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.

Every manifold is orientable

Statement

False. The Möbius band is nonorientable.

Facts & Assumptions

Given: The Möbius band presented as B=[0,1]×[−1,1]/(0,s)∼(1,−s).

[L1]

An orientation is a smooth choice of a determinant-line ray at every point (Oriented smooth manifolds and oriented charts).

Refutation

technique · direct
1.1

The gluing transition near the core has coordinates (t,s)↦(t−1,−s) and derivative diag⁡(1,−1), so transporting a local determinant ray once around the core reverses it.

givenalgebra
2.1

A global orientation in the sense of [L1] would return the chosen ray unchanged after this loop, contradicting step 1.1. Hence the Möbius band is a manifold that is not orientable, refuting the universal claim.

L1step 1.1contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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