ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

A periodic resolution for a finite cyclic group

Example

For $C_{m} = ⟨ t ⟩$ , the augmented complex with alternating maps $t - 1$ and $N = 1 + t + \dots + t^{m - 1}$ is a free periodic resolution of $Z$ .

Verification

Given: The group ring $Z [C_{m}]$ .

1.1

$(t - 1) N = N (t - 1) = t^{m} - 1 = 0$ , so this is a complex.

given

2.1

Writing an element as $\sum a_{i} t^{i}$ shows $\ker ε = (t - 1) Z [C_{m}]$ , $\ker (t - 1) = N Z [C_{m}]$ , and $\ker N = (t - 1) Z [C_{m}]$ . Hence it is exact and every term is free.

step 1.1∎

Depends on

Integral group modules and the trivial module

Used by

Cohomology of a finite cyclic group Example

Dependency tree · two levels

4 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

Löh, Proposition 1.6.11 (standard reference, not scraped)