ExampleConstruction: AI-generatedVerification: AI-generatedprecheck 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.

Distinct continuous functions can share one germ, but equal germs everywhere force equality

Example

On the sheaf of continuous real-valued functions on $R$ , the zero function and the function $g (x) = \max (1 - ∣ x - 2 ∣, 0)$ have the same germ at $0$ but are not equal globally. On the other hand, if two continuous functions on an open set $U$ have the same germ at every point of $U$ , then they are equal.

Facts & Assumptions

Given: Continuous functions $f, g : U \to R$ on an open set $U$ .

[F1]

Two germ representatives are equal exactly when they agree on a smaller neighbourhood (Equality on a smaller neighbourhood defines the germ equivalence relation).

[L1]

Sheaf morphisms are determined by stalk maps (Morphisms of sheaves are determined by their maps on stalks).

Verification

technique · direct

1.1F1given

The function $g$ vanishes on the neighbourhood $(- 1, 1)$ of $0$ , so its germ at $0$ equals the germ of the zero function by [F1]. But $g (2) = 1$ , so the two functions are not equal globally.

1.2F1given

If $f_{x} = g_{x}$ for every $x \in U$ , then [F1] gives for each $x$ an open neighbourhood $V_{x} \subseteq U$ on which $f ∣_{V_{x}} = g ∣_{V_{x}}$ . The sets $V_{x}$ cover $U$ , so $f$ and $g$ agree at every point of $U$ and hence are equal.

2.1L1step 1.1step 1.2∎

This pointwise-germ criterion is the section-level instance behind [L1]: one stalk does not determine a section, but all stalks together do.

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

The Stacks Project, Sheaves on Spaces, Section 11 (standard reference, not scraped)