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 finite discrete manifold

Example

An $m$ -point discrete manifold has de Rham ring $R^{m}$ in degree zero with componentwise multiplication and no other nonzero degrees, including $m = 0$ .

Facts & Assumptions

Given: $M = {1, \dots, m}$ , $m \geq 0$ .

[F1]

De rham cohomology of a finite disjoint union is the direct sum: For a finite disjoint union $M = ∐_{j = 1}^{m} M_{j}$ , restrictions give $H_{d R}^{k} (M) ≅ ⨁_{j = 1}^{m} H_{d R}^{k} (M_{j})$ .

[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 function is a tuple $(a_{1}, \dots, a_{m})$ ; every function is locally constant, and each tangent space is zero, so $d f = 0$ and all positive-degree form spaces vanish. The quotient has $B^{0} = 0$ .

F2given

2.1

Restriction to the finite disjoint points is the direct-sum identification. For tuples $a, b$ , $(a b)_{j} = a_{j} b_{j}$ , so multiplication is coordinatewise. For $m = 0$ this is the zero algebra and for $m = 1$ it is $R$ .

F1step 1.1∎

Source locator

Lee, Proposition 17.6, p.443, and Proposition 17.5, pp.442–443, disjoint unions; here the union is finite.

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)