ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge 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.

Kunneth over a field

Statement

For $C = (0 \to k \overset{0}{\to} k \to 0)$ and $D = (0 \to k \to 0)$ concentrated in degree $0$ , cross product identifies $H_{*} (C \otimes_{k} D)$ with $H_{*} (C) \otimes_{k} H_{*} (D)$ .

Verification

Given: the displayed complexes of $k$ -vector spaces.

1.1given

The zero differential makes $H_{0} C ≅ H_{1} C ≅ k$ and $H_{0} D ≅ k$ .

2.1step 1.1∎

The tensor complex has $k$ in total degrees zero and one with zero differential, matching the two cross-product summands; no Tor term occurs over $k$ .

Depends on

Kunneth over a field

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

Weibel, An Introduction to Homological Algebra (standard reference, not scraped)