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 a point

Example

The de Rham ring of a point is $R$ in degree zero only.

Facts & Assumptions

Given: $M = {p}$ with its zero-dimensional smooth structure.

[F1]

Zero and out of range de rham cohomology: $H_{d R}^{k} (M) = 0$ if $k < 0$ or $k > \dim M$ . If $M = \emptyset$ , its cohomology vanishes in every degree.

[F2]

Zero th de rham cohomology is locally constant functions: $H_{d R}^{0} (M)$ is the algebra of locally constant real functions. For nonempty connected $M$ it is canonically $R$ .

Verification

technique · direct

1.1

A smooth function is uniquely its value at $p$ , so $Ω^{0} (M) = R$ . There are no nonzero cotangent vectors, hence no positive-degree forms, and $d = 0$ .

F1given

2.1

The degree-zero quotient has no boundaries, and evaluation at $p$ sends $[a] [b]$ to $a b$ and $[1]$ to $1$ . Thus it is the algebra $R$ ; every other group is zero.

F2step 1.1∎

Source locator

Lee, p.441, the cycle quotient, and Proposition 17.6, p.443, degree zero; the point has no positive-degree forms.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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)