Alphabeta Math
CounterexampleConstruction: 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 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 S1⊆R2 and the center point 0∈R2.

[L1]

Nearest-point projection agrees with the tubular retraction only after one shrinks to a sufficiently small tube (Nearest-point projection is the tubular retraction after shrinking).

Counterexample

technique · direct
1.1

Every point of S1 is distance 1 from the origin. Hence the origin has infinitely many nearest points on S1.

givenalgebra
2.1

Therefore nearest-point projection is not uniquely defined at the origin, which lies outside every sufficiently small annular tubular neighbourhood. This is exactly the boundary described in [L1].

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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