Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01
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.

Transverse maps are dense in the strong smooth topology

Statement

For a fixed closed embedded submanifold Z⊆N, the smooth maps M→N that are transverse to Z are dense in the strong smooth topology.

Facts & Assumptions

Given: Smooth manifolds M,N and a closed embedded submanifold Z⊆N.

[L1]

Every strong neighbourhood of a smooth map contains a transverse map (Strong Whitney approximation by transverse maps).

Proof

technique · direct
1.1

Let f:M→N be smooth and let U be any strong neighbourhood of f. By [L1], U contains a transverse smooth map.

L1given
2.1

Since this holds for every f and every neighbourhood U, the transverse maps are dense.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

1 result within one dependency step 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