Alphabeta Math
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 punctured three space

Example

Under countable choice, R3∖{0} has de Rham cohomology R in degrees zero and two only.

Facts & Assumptions

Given: Assume countable choice. The punctured three-dimensional Euclidean space.

[F1]

De rham cohomology of punctured euclidean space: Under countable choice, Rn∖{0} has cohomology R in degrees 0,n−1 only for n≥2. For n=1 it has R2 in degree zero only, and for n=0 all groups vanish.

Verification

technique · direct
1.1

The radial map r(x)=x/∣x∣ retracts onto S2. The homotopy F(x,t)=((1−t)+t/∣x∣)x has norm (1−t)∣x∣+t>0; it begins at x, ends at r(x) and fixes points of S2.

givenalgebra
2.1

This is the n=3 instance of the punctured-space theorem, so its only nonzero groups are H0=H2=R. Products of positive-degree classes vanish because their degree is at least four.

F1step 1.1∎

Source locator

Lee, Corollary 17.23, p.451; the radial maps and the degree-two generator are displayed in the verification.

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