Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04
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 set-valued sheaf has a unique section over the empty open set

Statement

Let F be a sheaf of sets on a topological space X. Then F(∅) is a singleton.

Facts & Assumptions

Given: A sheaf F on X.

[L1]

A sheaf satisfies locality and gluing for every open cover, including the empty cover of ∅ (A sheaf on a topological space).

Proof

technique · direct
1.1

Apply [L1] to the empty cover of ∅. The empty family of local sections is vacuously compatible, so gluing produces at least one section s∈F(∅).

L1given
2.1

Let t∈F(∅). Then t also restricts to the same empty family on the empty cover. By the uniqueness part of [L1], one has t=s. Therefore F(∅)={s}.

L1step 1.1∎

Depends on

Used by

Dependency tree · two levels

2 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