PropositionStatement: Literature-sourcedProof: 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.

The octahedral axiom gives a triangle relating the cones of f, g, and gf

Statement

For composable $X \overset{f}{\to} Y \overset{g}{\to} Z$ , choices of cone objects $Q_{f}, Q_{g f}, Q_{g}$ fit into a distinguished triangle $Q_{f} ⟶ Q_{g f} ⟶ Q_{g} ⟶ Q_{f} [1] .$

Facts & Assumptions

Given: Composable maps and distinguished completions for $f$ , $g$ , and $g f$ .

Proof

1.1given

Apply TR4 to precisely these three completions.

2.1step 1.1given∎

Its fourth face is the displayed distinguished triangle; changing a chosen completion only replaces its cone object by a noncanonical isomorphic one.

Depends on

Dependency tree · 2 levels

Sources

The Stacks Project, Definition 13.3.2