CounterexampleConstruction: Literature-sourcedVerification: Literature-sourcedPipeline-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.

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Nonuniqueness of a TR3 completion

Statement refuted

TR3 completions need not be unique. In $K (Z -Mod)$ , take the standard cone triangle of the zero map $S^{0} Z \to S^{0} Z [1]$ . The zero square on its first two terms has two distinct completions.

Counterexample

Given: The displayed data.

1.1given

The cone is $S^{0} Z [1] \oplus S^{0} Z [1]$ . Besides the zero third component, take the chain map whose matrix has the identity from the second summand to the first and zero in all other entries.

2.1step 1.1given∎

Both third components kill the cone inclusion and are killed by the cone projection, so each completes the same zero square; they differ on the second summand and hence are distinct in $K (Z -Mod)$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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, Derived Categories, Remark 13.9.11 (standard reference, not scraped)