ExampleConstruction: AI-generatedVerification: AI-generatedverified 2026-09-23 (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.

A two-function smooth partition on the circle

Example

Let $U_{1} : = S^{1} ∖ {(- 1, 0)}$ and $U_{2} : = S^{1} ∖ {(1, 0)}$ . Then there exist smooth functions $ϕ_{1}, ϕ_{2} : S^{1} \to [0, 1]$ such that $ϕ_{1} + ϕ_{2} = 1$ , $supp (ϕ_{1}) \subseteq U_{1}$ , and $supp (ϕ_{2}) \subseteq U_{2}$ .

Facts & Assumptions

Given: The two-set open cover $U_{1}, U_{2}$ of the circle.

[F1]

The function $h (u) = e^{- 1 / u}$ for $u > 0$ and $h (u) = 0$ for $u \leq 0$ is smooth: every derivative on $u > 0$ is a polynomial in $u^{- 1}$ times $e^{- 1 / u}$ and tends to zero as $u ↓ 0$ .

Verification

technique · direct

1.1F1

Write a point of $S^{1}$ as $(x, y)$ and set $b (t) = h (t + 1 / 2)$ using [F1]. Then $b$ is nonnegative and smooth, is zero for $t \leq - 1 / 2$ , and is positive for $t > - 1 / 2$ .

2.1step 1.1

Put $g_{1} (x, y) = b (x)$ and $g_{2} (x, y) = b (- x)$ . At least one of $x, - x$ is nonnegative, so $g_{1} + g_{2} > 0$ everywhere. Define $ϕ_{i} = g_{i} / (g_{1} + g_{2})$ . These functions are smooth, take values in $[0, 1]$ , and sum to one.

3.1step 1.1step 2.1∎

The support of $ϕ_{1}$ is contained in the closed arc ${x \geq - 1 / 2}$ , which misses $(- 1, 0)$ , and the support of $ϕ_{2}$ is contained in ${x \leq 1 / 2}$ , which misses $(1, 0)$ . Thus $supp (ϕ_{i}) \subseteq U_{i}$ as required, with no use of the general partition theorem.

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