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CorollaryStatement: Literature-sourcedProof: 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.
  • 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.

Positive entire harmonic functions are constant

Statement

Assume the Axiom of Countable Choice ACω. For n≥2, every nonnegative harmonic function on Rn is constant; this includes the zero function as well as strictly positive functions.

Facts & Assumptions

Given: Countable Choice and the objects and hypotheses in the statement (The Axiom of Countable Choice (ACω)).

[F1]

A real entire harmonic function bounded above or bounded below is constant. (Liouville theorem for bounded harmonic functions).

Proof

technique · direct
1.1given

Nonnegativity is a global lower bound by the finite real number zero.

2.1F1step 1.1∎

The one-sided Liouville theorem therefore applies and yields constancy. The constant may be zero.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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