CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

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A smooth approximation without relative control destroys prescribed values

Statement refuted

An arbitrarily close smooth approximation automatically preserves values on a closed set where the original map was already fixed.

Facts & Assumptions

Given: The zero map $f : R \to R$ and the closed set $A : = {0}$ .

[L1]

Relative Whitney approximation is the theorem that guarantees preservation near the closed set (Relative Whitney approximation for Euclidean-valued maps).

Counterexample

technique · direct

1.1

For every $ϵ > 0$ , the smooth function $g_{ϵ} (x) : = ϵ e^{- x^{2}}$ satisfies $\sup_{x} ∣ g_{ϵ} (x) - f (x) ∣ \leq ϵ$ . So $g_{ϵ}$ is an arbitrarily close smooth approximation to $f$ .

givenalgebra

2.1

But $g_{ϵ} (0) = ϵ \neq 0 = f (0)$ , so this approximation does not preserve the prescribed value on $A$ . Thus closeness alone is weaker than the relative conclusion in [L1].

L1step 1.1∎

Depends on

Relative Whitney approximation for Euclidean-valued maps

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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, 2nd ed., Smooth Approximation of Maps Between Manifolds (standard reference, not scraped)