ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 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.

A smooth map with a nonclosed critical-value set

Example

Fix a smooth bump $β : R \to R$ supported in $[- 1, 1]$ with $β (0) = 1$ , $β^{'} (0) = 0$ , and $0 < β (t) < 1$ for $0 < ∣ t ∣ < 1$ . Then

f (x) : = \sum_{k = 1}^{\infty} (1 - \frac{1}{k}) β (2^{k} (x - k))

is smooth, has critical values $1 - \frac{1}{k}$ for all $k$ , and has regular value $1$ . Hence its critical value set is not closed.

Facts & Assumptions

Given: The smooth bump $β$ and the function $f$ above.

[F1]

The critical value set is the image of the critical locus (The critical locus and critical value set).

[L1]

The critical value set need not be closed (The critical-value set need not be closed).

Verification

technique · direct

1.1

The supports of the summands are pairwise disjoint, so near each $x$ only finitely many summands are nonzero. Therefore $f$ is smooth. At each center $x = k$ , the derivative of the $k$ th summand vanishes and every other summand is zero, so $k$ is a critical point with critical value $f (k) = 1 - \frac{1}{k}$ .

givenalgebra

2.1

The sequence $1 - \frac{1}{k}$ tends to $1$ . But $f (x) \neq 1$ for every $x$ : on each bump support the value is at most $1 - \frac{1}{k} < 1$ , and away from the supports the value is $0$ . Thus $1$ has empty fibre and is therefore regular.

step 1.1algebra

3.1

By [F1], the critical value set contains every $1 - \frac{1}{k}$ but not the limit $1$ , so it is not closed. This is exactly the phenomenon noted in [L1].

F1L1step 2.1∎

Depends on

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

Marco Gualtieri, Topology I: Smooth Manifolds, cumulative notes (standard reference, not scraped)