Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: Literature-sourcedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05
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 unit-ideal expression gives a finite distinguished-open cover

Example

In Z one has 1=(−1)⋅2+1⋅3. Therefore Spec⁡(Z)=D(2)∪D(3).

Facts & Assumptions

Given: The ring R=Z.

[L1]

A finite unit expression yields a finite distinguished-open cover of the spectrum (A finite unit-ideal expression yields a finite distinguished-open subcover).

Verification

technique · direct
1.1

The displayed identity is a unit expression of the form required by [L1], with f1=2 and f2=3.

L1given
2.1

Applying [L1] gives Spec⁡(Z)=D(2)∪D(3). Concretely, every prime ideal of Z omits at least one of 2 and 3, because no prime ideal contains their linear combination 1.

L1step 1.1algebra
3.1

Thus one explicit unit-ideal expression produces a finite distinguished-open cover.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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