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 real projective space

Example

${R P}^{m}$ has one cell in every dimension $0 \leq n \leq m$ , with $d_{n} = 0$ for odd $n$ and $d_{n} = 2$ for positive even $n$ . Thus $H_{0} ≅ Z$ , $H_{n} ≅ Z / 2$ for odd $0 < n < m$ , and the top group is $Z$ if $m$ is odd and $0$ if $m$ is positive even.

Facts & Assumptions

Given: The standard quotient CW structure of ${R P}^{m}$ .

Verification

technique · direct

1.1given

The attaching map of the $n$ -cell has incidence degree $1 + (- 1)^{n}$ , so Cellular boundary is the incidence degree matrix gives the stated alternating $0, 2$ differentials.

2.1step 1.1∎

Taking kernels modulo images gives the listed groups, including ${R P}^{0}$ where only $C_{0} = Z$ occurs; Cellular homology computes singular homology identifies them with singular homology.

Depends on

Used by

Cellular homology of an infinite-dimensional projective space Example

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 2.42 (standard reference, not scraped)