CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-02

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 flat refinement has a strict minimum on every line through the origin but no local extremum

Statement refuted

Refuted: smoothness together with a strict minimum on every line through a point forces a local minimum.

Facts & Assumptions

Given: $r (0) = 0$ , $r (x) = e^{- 1 / x^{2}}$ for $x \neq 0$ , and $q (x, y) = (y - 3 r (x)) (y - r (x))$ .

[L1]

The exponential dominates every polynomial at infinity (The exponential dominates every fixed nonnegative integer power at $+ \infty$ ).

Counterexample

Proof

technique · direct

1.1

The flat function $r$ is smooth at $0$ : every derivative is a polynomial in $1 / x$ times $e^{- 1 / x^{2}}$ off $0$ and tends to $0$ by [L1].

L1algebra

1.2

Along $y = 2 r (x)$ , $q (x, 2 r (x)) = - r (x)^{2} < 0$ for $x \neq 0$ .

givenalgebra

2.1

On each line $y = m x$ , the factor $r (x)$ is smaller than every positive power of $∣ x ∣$ , so $q (x, m x) > 0$ for sufficiently small nonzero $x$ ; the same holds on $y = 0$ .

step 1.1L1algebra

3.1

Thus $q$ is smooth and linewise strictly minimal at the origin but has no local minimum there.

step 1.1step 2.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

No minimum at the origin but a minimum among all lines (standard reference, not scraped)