LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-30

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 smooth bump between concentric Euclidean balls

Statement

Let $0 < r < R$ and $n \geq 1$ . Then there exists a smooth function $ρ : R^{n} \to [0, 1]$ such that $ρ = 1$ on ${\overline{B}}_{r} (0)$ and $supp (ρ) \subseteq B_{R} (0)$ .

Facts & Assumptions

Given: Real numbers $0 < r < R$ .

[F1]

The standard smooth step function $σ$ is smooth, vanishes on $(- \infty, 0]$ , and equals $1$ on $[1, \infty)$ (The standard smooth step function).

[L1]

Total derivatives satisfy the Euclidean chain rule and are stable under sums and scalar multiples (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Sums and scalar multiples of totally differentiable maps are totally differentiable with the expected derivatives).

[A1]

The function $q (x) : = ∥ x ∥^{2}$ is smooth on $R^{n}$ .

Proof

technique · direct

1.1

Define $u (x) : = (R^{2} - ∥ x ∥^{2}) / (R^{2} - r^{2})$ and $ρ (x) : = σ (u (x))$ ; then $u$ is smooth by [A1] and [L1], so $ρ$ is smooth by [F1] and [L1].

A1F1L1construct

2.1

If $∥ x ∥ \leq r$ , then $u (x) \geq 1$ , so $ρ (x) = 1$ by [F1]; if $∥ x ∥ \geq R$ , then $u (x) \leq 0$ , so $ρ (x) = 0$ by [F1].

F1step 1.1

3.1

Thus $ρ$ maps into $[0, 1]$ , equals $1$ on ${\overline{B}}_{r} (0)$ , and vanishes on $R^{n} ∖ B_{R} (0)$ , so $supp (ρ) \subseteq B_{R} (0)$ .

step 2.1∎

Depends on

Used by

A radial bump on Euclidean space Example
A Euclidean bump for a compact set inside an open set Lemma

Dependency tree · two levels

9 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 (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)