LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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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.

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Derivatives of harmonic functions are harmonic

Statement

Every partial derivative of a smooth harmonic function is smooth harmonic. If $Δ T = 0$ , every distributional derivative of $T$ is induced by the corresponding smooth harmonic derivative.

Proof

Given: $Δ h = 0$ .

1.1

Constant-coefficient derivatives commute, so $Δ (\partial^{α} h) = \partial^{α} Δ h = 0$ [given].

2.1

For $T = T_{h}$ from Weyl's lemma for the Laplacian, the derivative definition gives $\partial^{α} T = T_{\partial^{α} h}$ [step 1.1]. ∎

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