Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05
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 smooth distribution is integrable

Statement

Every smooth distribution is integrable.

Facts & Assumptions

Given: On R3, let X1:=∂x+y∂z,X2:=∂y, and let D:=span⁡(X1,X2).

[A1]

This is the standard contact plane field.

Refutation

technique · direct
1.1

The fields X1 and X2 are smooth and pointwise independent, so [given] D is a smooth rank-2 distribution.

given
1.2

Their bracket is [given] [X1,X2]=−∂z, which does not lie in the span of X1 and X2 at any point. Hence the distribution is not involutive.

givenalgebra
1.3

A rank-2 integral manifold would force brackets of tangent vector fields [given] to remain tangent, so such manifolds cannot realize this distribution. Thus the distribution is smooth but not integrable.

given
2.1

Therefore the universal statement is false. [given] ∎

given

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