CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

One centred ball-mean identity does not force harmonicity

Statement refuted

One ball-mean identity at one centre forces harmonicity.

Counterexample

Given: $n \geq 1$ , $R > 0$ , and $u : R^{n} \to R$ defined by $u (x) = ∣ x ∣^{2} (∣ x ∣^{2} - \frac{n + 2}{n + 4} R^{2}) .$

1.1

Polar coordinates Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma give $\frac{1}{∣ B_{R} ∣} \int_{B_{R}} ∣ x ∣^{2} = \frac{n}{n + 2} R^{2}$ and $\frac{1}{∣ B_{R} ∣} \int_{B_{R}} ∣ x ∣^{4} = \frac{n}{n + 4} R^{4}$ , hence $\frac{1}{∣ B_{R} ∣} \int_{B_{R}} u = 0 = u (0)$ [given, algebra].

2.1

Yet $Δ u = 4 (n + 2) ∣ x ∣^{2} - 2 n \frac{n + 2}{n + 4} R^{2}$ , which is not identically zero [given, algebra]. ∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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)