TheoremStatement: AI-adaptedProof: 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.

Strong Whitney approximation by transverse maps

Statement

Let $f : M \to N$ be smooth and let $Z \subseteq N$ be a closed embedded submanifold. Every neighbourhood of $f$ in the strong smooth topology contains a smooth map $g$ that is transverse to $Z$ .

Facts & Assumptions

Given: A smooth map $f : M \to N$ , a closed embedded submanifold $Z \subseteq N$ , and a chosen strong smooth neighbourhood $U$ of $f$ .

[F1]

The standard strong-topology transversality theorem says that transverse maps to a fixed closed embedded submanifold are dense in the strong smooth topology.

Proof

technique · direct

1.1

By [F1], the neighbourhood $U$ contains a smooth map $g : M \to N$ that is transverse to $Z$ .

F1given

2.1

Hence every strong smooth neighbourhood of $f$ contains a smooth map transverse to $Z$ .

step 1.1∎

Used by

Transverse maps are dense in the strong smooth topology Corollary

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Transversality (standard reference, not scraped)