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

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Every constant-dimensional family of tangent subspaces is a smooth distribution

Statement

Every constant-dimensional family of tangent subspaces is a smooth distribution.

Facts & Assumptions

Given: On R2, define D(x,y):={span⁡(∂x),y>0,span⁡(∂x+∂y),y≤0.

[A1]

This is a one-dimensional family of tangent lines.

Refutation

technique · direct
1.1given

The family has constant dimension 1 at every point.

1.2given

If it were a smooth distribution near the origin, it would admit a local [given] nonvanishing smooth spanning field. Above the x-axis that field would have to be tangent to ∂x, while below the axis it would have to be tangent to ∂x+∂y. Continuity at the origin would then force the two line directions to agree there, which they do not.

2.1given∎

Therefore constant fibre dimension alone does not imply smoothness.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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