CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passverified 2026-09-23 (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.
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Liouville needs one sided boundedness

Statement refuted

For $n \geq 2$ , an entire real harmonic function need not be constant without a one-sided bound. The coordinate function $u (x) = x_{1}$ is a counterexample.

Facts & Assumptions

Given: The objects and hypotheses in the statement refuted.

Counterexample

technique · direct

1.1givenalgebra

All second derivatives vanish, so $u$ is entire harmonic, and $u (e_{1}) = 1 \neq 0 = u (0)$ .

2.1step 1.1algebra∎

As $t \to \infty$ , $u (t e_{1}) = t \to \infty$ and $u (- t e_{1}) = - t \to - \infty$ . Thus it has neither a global upper nor a global lower bound, while step 1.1 already shows it is nonconstant.

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Gantumur, Harmonic functions (standard reference, not scraped)