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.
Frobenius applies to any variable-rank family of subspaces
Statement
Frobenius applies to any variable-rank family of subspaces.
Facts & Assumptions
Given: On , let
The rank jumps from at the origin to elsewhere.
Refutation
This family is not a smooth distribution in the regular sense, because a [given] smooth distribution is by definition a smooth vector subbundle of constant rank.
Since the regular Frobenius theorem starts from smooth constant-rank [given] distributions, its hypotheses do not even apply to this variable-rank family.
Therefore the statement is false: singular families require a different [given] theory.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
6 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
- John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)