DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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.

Exterior of a smooth proper disk

Definition

Assume AC. Let $D \subset B^{4}$ be a smooth proper disk. Give $B^{4}$ its standard orientation, choose an orientation of $D$ , and give $K = \partial D \subset S^{3}$ the induced boundary orientation. Choose a closed tubular neighborhood $N (D) ≅ D^{2} \times D^{2}$ meeting $S^{3}$ in $\partial D \times D^{2}$ , and let $X_{D}$ be the closure of $B^{4} ∖ int N (D)$ , with corners rounded. The positive meridian is the oriented circle ${x} \times \partial D^{2}$ in the overlap $D^{2} \times S^{1}$ , oriented by the ambient and disk orientations.

Definition references

Proper surfaces and boundary collars

Tubular neighborhoods in smooth manifolds

Axiom of Choice

Depends on

Used by

Dependency tree · two levels

21 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

A. J. Casson and C. McA. Gordon, Cobordism of Classical Knots (1986), with appendix by P. M. Gilmer (standard reference, not scraped)