Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-sol)audited 2026-09-30
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.
Proper morphisms
Definition
A morphism of schemes is proper if and only if it is separated, of finite type, and universally closed. Here separatedness has the meaning of Separated morphism of schemes, finite type has the meaning of Locally finite type and finite type morphisms, and universally closed has the meaning of Universally closed morphisms. The definition applies to arbitrary schemes: it assumes neither Noetherian hypotheses nor finite presentation.