Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-07
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.

Diffeomorphisms preserve interior and boundary

Statement

Every diffeomorphism of manifolds with boundary maps ∂M onto ∂N and Int⁡M onto Int⁡N.

Facts & Assumptions

Given: A diffeomorphism F:M→N between smooth manifolds with boundary.

[L1]

A smooth diffeomorphism between relatively open half-space sets preserves their face and relative interior (Smooth invariance of the manifold boundary).

[L2]

Smoothness between manifolds with boundary is tested in boundary charts (Smooth maps between manifolds with boundary).

Proof

technique · direct
1.1givenL2

By [L2], every coordinate representative of F and its inverse in boundary charts is a smooth half-space diffeomorphism.

2.1L1step 1.1∎

Applying [L1] to those representatives shows that F maps boundary points exactly to boundary points and interior points exactly to interior points. Bijectivity then gives F(∂M)=∂N and F(Int⁡M)=Int⁡N.

Depends on

Used by

Dependency tree · two levels

8 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