TheoremStatement: 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.

Long exact Hom sequences of a distinguished triangle

Statement

For every $W$ and distinguished triangle $X \to Y \to Z \to X [1]$ , both $\dots \to T (W, X) \to T (W, Y) \to T (W, Z) \to T (W, X [1]) \to \dots$ and the oppositely oriented sequence with $T (-, W)$ are exact.

Facts & Assumptions

Given: A distinguished triangle and an object $W$ .

Proof

1.1

The two representable functors have the required three-term exactness on the given triangle.

given

2.1

Apply that exactness to every translate, using signed rotations to identify consecutive three-term portions; concatenating them gives exactly the two displayed long sequences.

step 1.1given∎

Depends on

Used by

The triangulated five lemma Corollary
The long exact Hom sequence of a cone triangle Example
A distinguished triangle with zero first map is split Proposition

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, Lemma 13.4.2 (standard reference, not scraped)