ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

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.

A nonneutral periodic source has no periodic Poisson solution

Example

On any three-dimensional torus, a single point charge of $1$ is nonneutral and cannot be the right side of a periodic Poisson equation $- Δ u = 4 π ρ$ .

Facts & Assumptions

Given: A full-rank three-dimensional cell, a point $x_{1}$ in its torus, and the one-charge source distribution $ρ$ with $q_{1} = 1$ .

[F1]

$⟨ ρ, φ ⟩ = \sum_{i} q_{i} φ (x_{i})$ and the distributional Laplacian satisfies $⟨ - Δ u, φ ⟩ = ⟨ u, - Δ φ ⟩$ (Neutral point sources and Fourier data on a three-dimensional torus).

Verification

technique · direct

1.1givenF1algebra

Test a hypothetical periodic distribution solution against the constant smooth periodic function $φ \equiv 1$ . Since $- Δ 1 = 0$ , the left side is $⟨ u, - Δ 1 ⟩ = 0$ by [F1].

2.1givenF1step 1.1algebra∎

The right side would be $4 π ⟨ ρ, 1 ⟩ = 4 π q_{1} = 4 π$ by [F1], contradicting step 1.1. Therefore no periodic distribution solution exists for this nonneutral source.

Depends on

Neutral point sources and Fourier data on a three-dimensional torus

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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