Alphabeta MathHow 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 one-form with has no local exact multiple
Statement refuted
Every nowhere-zero one-form on an open subset of has a nonzero local integrating factor making it exact.
Facts & Assumptions
Given: A local integrating factor means a nowhere-zero function with for some scalar on a neighborhood.
A nonsingular Pfaff one-form and its horizontal paths: the coordinate exterior derivative and wedge product are defined by antisymmetrized partials and alternating products.
An exact nonzero multiple satisfies the Pfaff condition: any such exact nonzero multiple satisfies .
Counterexample
On with coordinates take . It is smooth and nowhere zero because the coefficient is .
Its exterior derivative is , so , a nonzero coordinate volume form.
If a local as in the refuted claim existed on any open neighborhood, [F2] would force there, contradicting step 2.1 at every point. Therefore no such local exact multiple exists.