Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck 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.

Direct image along an open immersion is restriction-compatible intersection

Example

Let j:U↪X be an open immersion, and let F be a sheaf on U. Then for every open set V⊆X one has

(j∗F)(V)=F(V∩U).

Facts & Assumptions

Given: An open inclusion j:U↪X, a sheaf F on U, and an open set V⊆X.

[F1]

Direct image is computed by preimage opens: (j∗F)(V)=F(j−1(V)) (Direct image of a sheaf along a continuous map).

[F2]

Restriction to an open subspace is inverse image along the inclusion (Restriction of a sheaf to an open subspace).

Verification

technique · direct
1.1given

Because j is the inclusion of U into X, one has j−1(V)=V∩U.

2.1F1F2step 1.1∎

Substituting step 1.1 into [F1] gives (j∗F)(V)=F(V∩U). This is exactly the announced formula, and it is compatible with the restriction interpretation in [F2].

Depends on

Used by

Dependency tree · two levels

4 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