Alphabeta Math
TheoremStatement: Literature-sourcedProof: 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.

Orientability is equivalent to a nowhere-vanishing top form

Statement

Assume ACω. A smooth manifold is orientable if and only if it has a nowhere-vanishing smooth top-degree form.

Facts & Assumptions

Given: ACω and a smooth manifold.

Proof

technique · direct
1.1

A nonzero top form selects the determinant ray on which it is positive, producing an orientation.

given
2.1

Conversely choose positive local top forms for an orientation and multiply them by a subordinate partition of unity. At every point all nonzero summands lie in the same positive ray, so their sum is nonzero and is a global top form.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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