ExampleConstruction: AI-generatedVerification: 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.

A transverse section of a line bundle has a hypersurface zero set

Example

For the trivial line bundle $R^{2} \times R \to R^{2}$ , the transverse section

s (x, y) = x

is transverse to the zero section. Its zero set is the line ${x = 0}$ , a codimension-one submanifold.

Facts & Assumptions

Given: The section $s (x, y) = x$ of the trivial line bundle $R^{2} \times R \to R^{2}$ .

Verification

technique · direct

1.1

The derivative of the scalar function $s (x, y) = x$ is $d s_{(x, y)} (u, v) = u$ . At every zero $(0, y)$ this is surjective onto the fibre $R$ , so $s$ is transverse to the zero section.

givenalgebra

2.1

Therefore the conclusion of A smooth section transverse to the zero section has a submanifold zero set holds in this concrete case, and the zero locus is the hypersurface ${x = 0}$ .

step 1.1∎

Depends on

A smooth section transverse to the zero section has a submanifold zero set

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

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