ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-01

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.

A noncompact embedded curve with no uniform tubular radius

Example

Construct a smooth embedding $γ : R \to R^{2}$ by concatenating successively farther-right smoothed hairpins, where the $n$ th hairpin contains two nearly parallel strands at distance $2^{- n}$ and is joined to the next one by a long horizontal segment. The image is a noncompact embedded curve.

Facts & Assumptions

Given: The smooth hairpin curve described above.

Verification

technique · direct

1.1

Each hairpin occupies a region disjoint from all the previous ones except for one joining segment, and the joins can be smoothed so that the velocity never vanishes. Therefore the concatenated curve is a smooth embedding of $R$ .

givenconstruct

2.1

Fix $r > 0$ and choose $n$ with $2^{- n} < 2 r$ . In the $n$ th hairpin the two nearly parallel strands are closer than $2 r$ , so normal discs of radius $r$ based on opposite strands intersect. Hence no tubular neighbourhood of constant radius $r$ can be injective there.

step 1.1algebra

3.1

This realizes the failure asserted in FALSE: every noncompact submanifold has a uniform-radius tubular neighbourhood while remaining compatible with the variable-radius theorem The Euclidean tubular neighbourhood theorem.

step 2.1∎

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

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