LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-23 (gpt-6-sol)

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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.
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Derivatives of harmonic functions are harmonic

Statement

Every partial derivative of a smooth harmonic function is smooth harmonic. Under the Axiom of Countable Choice ${A C}_{ω}$ , if $Δ T = 0$ , every distributional derivative of $T$ is induced by the corresponding smooth harmonic derivative.

Proof

Given: $Δ h = 0$ ; for the distributional assertion, Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )) and $Δ T = 0$ .

1.1

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

2.1

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

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