ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge 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.

Harmonic function attaining only boundary extrema

Example

For $n \geq 2$ , $u (x) = x_{1}$ on ${\overline{B}}_{1} (0) \subset R^{n}$ is harmonic. Its maximum $1$ and minimum $- 1$ occur only at $e_{1}$ and $- e_{1}$ , respectively, on the boundary. At $e_{1}$ its outward sphere derivative is $1$ .

Facts & Assumptions

Given: The objects and hypotheses in the example.

[F1]

The weak maximum principle places closure maxima of continuous-closure subharmonic functions on bounded open sets on the boundary. (Weak maximum principle for the laplacian).

[F2]

Under an interior tangent ball, strict interior inequality and continuity on its closure, an existing outward derivative at the boundary maximum is positive. (Hopf boundary point lemma for the laplacian).

Verification

technique · direct

1.1

Every second derivative of $u$ is zero, so $Δ u = 0$ . In the open ball, $∣ x_{1} ∣ \leq ∣ x ∣ < 1$ . On its closure, equality $x_{1} = 1$ forces $x = e_{1}$ , and $x_{1} = - 1$ forces $x = - e_{1}$ . Thus the explicit extrema have exactly the boundary location allowed by the weak principle.

F1givenalgebra

2.1

The ball itself is an interior tangent ball at $e_{1}$ , with outward direction $e_{1}$ . The quotient $(u (e_{1}) - u (e_{1} - t e_{1})) / t$ equals $1$ for $0 < t < 2$ . It has a strictly positive limit, agreeing with Hopf since $u < 1$ throughout the interior.

F2step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

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

Hunter, Notes on Partial Differential Equations (standard reference, not scraped)