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 of a lens space

Example

For $L (p, q)$ , the standard cellular complex is $0 \to Z \overset{0}{\to} Z \overset{p}{\to} Z \overset{0}{\to} Z \to 0$ . Hence $H_{0}, H_{3} ≅ Z$ , $H_{1} ≅ Z / p$ , and $H_{2} = 0$ .

Facts & Assumptions

Given: The standard one-cell-in-each-dimension CW structure of $L (p, q)$ , $p \geq 1$ .

Verification

technique · direct

1.1given

Its incidence degrees are $0, p, 0$ in dimensions $3, 2, 1$ , respectively, so the displayed complex follows from Cellular boundary is the incidence degree matrix.

2.1step 1.1∎

Its kernels and images are the stated groups (also for $p = 1$ , when $Z / p = 0$ ), and the cellular comparison transfers them to singular homology.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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, Example 2.43 (standard reference, not scraped)