Alphabeta Math
ExampleConstruction: 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.

The product foliation

Example

On a product manifold M×N, the connected components of the fibres M×{q} for q∈N form a regular foliation of leaf dimension dim⁡M.

Facts & Assumptions

Given: The product smooth structure on M×N.

[A1]

Product charts have the form (u,v).

Verification

technique · direct
1.1

In a product chart, the slices v=constant are exactly the local [given] pieces of the fibres M×{q}. Their connected components are therefore the local plaques.

given
1.2

Transition maps preserve the second coordinate up to a change depending only [given] on the old second coordinate, so these charts satisfy the defining condition of a regular foliation atlas.

given
2.1

Hence the connected components of the fibres of the projection [given] M×N→N form the product foliation.

given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

20 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