ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-09-24 (gpt-6-sol)

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 stem extension that is not universal

Example

The central extension C2→D8→C2×C2 is stem but not universal because its base is not perfect.

Verification

Given: The center and commutator subgroup of $D_{8}$ are both $⟨ r^{2} ⟩$ .

1.1given

Thus $C_{2} \to D_{8} \to C_{2} \times C_{2}$ is stem.

2.1step 1.1algebra∎

Let $V = C_{2} \times C_{2}$ be its base. If the displayed stem extension $u : D_{8} ↠ V$ were universal, the maps $x \mapsto (u (x), 0)$ and $x \mapsto (u (x), u (x))$ from $D_{8}$ to the split central extension $V \times V \to V$ would have to be equal. They are distinct because $u$ is surjective and $V \neq 0$ . Thus this stem extension is not universal, using only the choice-free uniqueness condition in the definition.

Depends on

Used by

Nothing in the library uses this result yet.

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

Clara Löh, Group Cohomology (standard reference, not scraped)