LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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 radial profile with a prescribed inner plateau

Statement

There exists $b \in C_{c}^{\infty} (R)$ with $0 \leq b \leq 1$ , $b (x) = 1$ for $∣ x ∣ \leq 1$ , and $b (x) = 0$ for $∣ x ∣ \geq 2$ , such that the profile $χ : [0, \infty) \to R$ defined by $χ (r) = b (4 r)$ is the restriction of a smooth function on $R$ , equals $1$ for $0 \leq r \leq 1 / 4$ , and equals $0$ for $r \geq 1 / 2$ .

Here $C_{c}^{\infty} (R)$ denotes the infinitely differentiable, compactly supported real-valued functions.

Facts & Assumptions

Given: A one-dimensional smooth cutoff with unit plateau radius $1$ and support radius $2$ as stated in [F1].

[F1]

In dimension one there exists $b \in C_{c}^{\infty} (R)$ with $0 \leq b \leq 1$ , $b = 1$ on $∣ x ∣ \leq 1$ , and $b = 0$ on $∣ x ∣ \geq 2$ (Explicit compactly supported smooth cutoffs).

[F2]

Composition of a smooth function with an affine map is smooth, by the chain rule (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

Proof

technique · direct

1.1F1choose

Choose $b$ supplied by [F1] and set $χ (r) = b (4 r)$ for $r \geq 0$ .

2.1step 1.1F1F2algebra∎

The function $x \mapsto b (4 x)$ is smooth by [F2], so its restriction is the stated smooth profile. If $0 \leq r \leq 1 / 4$ , then $∣ 4 r ∣ \leq 1$ and $b (4 r) = 1$ ; if $r \geq 1 / 2$ , then $∣ 4 r ∣ \geq 2$ and $b (4 r) = 0$ . The bounds on $b$ are inherited by composition.

Depends on

Used by

Exterior-sphere domains need not have global C2alpha Dirichlet solutions Counterexample

Dependency tree · two levels

13 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

Semyon Dyatlov, MIT 18.155 notes (source inherited through the local cutoff supplier; no new retrieval claimed) (standard reference, not scraped)