Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-09-07
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.

An orientable manifold has a canonical orientation

Statement

False. A nonempty connected orientable manifold has two orientations, exchanged by reversal.

Facts & Assumptions

Given: A nonempty connected orientable smooth manifold M.

[L1]

Such a manifold has exactly two orientations, a chosen orientation and its pointwise opposite (Nonempty connected orientable manifolds have exactly two orientations).

Refutation

technique · direct
1.1

Orientability supplies an orientation o, and [L1] supplies its distinct pointwise opposite −o.

givenL1choose
2.1

By [L1], these two choices are exhaustive. Since the definition of orientability specifies neither one, orientability alone does not determine a canonical orientation.

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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