ExampleConstruction: Literature-sourcedVerification: 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.

The spectrum of a field is a one-point affine scheme

Example

For a field $k$ , $Spec k$ has the sole point $(0)$ . Its local ring, residue field, and ring of global functions are all canonically $k$ .

Facts & Assumptions

Given: A field $k$ .

[F1]

The stalk at a prime $p$ of an affine spectrum is $A_{p}$ (The stalk of the affine structure sheaf at a prime is A_p).

[F2]

Global functions on $Spec A$ recover $A$ (Global functions on Spec A recover A).

Verification

technique · direct

1.1given

The only proper ideal of a field is $(0)$ , and it is prime; hence the spectrum has exactly that point.

1.2F1algebra

By [F1], its local ring is $k_{(0)} = k$ , whose residue field is $k$ .

2.1F2step 1.1step 1.2∎

By [F2], its global sections are $k$ .

Depends on

Used by

Polynomial Proj charts Example

Dependency tree · two levels

20 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

The Stacks Project, Section 26.5 (standard reference, not scraped)