ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-04‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

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]

The germ of a section records equality on some neighbourhood of the point (Germs of sections).

[L1]

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

Verification

technique · direct

1.1

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.

F1given

1.2

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.

F1given

2.1

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

L1step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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)