LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

projective variety cone irreducible

Statement

The affine cone over a classical projective variety is irreducible.

Proof

Given: A classical projective variety $X$ .

1.1

$I_{+} (X)$ is prime.

given

2.1

The cone coordinate ring is $k [x_{0}, \dots, x_{n}] / I_{+} (X)$ and is therefore a domain.

step 1.1algebra

3.1

The affine prime-coordinate-ring criterion makes $C (X)$ irreducible.

step 2.1∎

Depends on

Used by

affine cone over conic Example

Dependency tree · two levels

5 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

J. S. Milne, Algebraic Geometry, Chapter 6 (standard reference, not scraped)
Michael Artin, Algebraic Geometry, Chapter 3 (standard reference, not scraped)