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 $F (\emptyset)$ 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 $\emptyset$ (A sheaf on a topological space).

Proof

technique · direct

1.1L1given

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

2.1L1step 1.1∎

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

A set-valued sheaf has a unique section over the empty open set

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources