LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-12

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.

Every nonempty open subset of an affine variety is dense

Let $X$ be a classical affine variety. Every nonempty open subset of $X$ is dense in $X$ .

Given: A classical affine variety $X$ and a nonempty open subset $U \subseteq X$ .

[L1]

In a classical affine variety, irreducibility is equivalent to every pair of nonempty open subsets meeting, and also to every nonempty open subset being dense (Irreducibility is equivalent to every pair of nonempty open sets meeting).

technique · direct

1.1L1given

Because $X$ is a classical affine variety, it is irreducible. By [L1], an irreducible space is one in which every nonempty open subset is dense.

2.1step 1.1∎

Applying step 1.1 to the specific nonempty open subset $U$ shows that $U$ is dense in $X$ .

Depends on