CorollaryStatement: Literature-sourcedProof: 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.

Locally integrable weakly harmonic functions are smooth

Statement

If $f \in L_{l o c}^{1} (Ω)$ and $Δ T_{f} = 0$ , then $f = h$ almost everywhere for a unique smooth harmonic $h$ .

Proof

Given: $f \in L_{l o c}^{1} (Ω)$ and $Δ T_{f} = 0$ .

1.1

Weyl's lemma for the Laplacian supplies a smooth harmonic $h$ with $T_{f} = T_{h}$ [given].

2.1

Equality of regular distributions implies $f = h$ almost everywhere, and uniqueness is inherited [step 1.1]. ∎

Depends on

Used by

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Dependency tree · two levels

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Sources

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