ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

Stereographic projection and its inverse are explicit in coordinates

Example

Stereographic projection from the north pole has the explicit formulas $Σ (z) = (\frac{2 Re z}{1 + ∣ z ∣^{2}}, \frac{2 Im z}{1 + ∣ z ∣^{2}}, \frac{∣ z ∣^{2} - 1}{1 + ∣ z ∣^{2}}), Π (x, y, t) = \frac{x + i y}{1 - t},$ with $Σ (\infty) = (0, 0, 1)$ and $Π (0, 0, 1) = \infty$ .

Facts & Assumptions

Given: The stereographic formulas.

[L1]

The displayed formulas define inverse homeomorphisms between $\hat{C}$ and the unit sphere (Stereographic projection identifies the Riemann sphere with the unit two-sphere).

Verification

technique · direct

1.1L1given

Fact [L1] already gives the inverse formulas. Substituting $z = 0$ gives the south pole $(0, 0, - 1)$ , and the defining clause sends $\infty$ to the north pole $(0, 0, 1)$ .

2.1L1given∎

Substituting $(0, 0, - 1)$ into the inverse formula returns $0$ , while the exceptional north-pole clause returns $\infty$ . Thus the coordinate formulas behave exactly as claimed.

Depends on

Stereographic projection identifies the Riemann sphere with the unit two-sphere

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Lars V. Ahlfors, Complex Analysis, 3rd ed., Ch. 3 §§2.2-3.5 (standard reference, not scraped)
Matthias Weber, Complex Analysis, Ch. 1 §§1.3-1.4 (standard reference, not scraped)
Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 8 §§1-2 (standard reference, not scraped)