CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-09-23 (gpt-6-sol)

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 pointwise local ball mean property is enough

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ .

If $u \in C (Ω)$ and every $x \in Ω$ has $r_{x} > 0$ with $B_{r_{x}} (x) ⋐ Ω$ such that $u (y) = A_{u} (y, r)$ whenever $y \in B_{r_{x}} (x)$ and $0 < r < r_{x} - ∣ y - x ∣$ , then $u$ is harmonic.

Proof

Given: Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) and the displayed local ball-mean hypothesis.

1.1

Each $B_{r_{x}} (x)$ satisfies the full ball mean-value property for balls compactly contained in it, so Continuous ball-mean-value functions are harmonic applies there under the stated Countable Choice hypothesis [given].

2.1

Hence $Δ u = 0$ on these balls, which cover $Ω$ [given]. ∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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)