Alphabeta Math
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.

A basic open of the affine line

Example

For a commutative ring k, the basic open D(t) of Ak1=Spec⁡k[t] is affine and isomorphic to Spec⁡k[t,t−1].

Facts & Assumptions

Given: A commutative ring k and the polynomial variable t.

[F1]

A principal localization spectrum is the corresponding distinguished open as a locally ringed space (A principal localization identifies its spectrum with a distinguished open).

Verification

technique · direct
1.1

Localizing k[t] at t adjoins t−1, giving k[t]t≅k[t,t−1].

givenalgebra
1.2

By [F1], D(t)≅Spec⁡(k[t]t).

F1
2.1

Combining the two identifications proves the example.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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