CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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.

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The product of two harmonic functions need not be harmonic

Statement refuted

Refuted claim: the product of two harmonic functions is always harmonic.

The witness is $u (x, y) = x$ and $v (x, y) = x$ . Each factor is harmonic, but their product is $x^{2}$ , whose Laplacian is $2$ .

Facts & Assumptions

Given: The functions $u (x, y) = x$ and $v (x, y) = x$ .

Counterexample

technique · direct

1.1

As in the previous example, $u$ and $v$ are harmonic because both have vanishing second partial derivatives.

algebra

2.1

Their product is $u v = x^{2}$ , and $(x^{2})_{x x} + (x^{2})_{y y} = 2 + 0 = 2 \neq 0$ . So $u v$ is not harmonic.

step 1.1algebra∎

Depends on

Plane harmonic functions

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

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