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.
This page uses the standard upper-semicontinuous subharmonic convention
The subharmonic functions on this page take values in , are required to be upper semicontinuous, and are excluded from being identically on any connected component of the domain. This is the standard potential-theoretic convention used by the sources behind the page, and it is the one compatible with for a holomorphic function and with Perron's method for the Dirichlet problem.
The older convention that a subharmonic function is merely a continuous real-valued function satisfying the submean inequality is recovered as the special case where the function never takes the value and happens to be continuous. The harmonic comparison theorems on the page are written so that the harmonic notion from Plane harmonic functions remains the same while the subharmonic class is large enough to include logarithmic singularities.