TheoremStatement: 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.

Weyl's lemma for the Laplacian

Statement

If $T \in D^{'} (Ω)$ and $Δ T = 0$ , there is a unique smooth harmonic $h$ with $T = T_{h}$ .

Proof

Given: $Δ T = 0$ on the open set $Ω$ .

1.1

The distributional Laplacian commutes with local mollification makes $h_{ε} = T * ρ_{ε}$ smooth and harmonic on $Ω_{ε}$ [given].

2.1

Radial mean invariance and associativity of convolution show on every common shrunken domain that $h_{ε} * ρ_{δ} = h_{δ} * ρ_{ε}$ [step 1.1].

3.1

The nested-interior double-convolution equality makes the regularizations agree as their radii shrink; their common local value defines a smooth harmonic $h$ , and $T_{h_{ε}} \to T$ gives $T_{h} = T$ [step 2.1].

4.1

If $T_{h} = T_{k}$ , then $h = k$ almost everywhere; continuity makes $h = k$ everywhere [given]. ∎

Depends on

Used by

Dependency tree · two levels

9 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)