CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passverified 2026-09-26 (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 nearest-point projection need not be unique outside the tubular radius

Statement refuted

Nearest-point projection onto an embedded submanifold stays uniquely defined everywhere in the ambient space.

Facts & Assumptions

Given: The unit circle $S^{1} \subseteq R^{2}$ and the center point $0 \in R^{2}$ .

Counterexample

technique · direct

1.1givenalgebra

Every point of $S^{1}$ is distance $1$ from the origin. Hence the origin has infinitely many nearest points on $S^{1}$ .

2.1step 1.1given∎

Therefore nearest-point projection is not uniquely defined at the origin, which lies outside every sufficiently small annular tubular neighbourhood of the circle.

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, 2nd ed., Tubular Neighborhoods (standard reference, not scraped)