ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

De rham cohomology of the two sphere

Example

Under countable choice, the sphere $S^{2}$ has $H^{0} = H^{2} = R$ and $H^{1} = 0$ , with all other groups zero.

Facts & Assumptions

Given: Assume countable choice. The unit sphere in $R^{3}$ .

[F1]

De rham cohomology of spheres: Assume countable choice. For $n \geq 1$ , $H_{d R}^{k} (S^{n})$ is $R$ in degrees $0, n$ and zero otherwise. For $S^{0}$ it is $R^{2}$ in degree zero and zero otherwise.

Verification

technique · direct

1.1

Apply F1 with $n = 2$ . It gives $H^{0} (S^{2}) = H^{2} (S^{2}) = R$ and $H^{1} (S^{2}) = 0$ , with all remaining degrees zero.

F1given

2.1

The same theorem with $n = 2$ gives the constant-function group in degree zero and the zero group in degree one. A product of two degree-two classes has degree four and is zero, so the ring is $R [u] / (u^{2})$ with $∣ u ∣ = 2$ .

F1step 1.1∎

Source locator

Lee, Theorem 17.21, pp.450–451; the degree groups and products are computed for dimension two.

Depends on

De rham cohomology of spheres

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

John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)