LemmaStatement: 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.

The distributional Laplacian commutes with local mollification

Statement

For $T \in D^{'} (Ω)$ , put $Ω_{ε} : = {x \in Ω : B_{ε} (x) ⋐ Ω}$ and define $(T * ρ_{ε}) (x) = T (ρ_{ε} (x - \cdot))$ on $Ω_{ε}$ . Then $Δ (T * ρ_{ε}) = (Δ T) * ρ_{ε}$ there.

Proof

Given: $ϕ \in C_{c}^{\infty} (Ω_{ε})$ .

1.1

The support margin makes $\tilde{ϕ} (y) : = \int ρ_{ε} (x - y) ϕ (x) d x$ a test function in $C_{c}^{\infty} (Ω)$ [given].

2.1

Differentiating the kernel and using Distributional harmonicity and Poisson's equation on an open subset of Rn gives $⟨ Δ (T * ρ_{ε}), ϕ ⟩ = ⟨ Δ T, \tilde{ϕ} ⟩$ [step 1.1].

3.1

This is $⟨ (Δ T) * ρ_{ε}, ϕ ⟩$ for all $ϕ$ [step 2.1]. ∎

Depends on

Used by

The principle of descent and the logarithmic domination principle Theorem
Weyl's lemma for the Laplacian Theorem

Dependency tree · two levels

4 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)