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.
Quasi-finite classical morphisms
Definition
A morphism of classical varieties is quasi-finite if every closed-point fibre is a finite set; empty fibres are allowed. Classical morphisms here are of finite type: for an affine target chart and an affine source chart above it, any finite set of -algebra generators of the source ring also generates it over the target ring. The inverse image has a finite affine cover because it is an open of a Noetherian variety.
Work over a fixed algebraically closed field , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points.