ExampleConstruction: Literature-sourcedVerification: AI-generatedverified 2026-09-26 (gpt-6-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.

The sphere and its two-sided normal tube

Example

For the unit sphere $S^{n} \subseteq R^{n + 1}$ , the outward unit normal at $p$ is again $p$ . Hence the normal bundle is the trivial line bundle $S^{n} \times R$ , and the normal addition map is $E (p, t) = (1 + t) p .$ For $∣ t ∣ < 1 / 2$ , its image is the spherical shell ${x \in R^{n + 1} : 1 / 2 < ∥ x ∥ < 3 / 2} .$

Facts & Assumptions

Given: The unit sphere $S^{n} \subseteq R^{n + 1}$ with the Euclidean metric.

Verification

technique · direct

1.1givenalgebra

Since $T_{p} S^{n} = {v \in R^{n + 1} : ⟨ v, p ⟩ = 0}$ , the orthogonal complement is the one-dimensional span of $p$ . Thus $N^{⊥} S^{n} ≅ S^{n} \times R$ .

2.1step 1.1algebra∎

Under that identification, normal addition is $E (p, t) = p + t p = (1 + t) p$ . Since $1 + t > 0$ for $∣ t ∣ < 1 / 2$ , its image is exactly the shell $1 / 2 < ∥ x ∥ < 3 / 2$ : the inverse is $x \mapsto (x / ∥ x ∥, ∥ x ∥ - 1)$ , which is smooth there. Thus this is an explicit two-sided tubular neighbourhood.

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

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