ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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 Mobius-band line foliation

Example

Model the Mobius band as the quotient of $[0, 1] \times R$ by the identification $(0, t) \sim (1, - t)$ . The horizontal lines $[0, 1] \times {t}$ descend to a regular one-dimensional foliation.

The line $t = 0$ becomes the central circle leaf, while every leaf with $t \neq 0$ is a circle: one traversal of the horizontal segment changes the transverse sign, and a second traversal returns to the starting point.

Facts & Assumptions

Given: The strip model of the Mobius band and its horizontal lines.

[A1]

The gluing preserves horizontality.

Verification

technique · direct

1.1

The quotient map identifies horizontal tangent directions with horizontal [given] tangent directions, so the horizontal line field descends to a smooth one-dimensional distribution on the Mobius band.

given

1.2

Its local plaques are the images of small horizontal intervals, and the [given] quotient charts preserve that plaque structure. Hence the descended line field defines a regular foliation.

given

1.3

The line $t = 0$ closes up to the central circle. If $t \neq 0$ , then [given] the endpoint $(1, t)$ is identified with $(0, - t)$ , so one horizontal pass through the quotient changes the transverse sign; a second pass along the line $- t$ returns to the original class. Thus the leaf through $t \neq 0$ is also a circle.

given

2.1

This yields the standard line foliation of the Mobius band. [given] ∎

given

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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)