ExampleConstruction: AI-generatedVerification: AI-generatedverified 2026-09-23 (gpt-6-sol)

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.

Generic affine hyperplanes meet an embedded submanifold transversely

Example

For the unit circle $S^{1} \subseteq R^{2}$ , the vertical line $H_{a} = {x = a}$ meets $S^{1}$ transversely for every $a \in R ∖ {- 1, 1}$ . For $∣ a ∣ > 1$ the intersection is empty, and for $∣ a ∣ < 1$ it consists of two transverse points.

Facts & Assumptions

Given: The height map $h : S^{1} \to R$ , $h (x, y) = x$ , and a real parameter $a$ .

Verification

technique · direct

1.1givenalgebra

The fibre $h^{- 1} (a)$ is $S^{1} \cap H_{a}$ . When $∣ a ∣ < 1$ , the equation $x = a$ on $x^{2} + y^{2} = 1$ gives the two points $(a, \pm \sqrt{1 - a^{2}})$ . A tangent vector at $(x, y)$ is $(- y, x)$ , and $d h (- y, x) = - y$ , which is nonzero at both points because $y \neq 0$ .

2.1step 1.1

For $∣ a ∣ > 1$ the fibre is empty, hence transverse vacuously. For $a = \pm 1$ it consists of one point with $y = 0$ , where $d h$ vanishes and transversality fails. Thus the exceptional set is exactly the finite, hence null, set ${- 1, 1}$ .

3.1step 2.1∎

Thus generic affine hyperplanes meet the embedded circle transversely.

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

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

Sources

Marco Gualtieri, Topology I: Smooth Manifolds, cumulative notes (standard reference, not scraped)