ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Cellular homology and ring-independent groups of complex projective space

Example

${C P}^{m}$ has one cell in each even dimension $0, 2, \dots, 2 m$ . Hence for any abelian group $G$ , $H_{2 i} ({C P}^{m}; G) ≅ G$ for $0 \leq i \leq m$ and all odd-degree groups vanish.

Facts & Assumptions

Given: The standard one-even-cell-per-dimension CW structure.

Verification

technique · direct

1.1given

There are no adjacent-dimensional cells, so all cellular boundaries vanish by A CW complex with no cells in adjacent dimensions has zero cellular boundary.

2.1step 1.1∎

The cellular groups are one copy of $G$ in the listed even dimensions and zero otherwise, and the comparison theorem gives the asserted singular groups.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

Allen Hatcher, Algebraic Topology, Section 2.2 (standard reference, not scraped)