ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

Smoothing a continuous circle-valued map through an annular retraction

Example

Parametrize the circle by $e^{i t}$ with $t \in [- π, π]$ and define

F (e^{i t}) = e^{i ∣ \sin (t / 2) ∣} .

This is continuous on $S^{1}$ but not smooth at $t = 0$ . Embedding the target circle in $R^{2}$ , approximating the planar representative smoothly, and then retracting through the standard annulus produces a smooth circle-valued map homotopic to $F$ .

Facts & Assumptions

Given: The continuous map $F (e^{i t}) = e^{i ∣ \sin (t / 2) ∣}$ on $S^{1}$ .

Verification

technique · direct

1.1

The map $F$ is continuous and well defined because the endpoint values and derivatives at $t = \pm π$ agree. Near $t = 0$ , $∣ \sin (t / 2) ∣$ has an absolute-value cusp, so $F$ fails to be smooth there.

givenalgebra

2.1

The manifold-valued Whitney approximation theorem Whitney approximation for manifold-valued maps applies with target $S^{1}$ , using the annular tubular neighbourhood from the standard circle example. It therefore produces a smooth map $\tilde{F} : S^{1} \to S^{1}$ homotopic to $F$ .

step 1.1∎

Depends on

Whitney approximation for manifold-valued maps

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., Smooth Approximation of Maps Between Manifolds (standard reference, not scraped)