Alphabeta MathHow 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
For , every nonnegative harmonic function on is constant; this includes the zero function as well as strictly positive functions.
Facts & Assumptions
Given: The objects and hypotheses in the statement.
A real entire harmonic function bounded above or bounded below is constant. (Liouville theorem for bounded harmonic functions).
Proof
Nonnegativity is a global lower bound by the finite real number zero.
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
2 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
- Gantumur, Harmonic functions (standard reference, not scraped)