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.

Cohomology of a finite cyclic group

Example

For $C_{m} = ⟨ t ⟩$ and a left $C_{m}$ -module $M$ , put $N = 1 + t + \dots + t^{m - 1}$ . Then $H^{0} (C_{m}; M) = \ker (t - 1)$ , and for $q \geq 0$ , $H^{2 q + 1} (C_{m}; M) = \ker N / (t - 1) M, H^{2 q + 2} (C_{m}; M) = \ker (t - 1) / N M .$

Verification

Given: The periodic free resolution.

1.1given

Applying ${Hom}_{Z [C_{m}]} (-, M)$ identifies every cochain group with $M$ ; the coboundaries alternate between $t - 1$ and $N$ .

2.1step 1.1∎

Taking kernel modulo preceding image gives exactly the displayed groups, including the degree-zero kernel.

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

Löh, Corollary 1.6.13 (standard reference, not scraped)