Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30
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.

Polar coordinates do not give a chart at the origin

Statement refuted

Polar coordinates give a smooth chart on all of R2.

Facts & Assumptions

Given: The polar formulas (r,θ)↦(rcos⁡θ,rsin⁡θ).

[F1]

A smooth chart must be a homeomorphism from an open set of the manifold onto an open subset of Euclidean space (Chart maps are diffeomorphisms onto Euclidean open sets).

Counterexample

technique · direct
1.1

At the origin, the angle coordinate is not defined, and for r>0 the same point admits many values of θ differing by multiples of 2π.

given
2.1

Therefore the polar description is not a single-valued homeomorphism on any neighbourhood containing the origin, so it cannot be a chart there by [F1].

F1step 1.1
3.1

This refutes the statement.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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