ExampleConstruction: AI-generatedVerification: 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.

An octahedron for two composable maps of stalk complexes

Example

For $S^{0} (Z) \overset{m}{\to} S^{0} (Z) \overset{n}{\to} S^{0} (Z)$ , the three two-term cones of $m$ , $n m$ , and $n$ are joined by a distinguished triangle $Cone (m) \to Cone (n m) \to Cone (n) \to Cone (m) [1]$ .

Facts & Assumptions

Given: The displayed data.

[F1]

The three-cone calculation supplies the comparison maps $α$ and $β$ (The three-cone calculation for a composite chain map).

[F2]

The octahedral axiom supplies a signed distinguished triangle joining the three cone objects (The octahedral axiom gives a triangle relating the cones of f, g, and gf).

Verification

1.1given

The maps are composable chain maps of stalk complexes.

2.1F1F2step 1.1given∎

By [F1] the three-cone calculation provides the comparison maps $α$ and $β$ . Applying [F2] in $K (A b)$ supplies the signed fourth face and proves that the resulting triangle is distinguished.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

The Stacks Project, Proposition 13.10.3 (standard reference, not scraped)