ExampleConstruction: Literature-sourcedVerification: 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.

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

1.1

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

givenalgebra

2.1

Under that identification, normal addition is $E (p, t) = p + t p = (1 + t) p$ . The same computation as for the circle shows that the image for $∣ t ∣ < 1 / 2$ is exactly the shell $1 / 2 < ∥ x ∥ < 3 / 2$ . This is the Euclidean tubular neighbourhood in the sense of The Euclidean tubular neighbourhood theorem.

step 1.1∎

Depends on

The Euclidean tubular neighbourhood theorem

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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)