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.

Cohomological dimensions of $1$ and $Z$

Example

${cd}_{Z} (1) = 0$ and ${cd}_{Z} (Z) = 1$ .

Verification

Given: $Z [Z] = Z [t, t^{- 1}]$ .

1.1given

The trivial module for $1$ is free, so its projective dimension is zero. For $Z$ , $0 \to Z [t, t^{- 1}] \overset{t - 1}{\to} Z [t, t^{- 1}] \to Z \to 0$ is a free resolution.

2.1step 1.1∎

With trivial coefficients $Z$ , applying Hom sends $t - 1$ to zero, so $H^{1} (Z; Z) = Z \neq 0$ . Hence the resolution length one is minimal.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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, §4.2.1 (standard reference, not scraped)

Cohomological dimensions of 11 and ZZ