ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passverified 2026-09-08 (Codex)

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 radial bump on Euclidean space

Example

For $0 < r < R$ , put $s = (r + R) / 2$ . The function $ρ (x) : = σ ((s^{2} - ∥ x ∥^{2}) / (s^{2} - r^{2}))$ is a smooth radial bump on $R^{n}$ , $n \geq 1$ : it equals $1$ on ${\overline{B}}_{r} (0)$ and has support in ${\overline{B}}_{s} (0) \subset B_{R} (0)$ .

Facts & Assumptions

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

[L1]

The concentric-ball construction produces exactly such a smooth bump (A smooth bump between concentric Euclidean balls).

Verification

technique · direct

1.1L1given

The displayed function is the explicit construction used in [L1].

2.1L1step 1.1

By [L1] it is smooth, equals $1$ on the inner closed ball, and has support in ${\overline{B}}_{s} (0) \subset B_{R} (0)$ . It is radial because the formula depends only on $∥ x ∥$ .

3.1step 2.1∎

This is the required Euclidean example.

Depends on

A smooth bump between concentric Euclidean balls

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