ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

ex-an-acyclic-complex-that-becomes-zero-in-d-but-not-in-k.md

Example

The complex $X = (Z \overset{2}{\to} Z \to Z / 2)$ in degrees $0, 1, 2$ is zero in $D (A b)$ and is nonzero in $K (A b)$ .

Facts & Assumptions

Given: The complex $X = (Z \overset{2}{\to} Z \to Z / 2)$ in degrees $0, 1, 2$ is zero in $D (A b)$ and is nonzero in $K (A b)$ .

[F1]

A complex is zero in $D$ iff it is acyclic (A complex is zero in the derived category exactly when it is acyclic).

Verification

1.1

Reduction modulo two after multiplication by two is zero. The map two is monic, its image equals the kernel of reduction, and reduction is epic; all other terms are zero. Thus every cohomology group vanishes, and the acyclicity criterion gives $Q X = 0$ .

F1algebra

2.1

If $X$ were zero in $K$ , its identity would be nullhomotopic. The equation in degree two would give $1_{Z / 2} = d^{1} h^{2}$ . But $h^{2} : Z / 2 \to Z$ is zero, so this is impossible. Hence $X$ is nonzero in $K$ .

step 1.1algebra∎

Depends on

A complex is zero in the derived category exactly when it is acyclic

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

13.11.1–13.11.6 (standard reference, not scraped)