TheoremStatement: 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.

Global functions on Spec A recover A

The canonical map $A \to Γ (Spec A, O)$ is an isomorphism, including when $A = 0$ .

Given: The basic-open section calculation.

technique · direct

1.1given

$D (1) = Spec A$ .

2.1step 1.1

The basic-open calculation gives $Γ (D (1), O) = A_{1}$ .

3.1step 2.1∎

The canonical map $A \to A_{1}$ is an isomorphism, also for $A = 0$ .

Depends on