ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

The projected unit interval is not nullhomotopic in $R / Z$

Example

The loop $t \mapsto [t]$ in $R / Z$ is not nullhomotopic.

Facts & Assumptions

Given: The objects, hypotheses, and choice principles stated above.

[F1]

For the quotient by integer translation, $q : R \to R / Z$ is a covering map, and every deck transformation is a unique translation $x \mapsto x + n$ with $n \in Z$ . (The quotient $R \to R / Z$ is a covering with integer translations as deck transformations).

[F2]

Let $p : E \to B$ be a covering, $H : Y \times I \to B$ a homotopy, and ${\tilde{H}}_{0} : Y \to E$ a lift of $H (-, 0)$ . There is a unique lift $\tilde{H} : Y \times I \to E$ of $H$ extending ${\tilde{H}}_{0}$ . (Existence and uniqueness of homotopy lifts through a covering map).

[F3]

Endpoint-fixed homotopic paths in the base have lifts with the same endpoint whenever their lifts begin at the same point. (The endpoint of a lifted path depends only on its endpoint-fixed homotopy class).

Verification

technique · direct

1.1givenF3F1

The loop $t \mapsto [t]$ lifts from zero to the path $t \mapsto t$ , whose endpoint is one.

2.1step 1.1F3F2

If it had an endpoint-fixed nullhomotopy, homotopy lifting would keep the terminal lift in the discrete fibre while deforming the initial lift to the constant path at zero, forcing endpoints one and zero to agree.

3.1step 2.1F1

This proves essentiality without asserting $π_{1} (R / Z) ≅ Z$ .

4.1step 3.1∎

The preceding construction and implications establish the assertion.

Depends on

Used by

The Hawaiian earring is locally path-connected but has no universal cover Example

Dependency tree · two levels

15 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, §1.3 (standard reference, not scraped)
Marco Gualtieri, MAT1300 Week 4 Term 2, §1.6 (standard reference, not scraped)
Omar Antolín Camarena, Proper local homeomorphisms and covering maps (standard reference, not scraped)

The projected unit interval is not nullhomotopic in R/ZR/Z