ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 2026-09-26 (gpt-6-sol)

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.

Distributional harmonicity removes an apparent interior corner

Example

Let $n \geq 1$ and define $v : R^{n} \to R$ by $v (x) = ∣ x_{1} ∣$ . This apparent corner is not distributionally harmonic on $R^{n}$ ; indeed $Δ v = 2 δ_{{x_{1} = 0}}$ , where $δ_{{x_{1} = 0}} (ϕ) = \int_{R^{n - 1}} ϕ (0, x^{'}) d x^{'}$ (the evaluation $ϕ (0)$ when $n = 1$ ). In particular, $∣ x_{1} ∣$ is not distributionally harmonic on any open set meeting the hyperplane ${x_{1} = 0}$ .

Verification

Given: $n \geq 1$ , the domain $R^{n}$ , and the distributional derivative convention Distributional harmonicity and Poisson's equation on an open subset of Rn.

1.1givenalgebra

In one variable, split the compactly supported test integral at $0$ and integrate by parts twice. The ordinary second derivative vanishes on both half-lines, and the jump of the first derivative from $- 1$ to $1$ contributes $2 ϕ (0)$ , so $(∣ t ∣)^{''} = 2 δ_{0}$ . Applying this calculation at each fixed $x^{'}$ and integrating over the remaining variables gives $Δ v (ϕ) = 2 \int_{R^{n - 1}} ϕ (0, x^{'}) d x^{'}$ ; derivatives in the other coordinates vanish.

2.1step 1.1∎

On any open set meeting the hyperplane, choose a nonnegative smooth test function supported near a meeting point and positive there. Its hyperplane integral is positive, so step 1.1 makes $Δ v$ nonzero on that open set. Thus the corner fails the distributional harmonicity test there.

Depends on

Distributional harmonicity and Poisson's equation on an open subset of Rn

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

PDE source treatment (standard reference, not scraped)