic Theorem
Weyl's lemma for the Laplacian Theorem

Dependency tree · two levels

8 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)