CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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 $x\ne0$ , and $q(x,y)=(y-3r(x))(y-r(x))$ .

[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 $y=2r(x)$ , $q(x,2r(x))=-r(x)^2<0$ for $x\ne0$ .

givenalgebra

2.1

On each line $y=mx$ , the factor $r(x)$ is smaller than every positive power of $|x|$ , so $q(x,mx)>0$ 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 52 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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