ExampleConstruction: Literature-sourcedVerification: 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 harmonic functions away from the origin

Example

Let $n \geq 1$ and $U \in C^{2} ((0, \infty))$ . The radial function $u : R^{n} ∖ {0} \to R$ given by $u (x) = U (∣ x ∣)$ is harmonic precisely when $U^{''} (r) + \frac{n - 1}{r} U^{'} (r) = 0 (r > 0) .$ Thus the families are $U (r) = a + b r$ for $n = 1$ , $U (r) = a + b \log r$ for $n = 2$ , and $U (r) = a + b r^{2 - n}$ for $n \geq 3$ .

Verification

Given: $n \geq 1$ , $U \in C^{2} ((0, \infty))$ , and $r = ∣ x ∣ > 0$ .

1.1

Direct differentiation gives $Δ U (r) = U^{''} (r) + (n - 1) U^{'} (r) / r$ [given].

2.1

Multiplying by $r^{n - 1}$ gives $(r^{n - 1} U^{'})^{'} = 0$ , whose integrations give the listed cases [given, algebra]. ∎

Depends on

The Laplacian of a $C^2$ function and of a $C^2$ vector field

Used by

Harmonic on a punctured domain need not extend Counterexample

Dependency tree · two levels

5 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)