ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-06‡ 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.

Cover-small chains for the two-arc cover of a circle

Example

Fix an abelian group $G$ and an element $g \in G$ . Let $U, V$ be proper overlapping open arcs covering $S^{1}$ , so $U \cap V$ has two components. Choose one point $a_{i}$ in each overlap component. Parameterize the closed subarc from $a_{0}$ to $a_{1}$ carried by $U$ as a singular path $u$ , and the complementary closed subarc from $a_{1}$ to $a_{0}$ carried by $V$ as $v$ . Thus $z_{Z} = u + v$ traverses the circle once with the indicated orientation. With coefficients in $G$ , put $z_{g} = u \otimes g + v \otimes g \in C_{1}^{{U, V}} (S^{1}; G) .$ This is the oriented two-edge circle cycle with coefficient $g$ ; no distinguished element of $G$ is assumed.

Facts & Assumptions

Given: The two-arc cover, overlap points, paths, abelian group $G$ , and supplied coefficient $g$ above.

Verification

technique · direct

1.1

Each proper open arc is an interval in the cyclic order. The interval inside $U$ joining the overlap points is one of the two closed subarcs between them, and the complementary subarc lies in $V$ because the omitted parts of $U$ must be covered by $V$ . Both endpoints are in both open sets. Thus the entire images of $u$ and $v$ , including their endpoints, lie in their respective cover members. Also $\partial (u \otimes g) = ([a_{1}] - [a_{0}]) \otimes g$ and $\partial (v \otimes g) = ([a_{0}] - [a_{1}]) \otimes g$ , so $z_{g}$ is cover-small and is a cycle.

givenconstruct

2.1

By Mayer–Vietoris connecting class, $δ [z_{g}] = [([a_{1}] - [a_{0}]) \otimes g] \in H_{0} (U \cap V; G)$ . Identifying the two components in the order containing $a_{0}, a_{1}$ gives the coefficient pair $(- g, g)$ . Indeed, paths within each arc component identify its point classes, and no path crosses the components. Replacing the oriented integral cycle by its negative negates this class. If $g = 0$ (including $G = 0$ ), both chain and connecting class are zero.

step 1.1algebra∎

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

Allen Hatcher, Algebraic Topology, §2.2 (standard reference, not scraped)