ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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.

The differential of a map between spheres in stereographic coordinates

Example

For the smooth map $F : S^{1} \to S^{1}$ , $F (e^{i t}) = e^{2 i t}$ , the stereographic coordinate representative is the rational map $u \mapsto 2 u / (1 - u^{2})$ on the overlap where both sides are defined. Its differential is multiplication by the derivative $2 (1 + u^{2}) / (1 - u^{2})^{2}$ .

Facts & Assumptions

Given: The degree-two map on the circle and compatible stereographic charts.

[L1]

The differential in coordinates is given by the Jacobian of the coordinate representative (Coordinate formula for the differential).

Verification

technique · direct

1.1given

In stereographic coordinates, the map is $u \mapsto 2 u / (1 - u^{2})$ .

2.1L1step 1.1

Differentiating this rational function gives $2 (1 + u^{2}) / (1 - u^{2})^{2}$ , so [L1] identifies this scalar as the matrix of the differential in the chosen one-dimensional bases.

3.1step 2.1∎

Hence the differential is explicitly computed by the coordinate formula.

Depends on

Coordinate formula for the differential

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 (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)